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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#traditional_construction'>Traditional construction</a></li> <li><a href='#InHigherGeometry'>In higher geometry</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#smooth_structure'>Smooth structure</a></li> <li><a href='#group_extension'>Group extension</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#CoveringAnAffineSymplecticGroup'>Covering an affine symplectic group</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#SmoothManifoldStructure'>Smooth manifold structure</a></li> <li><a href='#examples_2'>Examples</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Given a (<a class="existingWikiWord" href="/nlab/show/presymplectic+form">pre</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>, its <em>quantomorphism group</em> is the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> that <a class="existingWikiWord" href="/nlab/show/Lie+integration">integrates</a> the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie bracket</a> inside the <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a> 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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math>. This is a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> of the group of <a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphisms">Hamiltonian symplectomorphisms</a>. It extends and generalizes the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> of a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a>.</p> <p>(Warning on terminology: A more evident name for the quantomorphism group might seem to be “Poisson group”. But this already means something different, see <em><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+group">Poisson Lie group</a></em>.)</p> <h3 id="traditional_construction">Traditional construction</h3> <p>Over 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> an explicit construction of the corresponding quantomorphism group is obtained by choosing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>→</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P \to X, \nabla)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, regarded with an <a class="existingWikiWord" href="/nlab/show/Ehresmann+connection">Ehresmann connection</a> 1-form <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> on <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 then defining</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QuantomorphismGroup</mi><mo>↪</mo><mi>Diff</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> QuantomorphismGroup \hookrightarrow Diff(P) </annotation></semantics></math></div> <p>to be the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/diffeomorphism+group">diffeomorphism group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>P</mi></mrow><annotation encoding="application/x-tex">P \stackrel{\simeq}{\to} P</annotation></semantics></math> on those <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a> that preserve <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>. In other words, the quantomorphism group is the group of equivalences of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">bundles with connection</a> that need not cover the identity <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> on the base 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>.</p> <p>Notice that the tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P,A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/regular+contact+manifold">regular contact manifold</a> (see the discussion there), and so the quantomorphism group is equivalently that of <a class="existingWikiWord" href="/nlab/show/contactomorphisms">contactomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P,A) \to (P,A)</annotation></semantics></math> of weight 0.</p> <p>This is an infinite-dimensional <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. References discussing its <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifold">infinite-dimensional manifold</a>-structure are collected <a href="#SmoothManifoldStructure">below</a>. But the group has immediately the structure of a group in <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (making it a <a class="existingWikiWord" href="/nlab/show/smooth+group">smooth group</a>) (<a href="#Souriau79">Souriau 79</a>).</p> <h3 id="InHigherGeometry">In higher geometry</h3> <p>This perspective lends itself to a more abstract description in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>: we may regard the <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a> as being modulated by a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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></div> <p>in the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, with <a class="existingWikiWord" href="/nlab/show/domain">domain</a> the given symplectic manifold and <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> the smooth <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> for <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundles with connection</a>. This in turn may be regarded as an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <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">\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a>. Then the quantomorphism group is the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><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></munder><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{QuantMorph}(X,\nabla) \coloneqq \underset{\mathbf{B}U(1)_{conn}}{\prod} \mathbf{Aut}(\nabla) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, or rather its <a class="existingWikiWord" href="/nlab/show/differential+concretification">differential concretification</a> (<a href="#FRS13">FRS 13</a>).</p> <p>From this it is clear what the quantomorphism <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> of an <a class="existingWikiWord" href="/nlab/show/n-plectic+%E2%88%9E-groupoid">n-plectic ∞-groupoid</a> should be: for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>:</mo><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><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}^n U(1)_{conn} </annotation></semantics></math></div> <p>the morphism modulating a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a>, the corresponding quantomorphism <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>-group is again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(\nabla)</annotation></semantics></math>, now formed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><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">\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}</annotation></semantics></math></p> <h2 id="properties">Properties</h2> <h3 id="smooth_structure">Smooth structure</h3> <p>The quantomorphism group for a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> may naturally be equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in <a class="existingWikiWord" href="/nlab/show/ILH+manifolds">ILH manifolds</a> (<a href="Omori">Omori</a>, <a href="RatiuSchmid">Ratiu-Schmid</a>), as well as in <a class="existingWikiWord" href="/nlab/show/convenient+manifolds">convenient manifolds</a> (<a href="#Vizman">Vizman, prop.</a>).</p> <h3 id="group_extension">Group extension</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <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/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> there is a <a class="existingWikiWord" href="/nlab/show/central+extension+of+groups">central extension of groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>QuantomorphismGroup</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><mi>HamiltonianSymplectomorphisms</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 1 \to U(1) \to QuantomorphismGroup(X,\omega) \to HamiltonianSymplectomorphisms(X,\omega) \to 1 \,. </annotation></semantics></math></div></div> <p>This is due to (<a href="#Kostant">Kostant</a>). It appears also a <a href="#Brylinski93">Brylinski, prop. 2.4.5</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="CoveringAnAffineSymplecticGroup">Covering an affine symplectic group</h3> <p>Given a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,\omega)</annotation></semantics></math> one may consider the restriction of its <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a> to the <a class="existingWikiWord" href="/nlab/show/affine+symplectic+group">affine symplectic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ASp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ASp(V,\omega)</annotation></semantics></math> (<a href="#RobbinSalamon93">Robbin-Salamon 93, corollary 9.3</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><mi>ESp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>QuantMorph</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ASp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>HamSympl</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow \\ ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) } </annotation></semantics></math></div> <p>Sometimes (e.g. <a href="#RobbinSalamon93">Robbin-Salamon 93, p. 30</a>) this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ESp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ESp(V,\omega)</annotation></semantics></math> is called the <em>extended symplectic group</em>, but maybe to be more specific one should at the very least say “<a class="existingWikiWord" href="/nlab/show/extended+affine+symplectic+group">extended affine symplectic group</a>” or “extended inhomogeneous symplectic group” (<a href="#ARZ06">ARZ 06, prop. V.1</a>).</p> <p>Notice that the further restriction to <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> regarded as the <a class="existingWikiWord" href="/nlab/show/translation+group">translation group</a> over itself is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Heis</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Heis(V,\omega)</annotation></semantics></math></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>Heis</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>ESp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>QuantMorph</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>ASp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>HamSympl</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Heis(V,\omega) &\hookrightarrow& ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow && \downarrow \\ V &\hookrightarrow& ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) } </annotation></semantics></math></div> <p>The group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ESp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ESp(V,\omega)</annotation></semantics></math> is that of those <a class="existingWikiWord" href="/nlab/show/quantomorphisms">quantomorphisms</a> which come from <a class="existingWikiWord" href="/nlab/show/quadratic+Hamiltonians">quadratic Hamiltonians</a>. Those elements covering elements in the <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> instead of the <a class="existingWikiWord" href="/nlab/show/affine+symplectic+group">affine symplectic group</a> come from <a class="existingWikiWord" href="/nlab/show/homogeneously+quadratic+Hamiltonians">homogeneously quadratic Hamiltonians</a> (e.g. <a href="#RobbinSalamon93">Robbin-Salamon 93, prop. 10.1</a>). In fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ESp</mi></mrow><annotation encoding="application/x-tex">ESp</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/semidirect+product">semidirect product</a> of the <a class="existingWikiWord" href="/nlab/show/metaplectic+group">metaplectic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mp(V,\omega)</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> (<a href="#ARZ06">ARZ 06, prop. V.1</a>, see also <a href="#Low12">Low 12</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ESp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Heis</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>⋊</mo><mi>Mp</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ESp(V,\omega) \simeq Heis(V,\omega) \rtimes Mp(V,\omega) \,. </annotation></semantics></math></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conserved+current">conserved current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/classical+anomaly">classical anomaly</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice</a>-<a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-groups">automorphism ∞-groups</a> in <a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a>:</th><th><a class="existingWikiWord" href="/nlab/show/Heisenberg+%E2%88%9E-group">Heisenberg ∞-group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-bisections">∞-bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-bisections">∞-bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heisenberg+L-%E2%88%9E+algebra">Heisenberg L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+L-%E2%88%9E+algebra">Poisson L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+L-%E2%88%9E+algebra">Courant L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;">twisted vector fields</td></tr> </tbody></table> </div><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a>:</th><th>standard <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th>groupoid version of <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>:</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_1"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_2"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_3"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathbb{G}_{conn}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> without top-degree connection form</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <h3 id="general">General</h3> <p>Original accounts are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Structure des systemes dynamiques</em> Dunod, Paris (1970)</p> <p>Translated and reprinted as (see section V.18 for the quantomorphism group):</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) doi:<a href="https://doi.org/10.1007/978-1-4612-0281-3">10.1007/978-1-4612-0281-3</a></p> </li> <li id="Kostant"> <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 doi:<a href="https://doi.org/10.1007/BFb0079068">10.1007/BFb0079068</a></p> </li> </ul> <p>A textbook account is in</p> <ul> <li id="Brylinski93"><a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, section II.4 <em>Loop spaces, characteristic classes and geometric quantization</em>, Birkhäuser (1993)</li> </ul> <p>and in</p> <ul> <li>Rudolf Schmid, <em>Infinite-dimensional Lie groups with applications to mathematical physics</em>, J. Geom. Symmetry Phys., Volume 1 (2004), 54-120. <a href="https://projecteuclid.org/euclid.jgsp/1495505067">Project Euclid</a></li> </ul> <p>The description in terms of automorphism in the slice <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>-topos over the moduli stack of (higher) connections is in</p> <ul> <li id="FRS13"><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 geometric prequantum theory</a></em> (<a href="http://arxiv.org/abs/1304.0236">arXiv:1304.0236</a>)</li> </ul> <p>and in section 4.4.17 of</p> <ul> <li id="Sch"><em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> <p>Another expository account, which spells out the details of the isomorphism from smooth functions with the Poisson bracket to infinitesimal quantomorphisms is given in</p> <ul> <li><span class="newWikiWord">Jennifer Vaughan<a href="/nlab/new/Jennifer+Vaughan">?</a></span>, <em><a href="https://hdl.handle.net/1807/77389">Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization</a></em>, PhD dissertation, University of Toronto, 2016.</li> </ul> <h3 id="SmoothManifoldStructure">Smooth manifold structure</h3> <p>The <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>-structure (<a class="existingWikiWord" href="/nlab/show/diffeological+group">diffeological group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+group">smooth group</a> structure) on the quantomorphism group is at least implicit in</p> <ul> <li id="Souriau79"><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Groupes différentiels</em>, in <em>Differential Geometrical Methods in Mathematical Physics</em> (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (<a href="http://www.ams.org/mathscinet-getitem?mr=607688">MathSciNet</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH group</a> structure on the quantomorphism group is discussed in</p> <ul> <li id="Omori"> <p>H. Omori, <em>Infinite dimensional Lie transformation groups</em>, Springer lecture notes in mathematics 427 (1974)</p> </li> <li id="RatiuSchmid"> <p>T. Ratiu, R. Schmid, <em>The differentiable structure of three remarkable diffeomorphism groups</em>, Math. Z. 177 (1981)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/convenient+manifold">regular convenient Lie group</a> structure is discussed in</p> <ul> <li id="Vizman">Cornelia Vizman, <em>Some remarks on the quantomorphism group</em> (<a class="existingWikiWord" href="/nlab/files/VizmanQuantomorphism.pdf" title="pdf">pdf</a>)</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/metric">metric</a>-structure on quantomorphism groups is discussed in</p> <ul> <li>Y. Eliashberg,; L. Polterovich, <em>Partially ordered groups and geometry of contact transformations</em>. Geom.Funct.Anal.10(2000),no.6, 1448-1476. doi:<a href="https://doi.org/10.1007/PL00001656">10.1007/PL00001656</a>, arXiv:<a href="https://arxiv.org/abs/math/9910065">math/9910065</a></li> </ul> <h3 id="examples_2">Examples</h3> <p>The quantomorphisms over elements of the <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> of a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> are discussed in</p> <ul> <li id="Segal63"> <p><a class="existingWikiWord" href="/nlab/show/Irving+Segal">Irving Segal</a>, <em>Transforms for operators and symplectic automorphisms over a locally compact abelian group</em>, Math. Scand. 13 (1963) 31-43</p> </li> <li id="RobbinSalamon93"> <p><a class="existingWikiWord" href="/nlab/show/Joel+Robbin">Joel Robbin</a>, <a class="existingWikiWord" href="/nlab/show/Dietmar+Salamon">Dietmar Salamon</a>, <em>Feynman path integrals on phase space and the metaplectic representation</em> in <a class="existingWikiWord" href="/nlab/show/Dietmar+Salamon">Dietmar Salamon</a> (ed.), <em>Symplectic Geometry</em>, LMS Lecture Note series 192 (1993) (<a class="existingWikiWord" href="/nlab/files/RobbinSalamonMetaplectic.pdf" title="pdf">pdf</a>)</p> </li> <li id="ARZ06"> <p><a class="existingWikiWord" href="/nlab/show/Sergio+Albeverio">Sergio Albeverio</a>, J. Rezende and J.-C. Zambrini, <em>Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics</em>, Journal of Mathematical Physics 47, 062107 (2006) (<a href="http://gfm.cii.fc.ul.pt/people/jczambrini/JMathPhys-47-062107.pdf">pdf</a>)</p> </li> <li id="Low12"> <p>Stephen G. Low, <em>Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group</em>, J. Math. Phys. 55, 022105 (2014) (<a href="http://arxiv.org/abs/1207.6787">arXiv:1207.6787</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 16, 2023 at 01:20:18. 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