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value="Search" 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="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> <li><a href='#refinements'>Refinements</a></li> <ul> <li><a href='#PrequantumCorrespondences'>Prequantum correspondences</a></li> <li><a href='#motivic_stabilization'>Motivic stabilization</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>When <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> is used to model <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, then a <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> encodes the <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> of a <a class="existingWikiWord" href="/nlab/show/mechanical+system">mechanical system</a> and a <a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2) </annotation></semantics></math></div> <p>encodes a process undergone by this system, for instance the time evolution induced by a <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a>.</p> <p>However, this is too restrictive for a notion of a morphism. Indeed, even at the level of <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector spaces</a>, a symplectic morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>ω</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi: (V,\omega)\to (W,\omega')</annotation></semantics></math> is required to satisfy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>ω</mi><mo>′</mo><mo>=</mo><mi>ω</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi^*\omega'=\omega. </annotation></semantics></math></div> <p>This implies that for any element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mtext>ker</mtext><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v\in\text{ker}(\phi)</annotation></semantics></math>, it holds that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo>′</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\omega(v,w) = \omega'(f(v),f(w)) = 0</annotation></semantics></math>, and since <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 non-degenerate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> must be injective.</p> <p>Now the <a class="existingWikiWord" href="/nlab/show/graph">graph</a> of a <a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \times X_2</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> with symplectic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mi>ω</mi> <mn>1</mn></msub><mo>−</mo><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup><msub><mi>ω</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_1^\ast \omega_1 - p_2^\ast \omega_2</annotation></semantics></math>. In other words, a symplectomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> as above constitutes a <a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondence">Lagrangian correspondence</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,\omega_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_2, \omega_2)</annotation></semantics></math>. See for instance (<a href="#CattaneoMnevReshetikhin12">Cattaneo-Mnev-Reshetikhin 12</a>) for a review.</p> <p>This suggests that instead of the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a> and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/symplectomorphisms">symplectomorphisms</a>, one might consider a kind of <a class="existingWikiWord" href="/nlab/show/category+of+correspondences">category of correspondences</a> whose objects are symplectic manifolds, and whose morphisms include <a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondences">Lagrangian correspondences</a>, so that <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is given by forming the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> along adjacent legs of <a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a> called this would-be category the <em>symplectic category</em> and suggested that it is the natural <a class="existingWikiWord" href="/nlab/show/domain">domain</a> for <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>.</p> <p>However, take at face value, symplectic manifolds with <a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondences">Lagrangian correspondences</a> between them do not quite form a <a class="existingWikiWord" href="/nlab/show/category">category</a>, since the usual <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is only well-defined when the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>L</mi> <mn>2</mn></msub><mo>∩</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><mi>Δ</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>X</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">L_1 \times L_2 \cap X_1 \times \Delta(X_2) \times X_3</annotation></semantics></math> is <a href="http://ncatlab.org/nlab/show/transversal+maps">transverse</a>.</p> <p>Proposals for how to rectify this are in <a href="#WehrheimWoodward">Wehrheim & Woodward</a> and in <a href="#Kitchloo">Kitchloo</a> (by turning this into an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a>).</p> <h2 id="refinements">Refinements</h2> <h3 id="PrequantumCorrespondences">Prequantum correspondences</h3> <p>A refinement of the symplectic category to <a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is the following (see <a href="#SyntheticQFT">S 13</a>).</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">\mathbf{B}U(1)_{conn}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of smooth <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>. Write <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> for the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><msub><mi>Grpd</mi> <mrow><mo stretchy="false">/</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></msub></mrow><annotation encoding="application/x-tex">Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a>. Finally write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Corr_1(Smooth\infty Grpd,\mathbf{B}U(1)_{conn}) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+correspondences">(∞,1)-category of correspondences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><msub><mi>Grpd</mi> <mrow><mo stretchy="false">/</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></msub></mrow><annotation encoding="application/x-tex">Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}</annotation></semantics></math>.</p> <p>An object in here is 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><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\nabla)</annotation></semantics></math> given by a map</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><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∇</mo></mpadded></msup></mtd></mtr> <mtr><mtd><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></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \\ \downarrow^{\mathrlap{\nabla}} \\ \mathbf{B}U(1)_{conn} } \,. </annotation></semantics></math></div> <p>Under the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo lspace="verythinmathspace">:</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><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl}</annotation></semantics></math> this maps to a <a class="existingWikiWord" href="/nlab/show/presymplectic+structure">presymplectic structure</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>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X,\omega) = (X, F_{\nabla}) \,. </annotation></semantics></math></div> <p>If here <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 non-degenerate, this is a symplectic structure as in Weinstein’s symplectic category.</p> <p>Moreover, a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mo>∇</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,\nabla_1) \to (X_2,\nabla_2)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</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></mtd> <mtd></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><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></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && Z \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ X_1 && \swArrow_{\eta} && X_2 \\ & {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}} \\ && \mathbf{B}U(1)_{conn} } \,, </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a> space (smooth <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>-groupoid) <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">Z</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> together with an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msubsup><mi>i</mi> <mn>2</mn> <mo>*</mo></msubsup><msub><mo>∇</mo> <mn>2</mn></msub><mover><mo>→</mo><mo>≃</mo></mover><msubsup><mi>i</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mo>∇</mo> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta \colon i_2^\ast \nabla_2 \stackrel{\simeq}{\to} i_1^\ast \nabla_1 \,. </annotation></semantics></math></div> <p>On the underlying <a class="existingWikiWord" href="/nlab/show/curvatures">curvatures</a> this implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>i</mi> <mn>2</mn> <mo>*</mo></msubsup><msub><mi>ω</mi> <mn>2</mn></msub><mo>=</mo><msubsup><mi>i</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mi>ω</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> i_2^\ast \omega_2 = i_1^\ast \omega_1 \,. </annotation></semantics></math></div> <p>Hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z \to X \times Y</annotation></semantics></math> is a maximal inclusion with this property, the above diagram is a prequantization of a morphism in the Weinstein symplectic category.</p> <h3 id="motivic_stabilization">Motivic stabilization</h3> <p><a class="existingWikiWord" href="/nlab/show/Nitu+Kitchloo">Nitu Kitchloo</a> defines the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable</a> symplectic category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\mathbb{S}</annotation></semantics></math>, which has as <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a>, and <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are certain <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a></p> <p>associated to <a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondences">Lagrangian correspondences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>M</mi><mo>¯</mo></mover><mo>×</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\overline{M} \times N</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>M</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{M}</annotation></semantics></math> denotes the conjugate with symplectic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">-\omega</annotation></semantics></math>. One can view this as a category of symplectic <a class="existingWikiWord" href="/nlab/show/motives">motives</a>.</p> <p>Considering an oriented version of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\mathbb{S}</annotation></semantics></math>, there is a canonical <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>M</mi><mo>↦</mo><mi>𝕊</mi><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F : M \mapsto \mathbb{S}(pt, M)</annotation></semantics></math>, and one may consider the <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois 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> of monoidal <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> 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>. (<a href="#Kitchloo">Kitchloo 12, question 8.6, p. 19</a>).</p> <p>It turns out to have a natural subgroup which is isomorphic to the quotient of the <a class="existingWikiWord" href="/nlab/show/Grothendieck-Teichm%C3%BCller+group">Grothendieck-Teichmüller group</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondences+and+category-valued+TFT">Lagrangian correspondences and category-valued TFT</a></p> </li> </ul> <h2 id="References">References</h2> <p>The way that <a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondences">Lagrangian correspondences</a> encode <a class="existingWikiWord" href="/nlab/show/symplectomorphisms">symplectomorphisms</a> in <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> and hence evolution in <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> is reviewed (and put in the broader context of <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>) in</p> <ul id="CattaneoMnevReshetikhin12"> <li><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Mnev">Pavel Mnev</a>, <a class="existingWikiWord" href="/nlab/show/Nicolai+Reshetikhin">Nicolai Reshetikhin</a>, <em>Classical and quantum Lagrangian field theories with boundary</em> (<a href="http://arxiv.org/abs/1207.0239">arXiv:1207.0239</a>)</li> </ul> <p>In his work on Fourier integral operators,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, <em>Fourier Integral Operators I.</em>, Acta Math. 127 (1971) 79–183. <p>14</p> </li> </ul> <p>following</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Victor+Maslov">Victor Maslov</a>, <em>Theory of Perturbations and Asymptotic Methods</em>, (in Russian) Moskov. Gos. Univ., Moscow (1965).</li> </ul> <p>observed that, under a transversality assumption, the set-theoretic composition of two <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifolds">Lagrangian submanifolds</a> is again a Lagrangian submanifold, and that this composition is a “<a class="existingWikiWord" href="/nlab/show/classical+limit">classical limit</a>” of the composition of certain <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a>.</p> <p>Shortly thereafter,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J%C4%99drzej+%C5%9Aniatycki">Jędrzej Śniatycki</a>, W.M. Tulczyjew, <em>Generating forms of Lagrangian submanifolds</em>, Indiana Univ. Math. J. <strong>22</strong> (1972) 267-275</li> </ul> <p>defined symplectic relations as <a class="existingWikiWord" href="/nlab/show/isotropic+submanifolds">isotropic submanifolds</a> of products and showed that this class of relations was closed under “clean” composition. Following in part some (unpublished) ideas of <a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Victor+Guillemin">Victor Guillemin</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>, <em>Some problems in integral geometry and some related problems in microlocal analysis</em>, Amer. J. Math. 101 (1979), 915–955 (<a href="http://www.jstor.org/stable/2373923">JSTOR</a>)</li> </ul> <p>observed that the linear canonical relations (i.e., lagrangian subspaces of products of <a class="existingWikiWord" href="/nlab/show/symplectic+vector+spaces">symplectic vector spaces</a>) could be considered as the morphisms of a category, and they constructed a partial <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of this category (in which lagrangian subspaces are enhanced by halfdensities.) The quantization of the linear symplectic category was part of a larger project of quantizing canonical relations (enhanced with extra structure, such as half-densities) in a functorial way, and this program was set out more formally</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>Symplectic Manifolds and Their Lagrangian Submanifolds</em>, Advances in Math. <strong>6</strong> (1971) 329346 [<a href="https://doi.org/10.1016/0001-8708(71)90020-X">doi:10.1016/0001-8708(71)90020-X</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>Symplectic geometry</em>, Bulletin Amer. Math. Soc. <strong>5</strong> (1981) 1-13 [<a href="http://dx.doi.org/10.1090/S0273-0979-1981-14911-9">doi:10.1090/S0273-0979-1981-14911-9</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>W.M.Tulczyjew, S.Zakrzewski, <em>The category of Fresnel kernels</em>, J. Geom. Phys. 1:3, 1984, 79–120 <a href="https://doi.org/10.1016/0393-0440(84)90021-4">doi</a></li> </ul> <p>Lecture notes reviewing these developments include</p> <ul> <li id="Weinstein09"><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>Symplectic Categories</em>, proceedings of Geometry Summer School, Lisbon, July 2009 (<a href="http://arxiv.org/abs/0911.4133">arXiv:0911.4133</a>)</li> </ul> <p>from the introduction of which parts of the commented list of references above is taken. Further review includes</p> <ul> <li id="Canez11">Santiago Canez, <em>Double Groupoids, Orbifolds, and the Symplectic Category</em> (<a href="http://arxiv.org/abs/1105.2592">arXiv:1105.2592</a>)</li> </ul> <p>Further refinements in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>:</p> <ul> <li id="WehrheimWoodward"> <p><a class="existingWikiWord" href="/nlab/show/Katrin+Wehrheim">Katrin Wehrheim</a>, <a class="existingWikiWord" href="/nlab/show/Chris+T.+Woodward">Chris T. Woodward</a>, <em>Functoriality for Lagrangian correspondences in Floer theory</em>, Quantum Topology <strong>1</strong> 2 (2010) 129-170 [<a href="https://doi.org/10.4171/qt/4">doi:10.4171/qt/4</a>, <a href="https://arxiv.org/abs/0708.2851">arXiv:0708.2851</a>]</p> </li> <li id="Kitchloo"> <p><a class="existingWikiWord" href="/nlab/show/Nitu+Kitchloo">Nitu Kitchloo</a>, <em>The Stable Symplectic Category and Geometric Quantization</em> [<a href="http://arxiv.org/abs/1204.5720">arXiv:1204.5720</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nitu+Kitchloo">Nitu Kitchloo</a>, <a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, <em>The Grothendieck–Teichmüller group and the stable symplectic category</em>, 2012 [<a href="http://arxiv.org/abs/1212.6905">arxiv:1212.6905</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Damien+Calaque">Damien Calaque</a>, <em>Three lectures on derived symplectic geometry and topological field theories</em>, Indagationes Mathematicae <strong>25</strong> 5 (2014) 926–947 [<a href="https://doi.org/10.1016/j.indag.2014.07.005">doi:10.1016/j.indag.2014.07.005</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rune+Haugseng">Rune Haugseng</a>, <em>Iterated spans and classical topological field theories</em>, Mathematische Zeitschrift <strong>289</strong> 3 (2018) 1427–1488 [<a href="https://arxiv.org/abs/1409.0837">arXiv</a>, <a href="https://doi.org/10.1007/s00209-017-2005-x">doi:10.1007/s00209-017-2005-x</a>]</p> </li> </ul> <p>A closed <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> version of the symplectic category and the observation that this hence is a <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> for <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> qua <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> is in</p> <ul> <li id="Slavnov05"> <p><a class="existingWikiWord" href="/nlab/show/Sergey+Slavnov">Sergey Slavnov</a>, <em>From proof-nets to bordisms: the geometric meaning of multiplicative connectives</em>, Mathematical Structures in Computer Science <strong>15</strong>:06 (2005) 1151–1178</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sergey+Slavnov">Sergey Slavnov</a>, <em>Geometrical semantics for linear logic (multiplicative fragment)</em>, Theoretical Computer Science 357, no. 1–3 (2006) 215–229 <a href="https://doi.org/10.1016/j.tcs.2006.03.020">doi</a></p> </li> </ul> <p>Remarks about refinements to correspondences of smooth <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>-groupoids in the slice over prequantum moduli is in</p> <ul> <li id="SyntheticQFT"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Synthetic+Quantum+Field+Theory">Synthetic Quantum Field Theory</a></em>, Talk at <a href="http://cms.math.ca/Events/summer13/">CMS Summer Meeting 2013</a></li> </ul> <p>String diagrams for the linear and affine Weinstein category using graphical linear algebra</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cole+Comfort">Cole Comfort</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, <em>A Graphical Calculus for Lagrangian Relations</em>, In Proceedings ACT 2021. <a href="https://doi.org/10.4204/EPTCS.372.24">doi</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 2, 2024 at 22:33:21. 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