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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="geometric_quantization">Geometric quantization</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong> <strong><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <em><a href="geometry+of+physics#LagrangiansAndActionFunctionals">Lagrangians and Action functionals</a></em> + <em><a href="geometry+of+physics#GeometricQuantization">Geometric Quantization</a></em></p> <h2 id="prerequisites">Prerequisites</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> </li> </ul> </li> </ul> <h2 id="prequantum_field_theory">Prequantum field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a> = <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a></p> <ul> <li> <p>prequantum 1-bundle = <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, regular<a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>,<a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> = lift of <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+0-bundle">prequantum 0-bundle</a> = <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> </li> </ul> </li> </ul> <h2 id="geometric_quantization">Geometric quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+leaf">Bohr-Sommerfeld leaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">geometric quantization by push-forward</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+non-integral+forms">geometric quantization of non-integral forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/coherent+state+%28in+geometric+quantization%29">coherent state</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil+theorem">Borel-Weil theorem</a>, <a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schubert+calculus">Schubert calculus</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/geometric+quantization+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> <h4 id="symplectic_geometry">Symplectic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#ingredients'>Ingredients</a></li> <li><a href='#HistoryAndVariants'>History and variants</a></li> <li><a href='#overview'>Overview</a></li> <li><a href='#basic_jargon'>Basic Jargon</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#GeometricPrequantization'>Geometric prequantization</a></li> <ul> <li><a href='#prequantum_line_bundle'>Prequantum line bundle</a></li> <li><a href='#prequantum_states'>Prequantum states</a></li> <li><a href='#prequantum_operators'>Prequantum operators</a></li> </ul> <li><a href='#GeometricQuantizationProper'>Geometric quantization</a></li> <ul> <li><a href='#QuantumStates'>Quantum states</a></li> <ul> <li><a href='#Polarizations'>Quantum state space as space of polarized sections</a></li> <li><a href='#EulerCharacteristicOfSheafCohomology'>Quantum state space as Euler characteristic of prequantum sheaf cohomology</a></li> <li><a href='#IndexOfDolbeaultDiracOperator'>Quantum state space as index of Dolbeault-Dirac operator</a></li> <li><a href='#AsIndexOfSpinCDiracOperator'>Quantum state spaces as index of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-Dirac operator</a></li> </ul> <li><a href='#QuantumOperators'>Quantum operators / observables</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#DependenceOnChoiceOfPolarization'>Functorial dependence on choices</a></li> <li><a href='#compatibility_of_quantization_with_symplectic_reduction'>Compatibility of quantization with symplectic reduction</a></li> <li><a href='#characteristic_central_extensions'>Characteristic central extensions</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ExamplesSchroedingerRepresentation'>Schrödinger representation</a></li> <li><a href='#khler_manifolds'>Kähler manifolds</a></li> <li><a href='#ExampleThe2Sphere'>The 2-sphere</a></li> <li><a href='#ExamplesTori'>Tori</a></li> <li><a href='#theta_functions'>Theta functions</a></li> <li><a href='#quantization_of_chernsimons_theory'>Quantization of Chern-Simons theory</a></li> <li><a href='#quantization_of_loop_groups__of_the_wzw_model'>Quantization of loop groups / of the WZW model</a></li> <li><a href='#quantization_in_gromovwitten_theory'>Quantization in Gromov-Witten theory</a></li> <li><a href='#quantization_of_the_bosonic_string_model'>Quantization of the bosonic string <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesGeneral'>General</a></li> <li><a href='#holographic_quantization'>Holographic quantization</a></li> <li><a href='#quantization'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>ℂ</mi></msup></mrow><annotation encoding="application/x-tex">Spin^{\mathbb{C}}</annotation></semantics></math>-Quantization</a></li> <li><a href='#examples_2'>Examples</a></li> <li><a href='#ReferencesRelationToDeformation'>Relation to deformation quantization</a></li> <li><a href='#relation_to_path_integral_quantization'>Relation to path integral quantization</a></li> <li><a href='#ReferencesBRST'>Geometric BRST quantization</a></li> <li><a href='#ReferencesSupergeometric'>Supergeometric version</a></li> <li><a href='#of_presymplectic_manifolds'>Of presymplectic manifolds</a></li> <li><a href='#of_generalized_complex_manifolds'>Of generalized complex manifolds</a></li> <li><a href='#in_higher_differential_geometry'>In higher differential geometry</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Geometric quantization is one formalization of the notion of <em><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></em> of a <a class="existingWikiWord" href="/nlab/show/classical+mechanical+system">classical mechanical system</a>/<a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> to a <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>/<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. In comparison to <em><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></em> it focuses on <a class="existingWikiWord" href="/nlab/show/spaces+of+states">spaces of states</a>, hence on the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>.</p> <h3 id="ingredients">Ingredients</h3> <p>With a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> regarded as a <a class="existingWikiWord" href="/nlab/show/classical+mechanical+system">classical mechanical system</a>, <em>geometric quantization</em> produces <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of this to a <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> by</p> <ol> <li> <p>realize the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> as the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> (which requires the form to have integral <a class="existingWikiWord" href="/nlab/show/periods">periods</a>): called the <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>;</p> </li> <li> <p>choose a <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a> – a splitting of the abstract <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> into “coordinates” and “momenta”;</p> </li> </ol> <p>and then form</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/states">states</a> as the space of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> which depend only on the “coordinates” (not on the “momenta”);</p> </li> <li> <p>associate with every function on the symplectic manifold – every <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> – a <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a> on this Hilbert space.</p> </li> </ol> <h3 id="HistoryAndVariants">History and variants</h3> <p>The approach is due to <a class="existingWikiWord" href="/nlab/show/Alexandre+Kirillov">Alexandre Kirillov</a> (“<a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a>”), <a class="existingWikiWord" href="/nlab/show/Bertram+Kostant">Bertram Kostant</a> and <a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>. See the <a href="#ReferencesGeneral">References</a> below. It is closely related to <a class="existingWikiWord" href="/nlab/show/Berezin+quantization">Berezin quantization</a> and the subject of <a class="existingWikiWord" href="/nlab/show/coherent+states">coherent states</a>.</p> <p><img src="http://ncatlab.org/nlab/files/SouriauPrequantumFlowChart.jpg" alt="Prequantum flow chart" /></p> <p>(<a href="#Souriau74">Souriau 74, fig 1</a>)</p> <p>In a long term project <a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a> and many of his students have followed the idea that the true story behind geometric quantization crucially involves <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+groupoids">symplectic Lie groupoids</a>: <a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a>. See <a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a> for more on this.</p> <p>More generally, there is <em><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></em>.</p> <h3 id="overview">Overview</h3> <blockquote> <p>This overview is taken from (<a href="#Baez">Baez</a>).</p> </blockquote> <p><em>Geometric quantization</em> is a tool for understanding the relation between <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> and <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum physics</a>. Here’s a brief sketch of how it goes.</p> <ol> <li> <p>We start with a <em>classical <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></em>: mathematically, this is a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>.</p> </li> <li> <p>Then we do <em>prequantization</em>: this gives us a Hermitian <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> whose curvature equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">i \omega</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is called the <strong><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a></strong>.</p> <p><strong>Warning:</strong> we can only do this step if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> satisfies the <em><a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+leaf">Bohr-Sommerfeld condition</a></em>, which says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">/</mo><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">\omega/2\pi</annotation></semantics></math> defines an <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> class. If this condition holds, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> are determined up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, but not canonically.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math> of square-integrable <a class="existingWikiWord" href="/nlab/show/section">section</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is called the <em>prequantum Hilbert space</em>. This is not yet the Hilbert space of our quantized theory – it’s too big. But it’s a good step in the right direction. In particular, we can <em>prequantize classical observables</em>: there’s a map sending any smooth function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to an operator on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math>. This map takes <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>s to <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>s, just as one would hope. The formula for this map involves the <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </li> <li> <p>To cut down the prequantum Hilbert space, we need to choose a <em><a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></em>, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>. What’s this? Well, for each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, a polarization picks out a certain subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">P_x</annotation></semantics></math> of the complexified <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent space</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. We define the <em>quantum Hilbert space</em>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, to be the space of all square-integrable sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> that give zero when we take their <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> at any point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the direction of any vector in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">P_x</annotation></semantics></math>. The quantum Hilbert space is a subspace of the prequantum Hilbert space.</p> <p><strong>Warning:</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> to be a polarization, there are some crucial technical conditions we impose on the subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">P_x</annotation></semantics></math>. First, they must be <em>isotropic</em>: the complexified symplectic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> must vanish on them. Second, they must be <em>Lagrangian</em>: they must be maximal isotropic subspaces. Third, they must vary smoothly with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. And fourth, they must be <em>integrable</em>.</p> </li> <li> <p>The easiest sort of polarization to understand is a <em>real polarization</em>. This is where the subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">P_x</annotation></semantics></math> come from subspaces of the tangent space by complexification. It boils down to this: a <a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</a> is an <a class="existingWikiWord" href="/nlab/show/integrable+distribution">integrable distribution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> on the classical phase space where each space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">P_x</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Lagrangian+subspace">Lagrangian subspace</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">T_x X</annotation></semantics></math>.</p> </li> <li> <p>To understand this rigamarole, one must study examples! First, it’s good to understand how good old <em>Schrödinger quantization</em> fits into this framework. Remember, in Schrödinger quantization we take our classical <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>M</mi></mrow><annotation encoding="application/x-tex">T^* M</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> called the <em>classical configuration space</em>. We then let our quantum Hilbert space be the space of all <a class="existingWikiWord" href="/nlab/show/square-integrable+functions">square-integrable functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> <p>Modulo some technical trickery, we get this example when we run the above machinery and use a certain god-given real polarization on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>M</mi></mrow><annotation encoding="application/x-tex">X = T^*M</annotation></semantics></math>, namely the one given by the vertical vectors.</p> </li> <li> <p>It’s also good to study the <em>Bargmann–Segal representation</em>, which we get by taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{C}^n</annotation></semantics></math> with its god-given symplectic structure (the imaginary part of the inner product) and using the god-given <em>Kähler polarization</em>. When we do this, our quantum Hilbert space consists of analytic functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^n</annotation></semantics></math> which are square-integrable with respect to a Gaussian measure centered at the origin.</p> </li> <li> <p>The next step is to <em>quantize classical observables</em>, turning them into linear operators on the quantum Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. Unfortunately, we can’t quantize all such observables while still sending <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>s to <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>s, as we did at the prequantum level. So at this point things get trickier and my brief outline will stop. Ultimately, the reason for this problem is that quantization is not a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the <a class="existingWikiWord" href="/nlab/show/category">category</a> of symplectic manifolds to the category of Hilbert spaces – but for that one needs to learn a bit about <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>.</p> </li> </ol> <h3 id="basic_jargon">Basic Jargon</h3> <p>Here are some definitions of important terms. Unfortunately they are defined using other terms that you might not understand. If you are really mystified, you need to read some books on <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> and the math of <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a> before proceeding.</p> <ul> <li> <p><strong>complexification</strong>: We can tensor a real vector space with the complex numbers and get a complex vector space; this process is called complexification. For example, we can complexify the tangent space at some point of a manifold, which amounts to forming the space of complex linear combinations of tangent vectors at that point.</p> </li> <li> <p><strong>distribution</strong>: The word “distribution” means many different things in mathematics, but here’s one: a “distribution” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> on a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a choice of a subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">V_x</annotation></semantics></math> of each tangent space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>p</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">T_p X</annotation></semantics></math>, where the choice depends smoothly on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </li> <li> <p><strong>Hamiltonian vector field</strong>: Given a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a symplectic structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, any smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f: X \to \mathbb{R}</annotation></semantics></math> can be thought of as a “Hamiltonian”, meaning physically that we think of it as the energy function and let it give rise to a flow on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> describing the time evolution of states. Mathematically speaking, this flow is generated by a vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v(f)</annotation></semantics></math> called the “Hamiltonian vector field” associated to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. It is the unique vector field such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex"> \omega(-, v(f)) = d f </annotation></semantics></math></div> <p>In other words, for any vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mi>f</mi></mrow><annotation encoding="application/x-tex"> \omega(u,v(f)) = d f(u) = u f </annotation></semantics></math></div> <p>The vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v(f)</annotation></semantics></math> is guaranteed to exist by the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is nondegenerate.</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/integrable+distribution">integrable distribution</a></strong>: A <a class="existingWikiWord" href="/nlab/show/distribution+of+subspaces">distribution of subspaces</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is “integrable” if at least locally, there is a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by submanifolds such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">V_x</annotation></semantics></math> is the tangent space of the submanifold containing the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> class</strong>: Any closed <a class="existingWikiWord" href="/nlab/show/differential+p-form">differential p-form</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> defines an element of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. This is a finite-dimensional vector space, and it contains a lattice called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>th integral <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. We say a cohomology class is integral if it lies in this lattice. Most notably, if you take any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on any Hermitian line bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, its curvature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form will define an integral cohomology class once you divide it by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">2 \pi i</annotation></semantics></math>. This cohomology class is called the first <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a>, and it serves to determine the line <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>s</strong>: Given a symplectic structure on a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and given two smooth functions on that manifold, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, there’s a trick for getting a new smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f,g\}</annotation></semantics></math> on the manifold, called the Poisson bracket of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.</p> <p>This trick works as follows: given any smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> we can take its differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math>, which is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-form. Then there is a unique vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v(f)</annotation></semantics></math>, the Hamiltonian vector field associated to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex"> \omega(-,v(f)) = d f </annotation></semantics></math></div> <p>Using this we define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \{f,g\} = \omega(v(f),v(g)) </annotation></semantics></math></div> <p>It’s easy to check that we also have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo><mo>=</mo><mi>d</mi><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>v</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">\{f,g\} = d g(v(f)) = v(f) g</annotation></semantics></math>. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f,g\}</annotation></semantics></math> says how much <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> changes as we differentiate it in the direction of the Hamiltonian vector field generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>In the familiar case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math> with momentum and position coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>q</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">q_i</annotation></semantics></math>, the Poisson brackets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> work out to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><msub><mi>p</mi> <mi>i</mi></msub></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>g</mi></mrow><mrow><mi>d</mi><msub><mi>q</mi> <mi>i</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><msub><mi>q</mi> <mi>i</mi></msub></mrow></mfrac><mfrac><mrow><mi>d</mi><mi>g</mi></mrow><mrow><mi>d</mi><msub><mi>p</mi> <mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \{f,g\} = \sum_i \frac{d f}{d p_i} \frac{d g}{d q_i} - \frac{d f}{d q_i}\frac{d g}{d p_i} </annotation></semantics></math></div></li> <li> <p><strong>square-integrable sections</strong>: We can define an inner product on the sections of a Hermitian line bundle over a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a symplectic structure. The symplectic structure defines a volume form which lets us do the necessary integral. A section whose inner product with itself is finite is said to be square-integrable. Such sections form a Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0</annotation></semantics></math> called the “prequantum Hilbert space”. It is a kind of preliminary version of the Hilbert space we get when we quantize the classical system whose phase space is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> <li> <p><strong>symplectic structure</strong>: A symplectic structure on a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> which is nondegenerate in the sense that for any nonzero tangent vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> at any point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, there is a tangent vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> at that point for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w(u,v)</annotation></semantics></math> is nonzero.</p> </li> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a></strong>: The <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> is the group of unit complex numbers. Given a complex line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> with an inner product on each fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">L_x</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> connection on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a connection such that parallel translation preserves the inner product.</p> </li> <li> <p><strong>vertical vectors</strong>: Given a bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, we say a tangent vector to some point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is vertical if it projects to zero down on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> </li> </ul> <h2 id="definition">Definition</h2> <p>Geometric quantization involves two steps</p> <ol> <li> <p><a href="#GeometricPrequantization">Geometric prequantization</a></p> </li> <li> <p><a href="#GeometricQuantizationProper">Geometric quantization proper</a>.</p> </li> </ol> <h3 id="GeometricPrequantization">Geometric prequantization</h3> <h4 id="prequantum_line_bundle">Prequantum line bundle</h4> <p>Given the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a> for it is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a> whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>.</p> <p>In other words, prequantization is a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> through the curvature-<a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> (see there).</p> <p>The multiple of the <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> of this line bundle is identified with the inverse <em><a class="existingWikiWord" href="/nlab/show/Planck+constant">Planck constant</a></em>.</p> <h4 id="prequantum_states">Prequantum states</h4> <p>A <em>prequantum state</em> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</a>.</p> <p>This becomes a <em><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></em> or <a class="existingWikiWord" href="/nlab/show/wavefunction">wavefunction</a> if <a class="existingWikiWord" href="/nlab/show/polarization">polarized</a> (…).</p> <h4 id="prequantum_operators">Prequantum operators</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla : X \to \mathbf{B} U(1)_{conn}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(E)</annotation></semantics></math> for its space of smooth <a class="existingWikiWord" href="/nlab/show/sections">sections</a>, the <em><a class="existingWikiWord" href="/nlab/show/prequantum+space+of+states">prequantum space of states</a></em>.</p> <div class="num_defn" id="PrequantumOperator"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/prequantum+operators">prequantum operators</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(X, \mathbb{C})</annotation></semantics></math> a function on phase space, the corresponding <strong><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator</a></strong> is the linear map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">^</mo></mover><mo lspace="verythinmathspace">:</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hat f \colon \Gamma_X(E) \to \Gamma_X(E) </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation" id="eq:FormulaForPrequantumOperator"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mo>∇</mo> <mrow><msub><mi>v</mi> <mi>f</mi></msub></mrow></msub><mi>ψ</mi><mo>+</mo><mi>f</mi><mo>⋅</mo><mi>ψ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">v_f</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><msub><mi>v</mi> <mi>f</mi></msub></mrow></msub><mo>:</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> of sections along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">v_f</annotation></semantics></math> for the given choice of prequantum connection;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E)</annotation></semantics></math> is the operation of degreewise multiplication pf sections.</p> </li> </ul> </div> <div class="num_remark" id="OriginOfTheFormulaForPrequantumOperators"> <h6 id="remark">Remark</h6> <p><strong>(origin of the formulas for <a class="existingWikiWord" href="/nlab/show/prequantum+operators">prequantum operators</a>)*</strong></p> <p>The formula <a class="maruku-eqref" href="#eq:FormulaForPrequantumOperator">(1)</a> may look a bit mysterious on first sight. The correction term to the <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> appearing in this formula is ultimately due to the fact that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> corresponding to a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex">H_v</annotation></semantics></math> via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi><mo>=</mo><mi>d</mi><msub><mi>H</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex"> \iota_v \omega = d H_v </annotation></semantics></math></div> <p>then the <a class="existingWikiWord" href="/nlab/show/Lie+derivative">Lie derivative</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> (the symplectic potentiation, related by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>θ</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \theta = \omega</annotation></semantics></math>) is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>θ</mi><mo>=</mo><mi>d</mi><msub><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{L}_v \theta = d \tilde H_v </annotation></semantics></math></div> <p>for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mi>v</mi></msub><mo>=</mo><msub><mi>H</mi> <mi>v</mi></msub><mo>+</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde H_v = H_v + \iota_v \theta \,. </annotation></semantics></math></div> <p>Here the second term on the right is what yields the <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> in <a class="maruku-eqref" href="#eq:FormulaForPrequantumOperator">(1)</a>, while the first summand is the correction term in <a class="maruku-eqref" href="#eq:FormulaForPrequantumOperator">(1)</a>.</p> <p>A derivation of these formulas from first principles is given in (<a href="#FiorenzaRogersSchreiber13a">Fiorenza-Rogers-Schreiber 13a</a>, example 3.2.3 and remark 3.3.16).</p> </div> <h3 id="GeometricQuantizationProper">Geometric quantization</h3> <p>Given a <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</a> as above, the actual step of genuine <em>geometric quantization</em> consists first of forming <em>half</em> its space of sections in a certain sense. Physically this means passing to the space of <a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> that depend only on <a class="existingWikiWord" href="/nlab/show/canonical+positions">canonical positions</a> but not on <a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a>. Second, <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> of the group of (exponentiated) <a class="existingWikiWord" href="/nlab/show/prequantum+operators">prequantum operators</a> are made to descend to this space of quantum states, these are the <a class="existingWikiWord" href="/nlab/show/quantum+operators">quantum operators</a> or <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>.</p> <h4 id="QuantumStates">Quantum states</h4> <p>Historically, the traditional way to formalize the formation of the <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> is as a 3-step process</p> <ol> <li> <p>choose a <a href="#Polarizations">Polarization</a>;</p> </li> <li> <p>choose a <a href="#MetaplecticCorrection">Metaplectic correction</a>;</p> </li> <li> <p>form the induced <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> as the space of polarized sections of the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> tensored a certain half-form bundle.</p> </li> </ol> <p>This we discuss at</p> <ul> <li><em><a href="#Polarizations">Quantum state space as space of polarized sections</a></em></li> </ul> <p>This traditional route via polarizations and metaplectic corrections has the disadvantage that mathematically it is not a very natural operation. However, under mild conditions it turns out to be <a class="existingWikiWord" href="/nlab/show/equivalence">equivalent</a> to the following mathematically very natural construction</p> <ol> <li> <p>choose a <a class="existingWikiWord" href="/nlab/show/KU-orientation">KU-orientation</a>, hence a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> compatible with the given prequantum bundle;</p> </li> <li> <p>take the space of quantum states to be the <a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward</a> in <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> to the point, hence the <a class="existingWikiWord" href="/nlab/show/index">index</a> of the <a class="existingWikiWord" href="/nlab/show/spin%5Ec+Dirac+operator">spin^c Dirac operator</a> twisted by the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>.</p> </li> </ol> <p>This general <em><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">geometric quantization by push-forward</a></em> is discussed below at</p> <ul> <li><em><a href="#AsIndexOfSpinCDiracOperator">Quantum space of states as index of a Dirac operator</a></em>.</li> </ul> <p>In the special case that the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> admits a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a> this <a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">push-forward quantization</a> has a direct expression in terms of the <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex</a> <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> and of the <a class="existingWikiWord" href="/nlab/show/Dolbeault+operator">Dolbeault operator</a> of the prequantum <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic line bundle</a>. Now the choice of <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a> is precisely a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> (a “<a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>”) and the <a class="existingWikiWord" href="/nlab/show/index">index</a> is now that of the <a class="existingWikiWord" href="/nlab/show/Dolbeault-Dirac+operator">Dolbeault-Dirac operator</a> which is equivalently just the <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of the holomorphic <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>. This complex Dolbeault quantization case we discuss in</p> <ul> <li><em><a href="#EulerCharacteristicOfSheafCohomology">Quantum state space as Euler characteristic of prequantum sheaf cohomology</a></em></li> </ul> <p>and</p> <ul> <li><em><a href="#IndexOfDolbeaultDiracOperator">Quantum state space as index of the Dolbeault-Dirac operator</a></em>.</li> </ul> <h5 id="Polarizations">Quantum state space as space of polarized sections</h5> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, choose a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a>, hence an involutive Lagrangian subbundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>⊂</mo><msub><mi>T</mi> <mi>𝒞</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{P} \subset T_{\mathcal{C}} X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>∩</mo><mover><mi>𝒫</mi><mo>¯</mo></mover><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathcal{P} \cap \overline{\mathcal{P}} = 0</annotation></semantics></math>. Choose moreover a <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math>. This defines the <a class="existingWikiWord" href="/nlab/show/half-density">half-density</a> bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><mi>Ω</mi><mi>𝒫</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega \mathcal{P}}</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math>.</p> <p>Let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L,\nabla)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>.</p> <div class="num_defn" id="PolarizedQuantumStates"> <h6 id="definition_3">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> of the prequantum bundle defined by the choice of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><mi>Ω</mi><mi>𝒫</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega \mathcal{P}}</annotation></semantics></math> is the space of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⊗</mo><msqrt><mrow><mi>Ω</mi><mi>𝒫</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">L \otimes \sqrt{\Omega \mathcal{P}}</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> covariantly constant with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒫</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\mathcal{P}}</annotation></semantics></math> and square integrable with respect to the induced <a class="existingWikiWord" href="/nlab/show/integration">integration</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mover><mi>𝒫</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">X/\overline{\mathcal{P}}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℋ</mi> <mi>pol</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>L</mi> <mn>2</mn></msup><mrow><mo>(</mo><mrow><mo>{</mo><mi>ψ</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>L</mi><mo>⊗</mo><msqrt><mrow><mi>Ω</mi><mi>𝒫</mi></mrow></msqrt><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msub><mo>∇</mo> <mover><mi>𝒫</mi><mo>¯</mo></mover></msub><mi>ψ</mi><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}_{pol} \colon L^2 \left( \left\{ \psi \in \Gamma(L \otimes \sqrt{\Omega \mathcal{P}}) \;|\; \nabla_{\overline{\mathcal{P}}} \psi = 0 \right\} \right) \,. </annotation></semantics></math></div></div> <p>For instance (<a href="#BatesWeinstein">Bates-Weinstein, def. 7.17</a>).</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>In the case of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a> it is useful to write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Omega^{n,0}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a> of the polarization and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega^{n,0}}</annotation></semantics></math> for the corresponding choice of <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a>/<a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>.</p> </div> <h5 id="EulerCharacteristicOfSheafCohomology">Quantum state space as Euler characteristic of prequantum sheaf cohomology</h5> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> be a <em><a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a></em> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">L_\omega</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega^{n,0}}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a>.</p> <div class="num_defn" id="EulerQuantumStates"> <h6 id="definition_4">Definition</h6> <p>The corresponding Euler quantum Hilbert space is the <a class="existingWikiWord" href="/nlab/show/virtual+vector+space">virtual vector space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℋ</mi> <mi>Euler</mi></msub><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mspace width="thinmathspace"></mspace><msup><mi>H</mi> <mrow><mn>0</mn><mo>,</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}_{Euler} \coloneqq \bigoplus_{k = 0}^{n} (-1)^k \, H^{0,k}(X,L_\omega \otimes \sqrt{\Omega^{n,0}}) \,, </annotation></semantics></math></div> <p>which is the alternating <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the <a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a> space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">L_\omega \otimes \sqrt{\Omega^{n,0}}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> with the <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a>/<a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>.</p> </div> <div class="num_remark" id="SectionsAndEulerQuantizationCoincidesOnPositiveBundles"> <h6 id="remark_3">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Kodaira+vanishing+theorem">Kodaira vanishing theorem</a> asserts that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub><msup><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">L_\omega \sqrt{\Omega^{n,0}}^{-1}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/positive+line+bundle">positive line bundle</a> then all the higher <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> in the above expression vanish. Therefore in this case the definition coincides with that via polarizations in def. <a class="maruku-ref" href="#PolarizedQuantumStates"></a> above.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo>⊗</mo><msup><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mi>positive</mi><mo>)</mo></mrow><mo>⇒</mo><mrow><mo>(</mo><msub><mi>ℋ</mi> <mi>pol</mi></msub><mo>≃</mo><msub><mi>ℋ</mi> <mi>Euler</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( L_\omega \otimes \sqrt{\Omega^{n,0}}^{-1} \; positive \right) \Rightarrow \left( \mathcal{H}_{pol} \simeq \mathcal{H}_{Euler} \right) \,. </annotation></semantics></math></div></div> <h5 id="IndexOfDolbeaultDiracOperator">Quantum state space as index of Dolbeault-Dirac operator</h5> <p>Suppose again that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo>,</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega, L_\omega)</annotation></semantics></math> is equipped with a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a>.</p> <p>We need the following general fact on <a class="existingWikiWord" href="/nlab/show/spin+structures">spin structures</a> over <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifolds">Kähler manifolds</a>.</p> <div class="num_prop" id="SpinStructuresOnKaehlerManifolds"> <h6 id="proposition">Proposition</h6> <p>A choice of <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> on the <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (of real dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2n</annotation></semantics></math>) is equivalently a choice of <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> (“<a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega^{n,0}}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Omega^{n,0}</annotation></semantics></math>.</p> <p>Given such a choice, there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_X</annotation></semantics></math> and the (anti-)holomorphic form bundle tensored by this square root</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>≃</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \simeq \Omega^{0,\bullet} \otimes \sqrt{\Omega^{n,0}} \,. </annotation></semantics></math></div> <p>Finally, the corresponding <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> is the <a class="existingWikiWord" href="/nlab/show/Dolbeault-Dirac+operator">Dolbeault-Dirac operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∂</mo><mo>¯</mo></mover><mo>+</mo><msup><mover><mo>∂</mo><mo>¯</mo></mover> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\overline{\partial} + \overline{\partial}^\ast</annotation></semantics></math>.</p> </div> <p>See at <em><a href="spin+structure#OverAKahlerManifold">spin structure – Over a Kähler manifold</a></em>.</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which is twisted by the <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum</a> <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">L_\omega</annotation></semantics></math> is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mo>∂</mo><mo>¯</mo></mover> <mo>∇</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mover><mo>∂</mo><mo>¯</mo></mover> <mo>∇</mo> <mo>*</mo></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mi>even</mi><mo stretchy="false">/</mo><mi>odd</mi></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mi>ω</mi></msub><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt><mo>)</mo></mrow><mo>→</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mi>odd</mi><mo stretchy="false">/</mo><mi>even</mi></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mi>ω</mi></msub><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overline{\partial}_\nabla + \overline{\partial}^\ast_\nabla \;\colon\; \Omega^{0,even/odd}\left(X, \;L_\omega \otimes \sqrt{\Omega^{n,0}}\right) \to \Omega^{0,odd/even}\left(X, \; L_\omega \otimes \sqrt{\Omega^{n,0}}\right) \,. </annotation></semantics></math></div> <p>Observe how from the point of view of just the <a class="existingWikiWord" href="/nlab/show/Dolbeault+operator">Dolbeault operator</a>, this is twisting not just by the prequantum line bundle itself but by the <em><a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectically corrected</a></em> prequantum line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">L_\omega \otimes \sqrt{\Omega^{n,0}}</annotation></semantics></math>, while from the point of view of the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> it is just twisting by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">L_\omega</annotation></semantics></math>, since tensoring with the square root line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega^{n,0}}</annotation></semantics></math> induces the isomorphism between the antiholomorphic differential form bundle and the actual <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> over the <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a>.</p> </div> <div class="num_defn" id="DolbeaultDiracSpacesOfStates"> <h6 id="definition_5">Definition</h6> <p>The <em>Dolbeault-Dirac</em> space of quantum states of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo>,</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega, L_\omega)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/index">index</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">L_\omega</annotation></semantics></math>-twisted <a class="existingWikiWord" href="/nlab/show/Dolbeault-Dirac+operator">Dolbeault-Dirac operator</a> (the <a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℋ</mi> <mi>DolbDir</mi></msub><mo>≔</mo><mi>index</mi><mo stretchy="false">(</mo><msub><mover><mo>∂</mo><mo>¯</mo></mover> <mo>∇</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mover><mo>∂</mo><mo>¯</mo></mover> <mo>∇</mo> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}_{DolbDir} \coloneqq index( \overline{\partial}_\nabla + \overline{\partial}_\nabla^\ast ) \,. </annotation></semantics></math></div></div> <div class="num_remark" id="DolbeaultDiracAgreesWithEuler"> <h6 id="remark_5">Remark</h6> <p>This definition agrees with that by <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> in def. <a class="maruku-ref" href="#EulerQuantumStates"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℋ</mi> <mi>Euler</mi></msub><mo>≃</mo><msub><mi>ℋ</mi> <mi>DolbDirac</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}_{Euler} \simeq \mathcal{H}_{DolbDirac} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>What before in the quantization prescription <a href="#Polarizations">by polarization</a> and <a href="#EulerCharacteristicOfSheafCohomology">by abelian sheaf cohomology</a> was the “<a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a>” introduced “by hand” is now naturally part of the <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of prop. <a class="maruku-ref" href="#SpinStructuresOnKaehlerManifolds"></a> which identifies the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> with the <a class="existingWikiWord" href="/nlab/show/Dolbeault-Dirac+operator">Dolbeault-Dirac operator</a>.</p> </div> <h5 id="AsIndexOfSpinCDiracOperator">Quantum state spaces as index of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-Dirac operator</h5> <p>Finally we come to the true definition of geometric quantization, the most general and at the same time most natural one which contains the above as special cases.</p> <p>The actual definition is def. <a class="maruku-ref" href="#KTheoreticQuantization"></a> below. Here we lead up to it by spelling out the ingredients.</p> <p>We need the following general facts about <a class="existingWikiWord" href="/nlab/show/spin%5Ec+Dirac+operators">spin^c Dirac operators</a>.</p> <div class="num_defn" id="SpinCAndDeterminantLineBundle"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/spin%5Ec">spin^c</a>-<a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(n)</annotation></semantics></math> is equivalently</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of the ordinary <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> over the <a class="existingWikiWord" href="/nlab/show/group+of+order+2">group of order 2</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></munder><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spin^c \simeq Spin(n) \underset{\mathbb{Z}_2}{\times} U(1) </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo>≃</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex"> Spin^c \simeq \Omega \mathbf{B} Spin^c </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of the universal smooth <a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a> and the mod-2 reduction of the universal <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> in <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoids">smooth infinity-groupoids</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>det</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin^c(n) &\stackrel{det}{\to}& \mathbf{B}U(1) \\ \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\mathbf{c}_1 \, mod \, 2}} \\ \mathbf{B}SO(n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,. </annotation></semantics></math></div></li> </ul> <p>Here the top horizontal map is called the universal <em><a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a></em> map.</p> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a></em> for more details.</p> <div class="num_remark" id="SpinCInComponents"> <h6 id="remark_7">Remark</h6> <p>It follows that if we represent elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></munder><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(n) \simeq SO(n) \underset{\mathbb{Z}_2}{\times} U(1)</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,c)</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> map of def. <a class="maruku-ref" href="#SpinCAndDeterminantLineBundle"></a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>↦</mo><mn>2</mn><mi>c</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> det \;\colon \; (g,c) \mapsto 2c \,, </annotation></semantics></math></div> <p>where on the right we write the group operation in the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> additively.</p> <p>This factor of 2 on the right is crucial in all of the following.</p> </div> <div class="num_defn" id="SpinCStructure"> <h6 id="definition_7">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. A smooth <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\hat \tau_X</annotation></semantics></math> of the classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_X \colon X \to \mathbf{B} SO(n)</annotation></semantics></math> of its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> through the left vertical map in def. <a class="maruku-ref" href="#SpinCAndDeterminantLineBundle"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B} Spin^c(n) \\ & {}^{\mathllap{\hat \tau_X}}\nearrow & \downarrow \\ X &\stackrel{\tau_X}{\to}& \mathbf{B}SO(n) } \,. </annotation></semantics></math></div></div> <div class="num_remark" id="SpinCWithDivisibleDetIsSpinTensorLine"> <h6 id="remark_8">Remark</h6> <p>In words this says that a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> on an oriented manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a choice of <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> or equivalenty of hermitian <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a> such that its <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> modulo 2 equals the <a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. If this second Stiefel-Whitney class vanishes (as an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2(X,\mathbb{Z}_2)</annotation></semantics></math>) this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has a genuine <em><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></em>. So in other words whenever the <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> of a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> (in the sens of def. <a class="maruku-ref" href="#SpinCAndDeterminantLineBundle"></a>) has a <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> that is divisible by 2, then there is an actual <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>.</p> <p>We can formalize this statement as follows: there is a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>2</mn><mo>⋅</mo><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msup><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msup></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}( Spin(n) \times U(1) ) &\stackrel{((2 \cdot)\circ p_2}{\to}& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 \, mod\, 2}} \\ \mathbf{B}SO(n) &\stackrel{\mathbf{w}^2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } </annotation></semantics></math></div> <p>similar to the one in def. <a class="maruku-ref" href="#SpinCAndDeterminantLineBundle"></a> but crucially different in that here we have just the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> in the top left. In the standard presentation of these objects this diagram commutes on the nose (filled by a canonical <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>) simply because both ways to go from the top left to the bottom right hit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0 \in \mathbb{Z}_2</annotation></semantics></math>: the left-bottom one because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin(n)</annotation></semantics></math> is essentially by definition such that the second SW class trivializes on it, and the top-right one because first multiplying by 2 and then reducing mod 2 is 0.</p> <p>Since the diagram commutes, the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the above <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> says that there is a canonically induced map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (p_1, p_2) \;\colon\; Spin(n) \times U(1) \to Spin^c(n) \,. </annotation></semantics></math></div> <p>Of course this is just the evident <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a> which on elements is simply the identity</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (g,c) \mapsto (g,c) \,, </annotation></semantics></math></div> <p>only that on the right we regard the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,c)</annotation></semantics></math> as a placeholder for its <a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a> in the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>.</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> says that every <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\hat \tau_c</annotation></semantics></math> whose underlying <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> is such that its <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> is divisible by 2 actually factors through this map, hence that <em>it is just the product of an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>-principal bundle with a circle bundle</em>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mn>2</mn><mo>⋅</mo><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>det</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>c</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}(Spin(n) \times U(1)) &\stackrel{p_2}{\to}& \mathbf{B}U(1) \\ {}^{\mathllap{(p_1,p_2)}}\downarrow && \downarrow^{\mathbf{B}(2 \cdot)} \\ \mathbf{B}Spin^c(n) &\stackrel{det}{\to}& \mathbf{B}U(1) \\ \downarrow &(pb)& \downarrow^{\mathrlap{c_1\,mod\,2}} \\ \mathbf{B}SO(n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,. </annotation></semantics></math></div> <p>This is the crucial relation by which the K-theoretic quantization will harmonize with the above Euler- and Dolbeault- quantization, discussed below.</p> </div> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> an even number, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>n</mi></msub><mo>≃</mo><msup><mi>ℂ</mi> <mrow><msup><mn>2</mn> <mrow><mi>n</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> \Delta_n \simeq \mathbb{C}^{2^{n/2}} </annotation></semantics></math></div> <p>for the canonical complex <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a>, which decomposes into two chiral <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>n</mi></msub><mo>≃</mo><msubsup><mi>Δ</mi> <mi>n</mi> <mo>+</mo></msubsup><mo>⊕</mo><msubsup><mi>Δ</mi> <mi>n</mi> <mo>−</mo></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_n \simeq \Delta_n^+ \oplus \Delta_n^- \,. </annotation></semantics></math></div> <p>Then the <strong>canonical <a class="existingWikiWord" href="/nlab/show/spin%5Ec">spin^c</a>-<a class="existingWikiWord" href="/nlab/show/representation">representation</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Δ</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>Δ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \rho \colon Spin^c(n) \times \Delta_n \to \Delta_n </annotation></semantics></math></div> <p>is the one given in the components of remark <a class="maruku-ref" href="#SpinCInComponents"></a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi></mrow><annotation encoding="application/x-tex"> ((g,c), \psi) \mapsto (g(\psi)) \cdot c </annotation></semantics></math></div> <p>(first act with the spin-component in the usual way and then multiply by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c \in U(1) \hookrightarrow \mathbb{C}</annotation></semantics></math>).</p> </div> <p>As discussed at <em><a class="existingWikiWord" href="/nlab/show/representation">representation</a></em> we may think of this as a morphism of <a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> (<a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℂ</mi><mstyle mathvariant="bold"><mi>Mod</mi></mstyle><mo>≃</mo><mi>ℂ</mi><mstyle mathvariant="bold"><mi>Vect</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho \colon \mathbf{B} Spin^c(n) \to \mathbb{C}\mathbf{Mod} \simeq \mathbb{C}\mathbf{Vect} \,. </annotation></semantics></math></div> <div class="num_defn" id="SpinCSpinorBundle"> <h6 id="definition_9">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat \tau_X \colon X \to \mathbf{B}Spin^c(n)</annotation></semantics></math>, def. <a class="maruku-ref" href="#SpinCStructure"></a>, the <strong><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a></strong> is the one modulated by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>c</mi></msup><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>→</mo><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>ρ</mi></mover><mi>ℂ</mi><mstyle mathvariant="bold"><mi>Vect</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^c \colon X \stackrel{\hat \tau_X}{\to} \mathbf{B}Spin^c(n) \stackrel{\rho}{\to} \mathbb{C}\mathbf{Vect} \,. </annotation></semantics></math></div></div> <p>Crucial for the comparison of the K-theoretic quantization to be defined in a moment and the above Euler/Dolbeault quantization is the following:</p> <div class="num_remark" id="SpinCSpinorsAsTensorProductWhenDetDivisible"> <h6 id="remark_9">Remark</h6> <p>By remark <a class="maruku-ref" href="#SpinCWithDivisibleDetIsSpinTensorLine"></a> it follows that in the case that the <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> of the <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> is divisible by 2, hence in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>det</mi><mo stretchy="false">(</mo><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo>∈</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_1(det(\hat \tau_X)) = 0 \,mod\, 2 \in H^2(X,\mathbb{Z}_2)</annotation></semantics></math>, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> of def. <a class="maruku-ref" href="#SpinCSpinorBundle"></a> is just the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of the ordinary underlying <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> with <em>half</em> the <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>det</mi><mo stretchy="false">(</mo><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msup><mi>S</mi> <mi>c</mi></msup><mo>≃</mo><mi>S</mi><mo>⊗</mo><msqrt><mrow><mi>det</mi><mo stretchy="false">(</mo><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></msqrt><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( c_1(det(\hat \tau_X)) = 0 \,mod\, 2 \right) \;\Rightarrow\; \left(S^c \simeq S \otimes \sqrt{det(\hat \tau_X)}\right) \,. </annotation></semantics></math></div></div> <p>Let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> be a (<a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">pre-</a>)<a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L_{\omega}, \nabla)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>.</p> <div class="num_defn" id="KTheoreticQuantization"> <h6 id="definition_10">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\hat \tau_X</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whose underlying <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> coincides, up to equivalence, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>ω</mi></msub><msup><mo stretchy="false">)</mo> <mrow><msup><mo>⊗</mo> <mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">(L_\omega)^{\otimes^2}</annotation></semantics></math>, then the <em><a class="existingWikiWord" href="/nlab/show/spin%5Ec+quantization">spin^c quantization</a></em> of this prequantum data is the <a class="existingWikiWord" href="/nlab/show/index">index</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/spin%5Ec+Dirac+operator">spin^c Dirac operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">D_{\mathbf{c}}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℋ</mi> <mi>K</mi></msub><mo>≔</mo><mi>index</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}_{K} \coloneqq index(D_{\mathbf{c}}) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_10">Remark</h6> <p>This is equivalently the <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">push-forward</a> in <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> to the point.</p> <p>Moreover, on <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> this push-forward is expressed by the traditional construction of <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">of quantum states</a>:</p> <p>let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a manifold equipped with a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, a <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> and a <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo lspace="verythinmathspace">:</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D \colon \Gamma(S) \to \Gamma(S)</annotation></semantics></math>. Then the corresponding <a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a> cycle is:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(S)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/square+integrable+function">square integrable</a> <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>;</p> </li> <li> <p>equipped with the <a class="existingWikiWord" href="/nlab/show/action">action</a> by fiberwise multiplication of the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">of functions</a> <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℬ</mi><mo stretchy="false">(</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C_0(X) \to \mathcal{B}(L^2(S)) </annotation></semantics></math></div> <p>(which is by <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a>)</p> </li> <li> <p>and equipped with the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a> defining the <a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a> class.</p> </li> </ol> <p>Together with is equivalent an element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [D, L^2(S)] \in KK(C_0(X), \mathbb{C}) \,. </annotation></semantics></math></div> <p>Postcomposition of K-theory classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ξ</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo>,</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\xi] \in KK(\mathbb{C}, C_0(X))</annotation></semantics></math> with this map is the <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">push-forward</a>/<a class="existingWikiWord" href="/nlab/show/index">index</a> map in K-theory.</p> </div> <div class="num_remark"> <h6 id="remark_11">Remark</h6> <p>This definition does not assume any choice of <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a>, nor any choice of <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> etc. On the other hand, every choice of <a class="existingWikiWord" href="/nlab/show/almost+complex+structure">almost complex structure</a> (hence in particular of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a>) does induce a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>, as discussed there. See also (<a href="#BorthwickUribe96">Borthwick-Uribe 96</a>).</p> </div> <p>So then we can compare:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>If a choice of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a> exists on the <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,L_\omega)</annotation></semantics></math>, then the K-theoretic quantization of def. <a class="maruku-ref" href="#KTheoreticQuantization"></a> coincides with the Dolbeault-Dirac quantization of def. <a class="maruku-ref" href="#DolbeaultDiracSpacesOfStates"></a> and hence, by remark <a class="maruku-ref" href="#DolbeaultDiracAgreesWithEuler"></a>, with the Euler-characteristic definition, def. <a class="maruku-ref" href="#EulerQuantumStates"></a>, so that all the respectives <a class="existingWikiWord" href="/nlab/show/spaces+of+quantum+states">spaces of quantum states</a> agree:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mi>Kaehler</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>ℋ</mi> <mi>K</mi></msub><mo>≃</mo><msub><mi>ℋ</mi> <mi>DolbDirac</mi></msub><mo>≃</mo><msub><mi>ℋ</mi> <mi>Euler</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(\left(X,\omega\right) \; Kaehler\right) \;\Rightarrow\; \left( \mathcal{H}_K \simeq \mathcal{H}_{DolbDirac} \simeq \mathcal{H}_{Euler} \right) \,. </annotation></semantics></math></div></div> <p>This is to some extent discussed for instance in (<a href="#Hochs08">Hochs 08, lemma 3.32</a>, <a href="#Paradan09">Paradan 09, prop. 2.2</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega^{n,0}}</annotation></semantics></math> be a choice of <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> (<a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>) of the <a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a>, which according to <a class="maruku-ref" href="#SpinStructuresOnKaehlerManifolds"></a> is a choice of <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> in the <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a>. Then by that same proposition the corresponding <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>≃</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex"> S \simeq \Omega^{0,\bullet} \otimes \sqrt{\Omega^{n,0}} </annotation></semantics></math></div> <p>and so the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">L_\omega</annotation></semantics></math>-twisted spinor bundle is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>S</mi><mo>⊗</mo><msub><mi>L</mi> <mi>ω</mi></msub></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup><mo>⊗</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt><mo>)</mo></mrow><mo>⊗</mo><msub><mi>L</mi> <mi>ω</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup><mo>⊗</mo><mrow><mo>(</mo><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt><mo>⊗</mo><msub><mi>L</mi> <mi>ω</mi></msub><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} S \otimes L_\omega & \simeq \left(\Omega^{0,\bullet} \otimes \sqrt{\Omega^{n,0}}\right) \otimes L_\omega \\ & \simeq \Omega^{0,\bullet} \otimes \left(\sqrt{\Omega^{n,0}} \otimes L_\omega\right) \end{aligned} \,, </annotation></semantics></math></div> <p>where we have re-bracheted just for emphasis of how the <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a> appears as part of the <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>.</p> <p>At the same time, by remark <a class="maruku-ref" href="#SpinCSpinorsAsTensorProductWhenDetDivisible"></a> we have that under this assumption that an actual spin structure exists, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> is isomorphic to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>c</mi></msup><mo>≃</mo><mi>S</mi><mo>⊗</mo><msub><mi>L</mi> <mi>ω</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^c \simeq S \otimes L_\omega \,. </annotation></semantics></math></div> <p>So the two spinor bundles agree. It is now sufficient to observe that under this identification both the <a class="existingWikiWord" href="/nlab/show/Dolbeault-Dirac+operator">Dolbeault-Dirac operator</a> as well as the <a class="existingWikiWord" href="/nlab/show/spin%5Ec+Dirac+operator">spin^c Dirac operator</a> have the same <a class="existingWikiWord" href="/nlab/show/symbol+of+a+differential+operator">symbol</a> to conclude that they have the same <a class="existingWikiWord" href="/nlab/show/index">index</a>.</p> </div> <div class="num_remark"> <h6 id="remark_12">Remark</h6> <p>The assumption in the above that we do have a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> is only for comparison with the Euler-/ Dolbeault-quantization. It is not necessary for the K-theoretic geometric quantization by <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>.</p> <p>In the general case the <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> of the <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> may not admit a square root. (Its failure of having a square root will be compensated precisely by the failure of there being a genuine <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>.) Still, by the above discussion the <a class="existingWikiWord" href="/nlab/show/index">index</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> is like a quantization of a would-be square root of the determinant line bundle in this case.</p> </div> <h4 id="QuantumOperators">Quantum operators / observables</h4> <p>Given a <a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a> on the <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> by a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, we can apply the above <a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">K-theoretic quantization by push-foward</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>, to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K_G(\ast) \simeq K(\ast//G) \simeq R(G) </annotation></semantics></math></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and hence yields not just a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, but a Hilbert space equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a>. This is the action by <a class="existingWikiWord" href="/nlab/show/quantum+operators">quantum operators</a>, quantizing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-actions. Generalized orientation theory gives the necessary condition for this quantization to exist: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> needs to be oriented not just in <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> (<a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>) but in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> (equivariant <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>).</p> <p>So the geometric quantization of observables is essentially what mathematically is known as <a class="existingWikiWord" href="/nlab/show/Dirac+induction">Dirac induction</a>.</p> <h2 id="properties">Properties</h2> <h3 id="DependenceOnChoiceOfPolarization">Functorial dependence on choices</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+connection">Knizhnik-Zamolodchikov connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hitchin+connection">Hitchin connection</a></p> </li> <li> <p>L. Charles, <em>Semi-classical properties of geometric quantization with metaplectic correction</em> (<a href="http://arxiv.org/abs/math/0602168">arXiv:math/0602168</a>)</p> </li> </ul> <p>Chapter 3 of</p> <ul id="Lauridsen10"> <li>Lauridsen, <em>Aspects of quantum mathematics – Hitchin connections and the AJ conjecture</em>, PhD thesis Aarhus 2010 (<a href="http://pure.au.dk/portal/files/41741925/imf_phd_2010_mrl.pdf">pdf</a>)</li> </ul> <h3 id="compatibility_of_quantization_with_symplectic_reduction">Compatibility of quantization with symplectic reduction</h3> <p>On the relation between geometric quantization and <a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Guillemin-Sternberg+geometric+quantization+conjecture">Guillemin-Sternberg geometric quantization conjecture</a></li> </ul> <h3 id="characteristic_central_extensions">Characteristic central extensions</h3> <p>To a large extent geometric quantization is realized by <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> of various Lie groups arising in <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a>/<a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>.</p> <div> <p><strong>higher and integrated <a class="existingWikiWord" href="/nlab/show/Kostant-Souriau+extensions">Kostant-Souriau extensions</a></strong>:</p> <p>(<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla) </annotation></semantics></math></div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></th><th>geometry</th><th>structure</th><th>unextended structure</th><th>extension by</th><th>quantum extension</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+%E2%88%9E-group">Hamiltonian symplectomorphism ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega \mathbb{G})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connections">flat ∞-connections</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonians">Hamiltonians</a> under <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+group">Hamiltonian symplectomorphism group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+2-algebra">Poisson Lie 2-algebra</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphism">Hamiltonian 2-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism 2-group</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphisms">Hamiltonian n-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></td></tr> </tbody></table> <p>(extension are listed for sufficiently connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>)</p> </div> <h2 id="examples">Examples</h2> <h3 id="ExamplesSchroedingerRepresentation">Schrödinger representation</h3> <p>We discuss how the standard <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+representation">Schrödinger representation</a> of the <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relation">canonical commutation relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">}</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\{q,p\} = -1</annotation></semantics></math> arises via geometric quantization:</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> with canonical <a class="existingWikiWord" href="/nlab/show/coordinate+functions">coordinate functions</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{q,p\}</annotation></semantics></math> and to be called the <em><a class="existingWikiWord" href="/nlab/show/canonical+coordinate">canonical coordinate</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> and its <a class="existingWikiWord" href="/nlab/show/canonical+momentum">canonical momentum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and equipped with the <a class="existingWikiWord" href="/nlab/show/left+invariant+differential+form">constant</a> <a class="existingWikiWord" href="/nlab/show/differential+2-form">differential 2-form</a> given in in <a class="maruku-eqref" href="#eq:CanonicalMomentumPresymplecticCurrent">(?)</a> by</p> <div class="maruku-equation" id="eq:R2SymplecticForm"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mi>d</mi><mi>p</mi><mo>∧</mo><mi>d</mi><mi>q</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega = d p \wedge d q \,. </annotation></semantics></math></div> <p>This is <a class="existingWikiWord" href="/nlab/show/closed+differential+form">closed</a> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \omega = 0</annotation></semantics></math>, and invertible in that the contraction of <a class="existingWikiWord" href="/nlab/show/tangent+vector+fields">tangent vector fields</a> into it (def. <a class="maruku-ref" href="#ContractionOfFormsWithVectorFields"></a>) is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> to <a class="existingWikiWord" href="/nlab/show/differential+1-forms">differential 1-forms</a>, and as such it is a <em><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a></em>.</p> <p>A choice of <a class="existingWikiWord" href="/nlab/show/presymplectic+potential">presymplectic potential</a> for this <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> is</p> <div class="maruku-equation" id="eq:CanonicalSymplecticPotentialOnR2"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mi>q</mi><mspace width="thinmathspace"></mspace><mi>d</mi><mi>p</mi></mrow><annotation encoding="application/x-tex"> \theta \;\coloneqq\; - q \, d p </annotation></semantics></math></div> <p>in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>θ</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \theta = \omega</annotation></semantics></math>. (Other choices are possible, notably <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>=</mo><mi>p</mi><mspace width="thinmathspace"></mspace><mi>d</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">\theta = p \, d q</annotation></semantics></math>).</p> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> A \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{C} </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> (an <a class="existingWikiWord" href="/nlab/show/observable">observable</a>), we say that a <em><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></em> for it (as in def. <a class="maruku-ref" href="#HamiltonianForms"></a>) is a <a class="existingWikiWord" href="/nlab/show/tangent+vector+field">tangent vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">v_A</annotation></semantics></math> whose contraction into the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> <a class="maruku-eqref" href="#eq:R2SymplecticForm">(2)</a> is the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:HamiltonianVectorFieldOnR2"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mi>A</mi></msub></mrow></msub><mi>ω</mi><mo>=</mo><mi>d</mi><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota_{v_A} \omega = d A \,. </annotation></semantics></math></div> <p>Consider the <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> of this phase space by constant-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>-slices</p> <div class="maruku-equation" id="eq:ConstantqSlicesOnR2"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mi>q</mi></msub><mo>=</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Lambda_q = \subset \mathbb{R}^2 \,. </annotation></semantics></math></div> <p>These are also called the <em><a class="existingWikiWord" href="/nlab/show/leaves">leaves</a></em> of a <em><a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</a></em> of the <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>.</p> <p>(Other choices of <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a> are possible, notably the constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-slices.)</p> <p>We says that a smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \psi \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{C} </annotation></semantics></math></div> <p>is <em><a class="existingWikiWord" href="/nlab/show/polarization">polarized</a></em> if its <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">i \theta</annotation></semantics></math> along the <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> vanishes; which for the choice of polarization in <a class="maruku-eqref" href="#eq:ConstantqSlicesOnR2">(5)</a> means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><msub><mo>∂</mo> <mi>p</mi></msub></mrow></msub><mi>ψ</mi><mo>=</mo><mn>0</mn><mphantom><mi>AAA</mi></mphantom><mo>⇔</mo><mphantom><mi>AAA</mi></mphantom><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>p</mi></msub></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>ψ</mi><mo>+</mo><mi>i</mi><mi>θ</mi><mi>ψ</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \nabla_{\partial_p} \psi = 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} \iota_{\partial_p} \left( d \psi + i \theta \psi \right) = 0 \,, </annotation></semantics></math></div> <p>which in turn, for the choice of <a class="existingWikiWord" href="/nlab/show/presymplectic+potential">presymplectic potential</a> in <a class="maruku-eqref" href="#eq:CanonicalSymplecticPotentialOnR2">(3)</a>, means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>p</mi></mrow></mfrac><mi>ψ</mi><mo>−</mo><mi>i</mi><mi>q</mi><mi>ψ</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{\partial}{\partial p} \psi - i q \psi = 0 \,. </annotation></semantics></math></div> <p>The solutions to this <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> are of the form</p> <div class="maruku-equation" id="eq:PolarizedFunctionsForBasicExampleOnR2"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>p</mi><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Psi(q,p) = \psi(q) \, \exp(+ i p q) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>ℝ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\psi \colon \mathbb{R} \to \mathbb{C}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, now called a <em><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></em>.</p> <p>This establishes a <a class="existingWikiWord" href="/nlab/show/linear+isomorphism">linear isomorphism</a> between polarized smooth functions and <a class="existingWikiWord" href="/nlab/show/wave+functions">wave functions</a>.</p> <p>By <a class="maruku-eqref" href="#eq:HamiltonianVectorFieldOnR2">(4)</a> we have the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>q</mi></msub><mo>=</mo><msub><mo>∂</mo> <mi>p</mi></msub><mphantom><mi>AAAA</mi></mphantom><msub><mi>v</mi> <mi>p</mi></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo>∂</mo> <mi>q</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> v_q = \partial_p \phantom{AAAA} v_p = -\partial_q \,. </annotation></semantics></math></div> <p>The corresponding <em><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></em> is</p> <div class="maruku-equation" id="eq:R2PoissonBracket"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">{</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">}</mo></mtd> <mtd><mo>≔</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mi>p</mi></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mi>q</mi></msub></mrow></msub><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>q</mi></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>p</mi></msub></mrow></msub><mi>d</mi><mi>p</mi><mo>∧</mo><mi>d</mi><mi>q</mi><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \{q,p\} & \coloneqq \iota_{v_p} \iota_{v_q} \omega \\ & = -\iota_{\partial_q} \iota_{\partial_p} d p \wedge d q = \\ & = - 1 \end{aligned} </annotation></semantics></math></div> <p>The action of the corresponding <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat q</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat p</annotation></semantics></math> on the polarized functions <a class="maruku-eqref" href="#eq:PolarizedFunctionsForBasicExampleOnR2">(6)</a> is as follows</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mover><mi>q</mi><mo stretchy="false">^</mo></mover><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mo>∇</mo> <mrow><msub><mo>∂</mo> <mi>p</mi></msub></mrow></msub><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><mi>q</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><munder><munder><mrow><mo>(</mo><munder><munder><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>p</mi></mrow></mfrac><mrow><mo>(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>q</mi><mi>p</mi></mrow></msup><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>i</mi><mi>q</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></munder><mo>−</mo><mi>i</mi><mi>q</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>+</mo><mi>q</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>q</mi><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mi>q</mi><mi>p</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \hat q \Psi(q,p) & = - i \nabla_{\partial_p}\Psi(q,p) + q \Psi(q,p) \\ & = -i \underset{ = 0 }{ \underbrace{ \left( \underset{ = i q \Psi(q,p) }{ \underbrace{ \frac{\partial}{\partial p} \left( \psi(q) e^{i q p} \right) } } - i q \Psi(q,p) \right) } } + q \Psi(q,p) \\ & = \left( q \psi(q) \right) e^{i q p} \end{aligned} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mover><mi>p</mi><mo stretchy="false">^</mo></mover><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>i</mi><msub><mo>∇</mo> <mrow><msub><mo>∂</mo> <mi>q</mi></msub></mrow></msub><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>q</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>q</mi><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>q</mi></mrow></mfrac><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mi>q</mi><mi>p</mi></mrow></msup><mo>+</mo><munder><munder><mrow><munder><munder><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mrow><mo>(</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>q</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mi>i</mi><mi>q</mi><mi>p</mi></mrow></msup><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></munder><mo>+</mo><mi>p</mi><mi>Ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>q</mi></mrow></mfrac><mi>ψ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mi>p</mi><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \hat p \Psi(q,p) & = i \nabla_{\partial_q} \Psi(q,p) + p \Psi(q,p) \\ & = i \frac{\partial}{\partial q} (\psi(q)e^{i q p}) + p \Psi(q,p) \\ & = \left( i \frac{\partial}{\partial q}\psi(q) \right) e^{i q p} + \underset{ = 0}{ \underbrace{ \underset{ = - p \Psi(q,p) }{ \underbrace{ \psi(q) \left( i \frac{\partial}{\partial q} e^{i q p} \right) } } + p \Psi(q,p) } } \\ & = \left( i \frac{\partial}{\partial q}\psi(q) \right) e^{i p q} \end{aligned} \,. </annotation></semantics></math></div> <p>Hence under the identification <a class="maruku-eqref" href="#eq:PolarizedFunctionsForBasicExampleOnR2">(6)</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>q</mi><mo stretchy="false">^</mo></mover><mi>ψ</mi><mo>=</mo><mi>q</mi><mi>ψ</mi><mphantom><mi>AAAA</mi></mphantom><mover><mi>p</mi><mo stretchy="false">^</mo></mover><mi>ψ</mi><mo>=</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>q</mi></mrow></mfrac><mi>ψ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat q \psi = q \psi \phantom{AAAA} \hat p \psi = i \frac{\partial}{\partial q} \psi \,. </annotation></semantics></math></div> <p>This is called the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+representation">Schrödinger representation</a> of the <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relation">canonical commutation relation</a> <a class="maruku-eqref" href="#eq:R2PoissonBracket">(7)</a>.</p> <h3 id="khler_manifolds">Kähler manifolds</h3> <p>If the <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> happens to be also a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> there are natural choices of prequantization:</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> is a <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic line bundle</a> since its <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>T</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow> <mo>*</mo></msubsup><mi>X</mi></mrow><annotation encoding="application/x-tex">\omega \in T^*_{(1,1)}X</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a> is given by the anti-holomorphic tangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">T^{0,1}X</annotation></semantics></math>;</p> </li> <li> <p>a polarized section is a <a class="existingWikiWord" href="/nlab/show/holomorphic+section">holomorphic section</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of the <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> is given by the <a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a>, see (<a href="#Hitchin">Hitchin</a>).</p> </li> </ul> <p>Applied to a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> this yields the <span class="newWikiWord">Bargmann-Fock representation<a href="/nlab/new/Bargmann-Fock+representation">?</a></span> of the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a>.</p> <h3 id="ExampleThe2Sphere">The 2-sphere</h3> <p>Consider the 2-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> with its canonical round <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> as a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>. This admits a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/diffeomorphism+group">diffeomorphism group</a> which preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(3)</annotation></semantics></math>. The corresponding <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+metric">Kähler metric</a> is a multiple of the standard round <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> on the 2-sphere.</p> <p>By the identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B U(1) \simeq K(\mathbb{Z},2)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/prequantum+bundles">prequantum bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> are classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi><mo>=</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>,</mo><mi>ℒ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H} = H^0(S^2, \mathcal{L}) \,, </annotation></semantics></math></div> <p>the space of <a class="existingWikiWord" href="/nlab/show/holomorphic+sections">holomorphic sections</a> of such a <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic line bundle</a>. The <a class="existingWikiWord" href="/nlab/show/universal+covering+space">universal cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(3)</annotation></semantics></math> naturally acts on this Hilbert space. The canonical <a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">x,y,z \colon S^2 \to \mathbb{R}</annotation></semantics></math> naturally act on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> and as such form a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+representation">Lie algebra representation</a> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a>.</p> <p>A way to reduce the number of choices in this example of geometric quantization is to proceed by <a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a>, see (<a href="#GulovWitten08">Gukov-Witten 08, p. 6 onwards</a>).</p> <p>For more see <em><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a></em>.</p> <h3 id="ExamplesTori">Tori</h3> <ul> <li>G. G. Athanasiu, E. G. Floratos, <a class="existingWikiWord" href="/nlab/show/Stam+Nicolis">Stam Nicolis</a>, <em>Holomorphic Quantization on the Torus and Finite Quantum Mechanics</em> (<a href="http://arxiv.org/abs/hep-th/9509098">arXiv:hep-th/9509098</a>)</li> </ul> <h3 id="theta_functions">Theta functions</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/theta+function">theta function</a></em>.</p> <h3 id="quantization_of_chernsimons_theory">Quantization of Chern-Simons theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> leads to invariants for 3-<a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>.(<a href="#Witten">Witten 89</a>).</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/quantization+of+Chern-Simons+theory">quantization of Chern-Simons theory</a></em> for more.</p> <h3 id="quantization_of_loop_groups__of_the_wzw_model">Quantization of loop groups / of the WZW model</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/quantization+of+loop+groups">quantization of loop groups</a></em>.</p> <h3 id="quantization_in_gromovwitten_theory">Quantization in Gromov-Witten theory</h3> <p>Geometric quantization appears in <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a>. See (<a href="#CladerPriddisShoemaker13">Clader-Priddis-Shoemaker 13</a>).</p> <h3 id="quantization_of_the_bosonic_string_model">Quantization of the bosonic string <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model</h3> <p>For discussion of the geometric quantization of the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> 2d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> see at <em><a href="string#ReferencesSymplecticGeometryAndGeometricQuantization">string – Symplectic geometry and geometric quantization</a></em> .</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+quantization">Bohr-Sommerfeld quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reductions+deformations+resolutions+in+physics">reductions deformations resolutions in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weinstein+symplectic+category">Weinstein symplectic category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><strong>geometric quantization</strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-c+quantization">spin-c quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metaplectic+quantization">metaplectic quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+non-integral+forms">geometric quantization of non-integral forms</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schubert+calculus">Schubert calculus</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></strong></p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/category">category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dual+category">dual category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a></mtext></mover><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a></mtext></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup><mo>,</mo><mi>comm</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">comm. C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncomm. topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCTopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">NCTopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>general <a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>Schemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}Schemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} </annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A} \phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncomm. algebraic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCSchemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">NCSchemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mrow><mi>fin</mi><mo>,</mo><mi>red</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">fin. gen.</a> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi></mrow><annotation encoding="application/x-tex">SmoothManifolds</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mi>comm</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>SuperSpaces</mi> <mi>Cart</mi></msub></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mphantom><mtext>Pursell's theorem</mtext></mphantom></mover></mtd> <mtd><msubsup><mi>Alg</mi> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom></mrow> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \overset{\phantom{\text{Pursell's theorem}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+moduli+problem">formal</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super Lie theory</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mi>Super</mi><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>fin</mi></msub></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mtext><a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a></mtext><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>sdgcAlg</mi> <mi>op</mi></msup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">FDAs</a>”)</td></tr> </tbody></table> <p><strong>in <a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>:</p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/En-algebras">En-algebras</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">BD</a>-<a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism representation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div><div> <p><strong><a class="existingWikiWord" href="/nlab/show/partition+functions">partition functions</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> as <a class="existingWikiWord" href="/nlab/show/index">indices</a>/<a class="existingWikiWord" href="/nlab/show/genera">genera</a>/<a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientations</a> in <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a></strong>:</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional QFT</th><th><a class="existingWikiWord" href="/nlab/show/supercharge">supercharge</a></th><th><a class="existingWikiWord" href="/nlab/show/index">index</a> in <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/genus">genus</a></th><th>logarithmic coefficients of <a class="existingWikiWord" href="/nlab/show/Hirzebruch+series">Hirzebruch series</a></th><th></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;">push-forward in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>: <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KO">KO</a><a class="existingWikiWord" href="/nlab/show/K-theory">-theory</a> <a class="existingWikiWord" href="/nlab/show/index">index</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>Spin</mi><mo>→</mo><mi>KO</mi></mrow><annotation encoding="application/x-tex">M Spin \to KO</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">endpoint of <a class="existingWikiWord" href="/nlab/show/2d+Poisson-Chern-Simons+theory">2d Poisson-Chern-Simons theory</a> string</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a> twisted by <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> of boundary <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>/<a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">M Spin^c \to KU</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">endpoint of <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a> twisted by <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Todd+genus">Todd genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bernoulli+numbers">Bernoulli numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+orientation">Atiyah-Bott-Shapiro orientation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">M Spin^c \to KU</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> in NS-R sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ochanine+elliptic+genus">Ochanine elliptic genus</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO+orientation+of+elliptic+cohomology">SO orientation of elliptic cohomology</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+superstring">heterotic superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac-Ramond+operator">Dirac-Ramond operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane+charge">M5-brane charge</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/w4-orientation+of+EO%282%29-theory">w4-orientation of EO(2)-theory</a></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <h3 id="ReferencesGeneral">General</h3> <p>Original references:</p> <ul> <li id="Souriau69"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em><a href="http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm">Structure des systemes dynamiques</a></em>, Dunod, Paris (1970)</p> <p>Translated and reprinted as (see section V.18 for geometric quantization):</p> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Structure of dynamical systems - A symplectic view of physics</em>, Brikhäuser (1997) (<a href="https://link.springer.com/book/10.1007/978-1-4612-0281-3">doi:10.1007/978-1-4612-0281-3</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertram+Kostant">Bertram Kostant</a>, <em>Quantization and unitary representations</em>, in <em>Lectures in modern analysis and applications III</em>, Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208</p> </li> <li id="Souriau74"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Modèle de particule à spin dans le champ électromagnétique et gravitationnel</em>, Annales de l’I.H.P. Physique théorique, 20 no. 4 (1974), p. 315-364 (<a href="http://www.numdam.org/item?id=AIHPA_1974__20_4_315_0">numdam</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertram+Kostant">Bertram Kostant</a>, <em>On the definition of quantization</em>, in <em>Géométrie Symplectique et Physique Mathématique</em>, Colloques Intern. CNRS, vol. 237,</p> <p>Paris (1975) 187—210</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Guillemin">Victor Guillemin</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>, <em>Geometric Asymptotics</em>, Math. Surveys no. 14, Amer. Math. Soc. (1977) (<a href="http://www.ams.org/online%5Fbks/surv14/">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+Kirillov">Alexandre Kirillov</a>, <em>Geometric quantization</em> Dynamical systems – 4, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, VINITI, Moscow, 1985, 141–176 (<a href="http://mi.mathnet.ru/eng/intf35">web</a>)</p> </li> </ul> <p>A fairly comprehensive textbook with modern developments is</p> <ul id="Woodhouse"> <li><a class="existingWikiWord" href="/nlab/show/Nicholas+Woodhouse">Nicholas Woodhouse</a>, <em>Geometric Quantization</em>, Oxford University Press (1997)</li> </ul> <p>Introductions and lecture notes:</p> <ul> <li> <p>William Gordon Ritter, <em>Geometric Quantization</em> (<a href="http://arxiv.org/abs/math-ph/0208008">arXiv:math-ph/0208008</a>)</p> </li> <li id="Blau"> <p><a class="existingWikiWord" href="/nlab/show/Matthias+Blau">Matthias Blau</a>, <em>Symplectic geometry and geometric quantization</em> (<a class="existingWikiWord" href="/nlab/files/BlauGeometricQuantization.pdf" title="pdf">pdf</a>)</p> </li> <li id="Lerman"> <p><a class="existingWikiWord" href="/nlab/show/Eugene+Lerman">Eugene Lerman</a>, <em>Geometric quantization; a crash course</em> (<a href="http://arxiv.org/abs/1206.2334">arXiv:1206.2334</a>)</p> </li> <li> <p>Andrea Carosso: <em>Geometric Quantization</em> [<a href="https://arxiv.org/abs/1801.02307">arXiv:1801.02307</a>]</p> </li> </ul> <p>Lecture notes with an emphasis on <a class="existingWikiWord" href="/nlab/show/semiclassical+states">semiclassical states</a>:</p> <ul> <li id="BatesWeinstein">Sean Bates, <a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em><a class="existingWikiWord" href="/nlab/show/Lectures+on+the+geometry+of+quantization">Lectures on the geometry of quantization</a></em>, AMS (1997) [<a href="http://www.math.berkeley.edu/~alanw/GofQ.pdf">pdf</a>]</li> </ul> <p>A careful discussion of the polarization-step from prequantization to quantization is in</p> <ul> <li id="Śniatycki75"> <p><a class="existingWikiWord" href="/nlab/show/J%C4%99drzej+%C5%9Aniatycki">Jędrzej Śniatycki</a>, <em>Wave functions relative to a real polarization</em>, Internat. J. Theoret. Phys., <strong>14</strong> 4 (1975) 277-288 [<a href="https://doi.org/10.1007/BF01807689">doi:10.1007/BF01807689)</a>]</p> </li> <li id="Śniatycki80"> <p><a class="existingWikiWord" href="/nlab/show/J%C4%99drzej+%C5%9Aniatycki">Jędrzej Śniatycki</a>, <em>Geometric Quantization and Quantum Mechanics</em>, Applied Mathematical Sciences <strong>30</strong>, Springer-Verlag (1980) [<a href="https://doi.org/10.1007/978-1-4612-6066-0">doi:10.1007/978-1-4612-6066-0</a>]</p> </li> <li id="Śniatycki16"> <p><a class="existingWikiWord" href="/nlab/show/J%C4%99drzej+%C5%9Aniatycki">Jędrzej Śniatycki</a>, <em>Lectures on Geometric Quantization</em>, in <em>Geom. Integrability & Quantization</em> (2016) 95-129 [<a href="https://projecteuclid.org/ebooks/geometry-integrability-and-quantization-proceedings-series/Lectures-on-Geometric-Quantization/chapter/Lectures-on-Geometric-Quantization/10.7546/giq-17-2016-95-129">doi:10.7546/giq-17-2016-95-129</a>]</p> </li> </ul> <p>The special case of Kähler quantization is discussed for instance in</p> <ul> <li id="Hitchin"><a class="existingWikiWord" href="/nlab/show/Nigel+Hitchin">Nigel Hitchin</a>, <em>Flat connections and geometric quantization</em>, Comm. Math. Phys. <strong>131</strong> (1990), no. 2, 347–380.</li> </ul> <p>and for almost Kähler structure in</p> <ul> <li id="BorthwickUribe96">David Borthwick, Alejandro Uribe, <em>Almost complex structures and geometric quantization</em> (<a href="https://arxiv.org/abs/dg-ga/9608006">arXiv:dg-ga/9608006</a>)</li> </ul> <p>Discussion with an emphasis of quantizing <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> on curved <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicholas+Woodhouse">Nicholas Woodhouse</a>, <em>Geometric quantization and quantum field theory in curved space-times</em>, Reports on Mathematical Physics <strong>12</strong>:1, (1977) 45–56</li> </ul> <p>(For more on geometric quantization of quantum field theories see also at <em><a href="multisymplectic+geometry#RefsonQuantization">Quantization of multisymplectic geometry</a></em>.)</p> <p>Aspects at least of geometric prequantization are usefully discussed also in section II of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, <em>Loop spaces, characteristic classes and geometric quantization</em>, Birkhäuser (1993) (<a href="https://www.springer.com/gp/book/9780817647308">doi:10.1007/978-0-8176-4731-5</a>)</li> </ul> <p>Further reviews include</p> <ul> <li> <p>A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, <em>Mathematical Foundations of Geometric Quantization</em> Extracta Math. 13 (1998) 135-238 (<a href="http://arxiv.org/abs/math-ph/9904008">arXiv:math-ph/9904008</a>)</p> </li> <li> <p>Nima Moshayedi, <em>Notes on Geometric Quantization</em> (<a href="https://arxiv.org/abs/2010.15419">arXiv:2010.15419</a>)</p> </li> </ul> <p>The above “Overview” and “Basic Jargon” sections are taken from</p> <ul> <li id="Baez"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Geometric Quantization</em> (<a href="http://math.ucr.edu/home/baez/quantization.html">web</a>)</li> </ul> <p>Some useful talk notes include</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eva+Miranda">Eva Miranda</a>: <em>From action-angle coordinates to geometric quantization and back</em> (2011) [<a class="existingWikiWord" href="/nlab/files/Miranda-ActionAngleCoordinates.pdf" title="pdf">pdf</a>]</li> </ul> <p>Discussion with an eye towards the <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gabriel+Catren">Gabriel Catren</a>, <em>Towards a Group-Theoretical Interpretation of Mechanics</em> (<a href="http://philsci-archive.pitt.edu/10116/">PhilSci Archive</a>)</li> </ul> <p>On geometric quantization of the <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>:</p> <ul> <li>José Luis Alonso, Carlos Bouthelier-Madre, Jesús Clemente-Gallardo, David Martínez-Crespo, <em>Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution</em> [<a href="https://arxiv.org/abs/2402.07953">arXiv:2402.07953</a>]</li> </ul> <p>Discussion of relation to other qusntization schemes:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joshua+Lackman">Joshua Lackman</a>, <em>Geometric Quantization Without Polarizations</em> [<a href="https://arxiv.org/abs/2405.01513">arXiv:2405.01513</a>]</li> </ul> <h3 id="holographic_quantization">Holographic quantization</h3> <p>The geometric <a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a> is discussed in</p> <ul id="GukovWitten08"> <li><a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Branes and Quantization</em>, Adv. Theor. Math. Phys. 13 (2009) 1–73, (<a href="http://arxiv.org/abs/0809.0305">arXiv:0809.0305</a>, <a href="http://projecteuclid.org/euclid.atmp/1282054099">euclid</a>)</li> </ul> <p>See also the <a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a> below, around (<a href="#Hawkins">Hawkins</a>).</p> <h3 id="quantization"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>ℂ</mi></msup></mrow><annotation encoding="application/x-tex">Spin^{\mathbb{C}}</annotation></semantics></math>-Quantization</h3> <ul id="Hochs08"> <li><a class="existingWikiWord" href="/nlab/show/Peter+Hochs">Peter Hochs</a>, <em>Quantisation commutes with reduction for cocompact Hamiltonian group actions</em> 2008 (<a href="http://www.math.ru.nl/~landsman/ThesisPeterHochs.pdf">pdf</a>)</li> </ul> <ul id="Paradan09"> <li><a class="existingWikiWord" href="/nlab/show/Paul-Emile+Paradan">Paul-Emile Paradan</a>, <em>Spin-quantization commutes with reduction</em>, J. Symplectic Geom. Volume 10, Number 3 (2012), 389-422. (<a href="http://arxiv.org/abs/0911.1067">arXiv:0911.1067</a>, <a href="http://projecteuclid.org/euclid.jsg/1350392491">Euclid</a>)</li> </ul> <p>See the references at <em><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">geometric quantization by push-forward</a></em>.</p> <h3 id="examples_2">Examples</h3> <p>The basic example of geometric quantization of a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> is discussed in pretty much every text on the matter for instance <a href="#Nohara">Nohara, starting with example 2.6</a>.</p> <p>Discussion of geometric quantization of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is for instance in</p> <ul id="Witten"> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a> (lecture notes by Lisa Jeffrey), <em>New results in Chern-Simons theory</em>, pages 81 onwards in: S. Donaldson, C. Thomas (eds.) <em>Geometry of low dimensional manifolds 2: Symplectic manifolds and Jones-Witten theory</em> (1989) (<a href="http://cs5538.userapi.com/u11728334/docs/06f79688de6f/S_K_Donaldson_Geometry_of_LowDimensional_Ma.pdf">pdf</a>)</li> </ul> <ul> <li>Scott Axelrod, Steve Della Pietra, and <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Geometric quantization of Chern-Simons gauge theory</em>, J. Differential Geom. Volume 33, Number 3 (1991), 787-902. (<a href="http://projecteuclid.org/euclid.jdg/1214446565">EUCLID</a>)</li> </ul> <p>Discussion of geometric quantization of <a class="existingWikiWord" href="/nlab/show/abelian+varieties">abelian varieties</a>, <a class="existingWikiWord" href="/nlab/show/toric+varieties">toric varieties</a>, <a class="existingWikiWord" href="/nlab/show/flag+varieties">flag varieties</a> and its relation to <a class="existingWikiWord" href="/nlab/show/theta+functions">theta functions</a> is in</p> <ul id="Nohara"> <li>Yuichi Nohara, <em>Independence of polarization in geometric quantization</em> (<a href="http://geoquant.mi.ras.ru/nohara.pdf">pdf</a>)</li> </ul> <ul id="Tyurin"> <li><a class="existingWikiWord" href="/nlab/show/Andrei+Tyurin">Andrei Tyurin</a>, <em>Quantization, classical and quantum field theory and theta functions</em> (<a href="http://arxiv.org/abs/math/0210466v1">arXiv:math/0210466v1</a>)</li> </ul> <ul> <li>N.J. Hitchin, <em>Flat connections and geometric quantization</em>, Comm. Math. Phys. <strong>131</strong>, n 2 (1990), 347-380, <a href="http://projecteuclid.org/euclid.cmp/1104200841">euclid</a></li> </ul> <p>An appearance of geometric quantization in <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a> is pointed out in</p> <ul> <li>Andrei Tyurin, <em>Geometric quantization and mirror symmetry</em>, (<a href="http://arxiv.org/abs/math/9902027">math.AG/9902027</a>)</li> </ul> <p>For discussion of the geometric quantization of the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> 2d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> see at <em><a href="string#ReferencesSymplecticGeometryAndGeometricQuantization">string – Symplectic geometry and geometric quantization</a></em>.</p> <p>Discussion of geometric quantization of <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> is in</p> <ul> <li>Samuel Monnier, <em>Geometric quantization and the metric dependence of the self-dual field theory</em> (<a href="http://arxiv.org/abs/1011.5890">arXiv:1011.5890</a>)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a>:</p> <ul id="CladerPriddisShoemaker13"> <li>Emily Clader, Nathan Priddis, Mark Shoemaker, <em>Geometric Quantization with Applications to Gromov-Witten Theory</em> (<a href="http://arxiv.org/abs/1309.1150">arXiv:1309.1150</a>)</li> </ul> <h3 id="ReferencesRelationToDeformation">Relation to deformation quantization</h3> <p>Discussion of the relation of geometric quantization to <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, <em>The Correspondence between Geometric Quantization and Formal Deformation Quantization</em> (<a href="http://arxiv.org/abs/math/9811049">arXiv:math/9811049</a>)</li> </ul> <p>based on</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, <em>Geometric Quantization of Vector Bundles</em> (<a href="http://arxiv.org/abs/math/9808116">arXiv:math/9808116</a>)</li> </ul> <p>See also</p> <ul> <li>Christoph Nölle, <em>Geometric and deformation quantization</em> (<a href="http://arxiv.org/abs/0903.5336">arXiv:0903.5336</a>)</li> </ul> <p>For a basic comparative review of both see also section 1 of</p> <ul id="GukovWitten08"> <li><a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Branes and Quantization</em>, Adv. Theor. Math. Phys. 13 (2009) 1–73, (<a href="http://arxiv.org/abs/0809.0305">arXiv:0809.0305</a>, <a href="http://projecteuclid.org/euclid.atmp/1282054099">euclid</a>)</li> </ul> <p>(which then develops geometric <a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a>).</p> <h3 id="relation_to_path_integral_quantization">Relation to path integral quantization</h3> <p>Relation to <a class="existingWikiWord" href="/nlab/show/path+integral+quantization">path integral quantization</a> is discussed in</p> <ul> <li>Laurent Charles, <em>Feynman path integral and Toeplitz Quantization</em> (<a href="http://ipht.cea.fr/DocsphtV2/articles/t98/093/public/publi.pdf">pdf</a>)</li> </ul> <h3 id="ReferencesBRST">Geometric BRST quantization</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> version of an <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> is given (dually) by a <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>. Quantization over a BRST complex is hence quantization over an infinitesimal action groupoid. (See at <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a>).</p> <p>Geometric quantization over BRST complexes is discussed in the following articles.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Takashi+Kimura">Takashi Kimura</a>, <em>BRST Quantization and Poisson Reduction</em>, PhD Thesis (1990) (<a href="http://adsabs.harvard.edu/abs/1990PhDT.......142K">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <a class="existingWikiWord" href="/nlab/show/Takashi+Kimura">Takashi Kimura</a>, <em>Geometric BRST quantization</em> (<a href="http://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/gq.pdf">pdf</a>)</p> <p><em>Geometric BRST quantization. I. Prequantization</em>, Comm. Math. Phys. Volume 136, Number 2 (1991), 209-229. (<a href="http://projecteuclid.org/euclid.cmp/1104202348">Euclid</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gijs+Tuynman">Gijs Tuynman</a>, <em>Geometric quantization of the BRST charge</em> Comm. Math. Phys. Volume 150, Number 2 (1992), 237-265. (<a href="http://projecteuclid.org/euclid.cmp/1104251864">Euclid</a>, <a href="http://math.univ-lille1.fr/~gmt/PaperFolder/GQofBRST.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ronald+Fulp">Ronald Fulp</a>, <em>BRST Extension of Geometric Quantization</em>, Foundations of Physics Volume 37, Number 1 (2007), (<a href="http://arxiv.org/abs/math/0604270">arXiv:math/0604270</a>)</p> </li> </ul> <h3 id="ReferencesSupergeometric">Supergeometric version</h3> <p>One can consider geometric quantization in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>.</p> <ul> <li> <p>S.-M. Fei, H. -Y. Guo und Y. Yu, <em>Symplectic geometry and geometric quantization on supermanifold with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> numbers</em>, Z. Phys, C - Particles and Fields 45, 339-344 (1989)</p> </li> <li> <p>Gijs M. Tuynman, <em>Super Symplectic Geometry and Prequantization</em> (2003) (<a href="http://arxiv.org/abs/math-ph/0306049">arXiv:math-ph/0306049</a>)</p> </li> </ul> <h3 id="of_presymplectic_manifolds">Of presymplectic manifolds</h3> <p>Discussion of quantization of <a class="existingWikiWord" href="/nlab/show/presymplectic+manifolds">presymplectic manifolds</a> is in</p> <ul> <li> <p>C. Günther, <em>Presymplectic manifolds and the quantization of relativistic particles</em>, Salamanca 1979, Proceedings, Differential Geometrical Methods In Mathematical Physics<em>, 383-400 (1979)</em></p> </li> <li> <p>Izu Vaisman, <em>Geometric quantization on presymplectic manifolds</em>, Monatshefte für Mathematik, vol. 96, no. 4, pp. 293-310, 1983</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ana+Canas+da+Silva">Ana Canas da Silva</a>, <a class="existingWikiWord" href="/nlab/show/Yael+Karshon">Yael Karshon</a>, Susan Tolman, <em>Quantization of Presymplectic Manifolds and Circle Actions</em> (<a href="http://arxiv.org/abs/dg-ga/9705008">arXiv:dg-ga/9705008</a>)</p> </li> </ul> <h3 id="of_generalized_complex_manifolds">Of generalized complex manifolds</h3> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a> is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Zambon">Marco Zambon</a>, <em>Variations on Prequantization</em> (<a href="http://arxiv.org/abs/math/0412502">arXiv:math/0412502</a>)</p> </li> <li> <p>Alexander Cardona, <em>Geometric quantization of generalized complex manifolds</em> (<a href="http://pentagono.uniandes.edu.co/~acardona/GQGCM.pdf">pdf</a>)</p> </li> </ul> <h3 id="in_higher_differential_geometry">In higher differential geometry</h3> <p>Geometric quantization in or with tools of <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>, notably with/over <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> is discussed in the following references.</p> <ul> <li id="Bos"><a class="existingWikiWord" href="/nlab/show/Rogier+Bos">Rogier Bos</a>, <em>Groupoids in geometric quantization</em> PhD Thesis (2007) <a href="http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf">pdf</a></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a> is accomplished in</p> <ul> <li id="Hawkins"> <p><a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, <em>A groupoid approach to quantization</em>, J. Symplectic Geom.. 6:61–125, <a href="http://arxiv.org/abs/math.SG/0612363">arXiv:math.SG/0612363</a></p> </li> <li id="Nuiten13"> <p><a class="existingWikiWord" href="/nlab/show/Joost+Nuiten">Joost Nuiten</a>, section 5.2.2 of <em><a class="existingWikiWord" href="/schreiber/show/master+thesis+Nuiten">Cohomological quantization of local boundary prequantum field theory</a></em>, MSc thesis, August 2013</p> </li> </ul> <p>Discussion of geometric prequantization in fully fledged <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> is in</p> <ul> <li id="FiorenzaRogersSchreiber13a"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Higher+geometric+prequantum+theory">Higher U(1)-gerbe connections in geometric prequantization</a></em>, Reviews in Mathematical Physics, Volume 28, Issue 06, July 2016 (<a href="http://arxiv.org/abs/1304.0236">arXiv:1304.0236</a>)</p> </li> <li id="FiorenzaRogersSchreiber13b"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> <em><a class="existingWikiWord" href="/schreiber/show/L-%E2%88%9E+algebras+of+local+observables+from+higher+prequantum+bundles">L-∞ algebras of local observables from higher prequantum bundles</a></em>, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 142 (<a href="http://arxiv.org/abs/1304.6292">arXiv:1304.6292</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 2, 2025 at 15:10:32. 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