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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" 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="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> <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> </div> </div> <blockquote> <p>under construction</p> </blockquote> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#motivation_and_survey_of_results'>Motivation and survey of results</a></li> <ul> <li><a href='#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices'>Ordinary prequantum geometry in terms of automorphisms in slices</a></li> <li><a href='#TheNeedForHigherPrequantumBundles'>The need for higher prequantum bundles</a></li> <li><a href='#brief_recollection_higher_geometry'>Brief recollection: Higher geometry</a></li> <li><a href='#higher_atiyah_groupoids'>Higher Atiyah groupoids</a></li> <li><a href='#the_central_theorem_quantomorphism_group_extensions'>The central theorem: Quantomorphism <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>-group extensions</a></li> <li><a href='#examples__and__as_heisenberg_groups'>Examples: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math> as Heisenberg <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>-groups</a></li> </ul> <li><a href='#constructions_in_higher_prequantum_geometry'>Constructions in higher prequantum geometry</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Traditional <a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is the <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> which are equipped with a <em><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twist</a></em> in the form of a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> and a circle-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a>. In the context of <em><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></em> of <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a> these arise as <em><a class="existingWikiWord" href="/nlab/show/prequantum+bundles">prequantum bundles</a></em>. Equivalently, prequantum geometry is the <em><a class="existingWikiWord" href="/nlab/show/contact+geometry">contact geometry</a></em> of the total spaces of these bundles, equipped with their <a class="existingWikiWord" href="/nlab/show/Ehresmann+connection">Ehresmann connection</a> <a class="existingWikiWord" href="/nlab/show/differential+1-form">differential 1-form</a> and thought of as <em><a class="existingWikiWord" href="/nlab/show/regular+contact+manifolds">regular contact manifolds</a></em>. Prequantum geometry notably studies the <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> of <a class="existingWikiWord" href="/nlab/show/prequantum+bundles">prequantum bundles</a> covering <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a> of the base – the <em><a class="existingWikiWord" href="/nlab/show/prequantum+operators">prequantum operators</a></em> or <em><a class="existingWikiWord" href="/nlab/show/contactomorphisms">contactomorphisms</a></em> – and the <a class="existingWikiWord" href="/nlab/show/action">action</a> of these on 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> – the <em><a class="existingWikiWord" href="/nlab/show/prequantum+states">prequantum states</a></em>. This is an intermediate step in the genuine <em><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></em> of the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <a class="existingWikiWord" href="/nlab/show/differential+2-form">differential 2-form</a> of these bundles, which is obtained by “dividing the above data in half” (<a class="existingWikiWord" href="/nlab/show/polarization">polarization</a>), important for instance in the the <em><a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a></em>.</p> <p>But prequantum geometry is of interest in its own right. For instance the above automorphism group naturally provides the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> of the underlying <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, together with the canonical injection into the <a class="existingWikiWord" href="/nlab/show/group+of+bisections">group of bisections</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the <em><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></em> which is associated with the given circle bundle, all of which are fundamental objects of interest in the study of <a class="existingWikiWord" href="/nlab/show/line+bundles">line bundles</a> over <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>.</p> <p>For a plethora of applications in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, one wants to generalize this to <em><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></em> (see at <em><a class="existingWikiWord" href="/nlab/show/motivation+for+higher+differential+geometry">motivation for higher differential geometry</a></em>) and accordingly study <em>higher prequantum geometry</em>.</p> <h2 id="motivation_and_survey_of_results">Motivation and survey of results</h2> <h3 id="OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices">Ordinary prequantum geometry in terms of automorphisms in slices</h3> <p>A sequence of time-honored traditional concepts in <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>/<a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>:</th><th><a class="existingWikiWord" href="/nlab/show/Heisenberg+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+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/gauge+group">gauge group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie 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+Lie+algebra">Poisson Lie 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 <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a></td></tr> </tbody></table> <p>For instance in the <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electrically</a> <a class="existingWikiWord" href="/nlab/show/charged+particle">charged</a> <a class="existingWikiWord" href="/nlab/show/particle">particle</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> we have a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle 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">P</annotation></semantics></math> 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><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> on a <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><mi>X</mi><mo>=</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>Y</mi></mrow><annotation encoding="application/x-tex">X = T^* Y</annotation></semantics></math> which is essentially the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>-bundle on <a class="existingWikiWord" href="/nlab/show/target+space">target</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <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>. Its <em><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></em> is the group of <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</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> of the total space of the prequantum bundle which preserve the connection (also called the <em><a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></em> of <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><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P,\nabla)</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/regular+contact+manifold">regular contact manifold</a>). For the following it is convenient to say this using the language of <em><a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a></em>: we may regard <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> as a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (a “<a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>”) and then moreover as a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <a class="existingWikiWord" href="/nlab/show/stack">stack</a> on this site (a “<a class="existingWikiWord" href="/nlab/show/smooth+groupoid">smooth groupoid</a>”) and make use of the tautological existence of the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <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+connections">principal connections</a>, which we 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> (we don’t need further details right now, but they can be found for instance at <em><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></em> for details). By definition this is such that for any <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> a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</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 \colon X \to \mathbf{B}U(1)_{conn}</annotation></semantics></math> is equivalently 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+connection">principal connection</a> and such that a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo>→</mo><msub><mo>∇</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\eta \colon \nabla_1 \to \nabla_2</annotation></semantics></math> between two such maps is equivalently a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between two such connections. With this formulation a <a class="existingWikiWord" href="/nlab/show/quantomorphism">quantomorphism</a> of the <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</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> is equivalently a diagram of the form as on the right of</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><mo>∇</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><mo>↙</mo></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><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{QuantMorph}(\nabla) = \left\{ \array{ X &&\underoverset{\simeq}{\phi}{\to}&& X \\ & \searrow &\swArrow_{\eta}& \swarrow \\ && \mathbf{B}U(1)_{conn} } \right\} </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>, namely a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon X \stackrel{\simeq}{\to} X</annotation></semantics></math> of the base space of the bundle together with a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> of <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+connections">principal connections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mover><mo>→</mo><mo>≃</mo></mover><mo>∇</mo></mrow><annotation encoding="application/x-tex">\eta \colon \phi^* \nabla \stackrel{\simeq}{\to} \nabla</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a> is naturally an (<a class="existingWikiWord" href="/nlab/show/infinite-dimensional+Lie+group">infinite dimensional</a>) <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. Its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> is the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. If <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 equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> itself (notably if it is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>), then the sub-Lie algebra of that on the <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant vectors</a> is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a> and the Lie group corresponding to that is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a>.</p> <p>One also says that a triangular diagram as above is an autoequivalence of the “modulating” map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</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 \colon X \to \mathbf{B}U(1)_{conn}</annotation></semantics></math> in the <em><a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-category">slice (2,1)-category</a></em> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>/<a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> over <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>.</p> <p>Such autoequivalences in slices are familiar from basic concepts of <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> theory. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>→</mo></mover><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, we may regard the inclusion of its manifold of objects as an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> being a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>𝒢</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo>→</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">p_\mathcal{G} \colon\mathcal{G}_0 \to \mathcal{G}</annotation></semantics></math>. Regarding this atlas as an object in the <a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-category">slice (2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>/<a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> over <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">\mathcal{G}</annotation></semantics></math>, its autoequivalences are diagrams as on the right of</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>BiSect</mi></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>𝒢</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>𝒢</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>𝒢</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒢</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BiSect}(p_{\mathcal{G}}) = \left\{ \array{ \mathcal{G}_0 &&\stackrel{\phi}{\to}&& \mathcal{G}_0 \\ & \searrow &\swArrow_\eta & \swarrow \\ && \mathcal{G} } \right\} \,. </annotation></semantics></math></div> <p>This is a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi \colon \mathcal{G}_0 \stackrel{\simeq}{\to} \mathcal{G}_0</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> equipped with a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> whose component map is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> that assigns to each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">q\in\mathcal{G}_0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</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">\mathcal{G}</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>q</mi></msub><mo lspace="verythinmathspace">:</mo><mi>q</mi><mo>→</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_q \colon q \to \phi(q)</annotation></semantics></math>. This collection of data is known as a <em><a class="existingWikiWord" href="/nlab/show/bisection">bisection</a></em> of a <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>. Bisections naturally form a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>𝒢</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{BiSect}(p_{\mathcal{G}})</annotation></semantics></math> , which is all the more manifest if we understand them as autoequivalences of the atlas in the slice, called the <a class="existingWikiWord" href="/nlab/show/group+of+bisections">group of bisections</a>.</p> <p>This perspective of regarding maps of <a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> as objects in the slice over their codomain (an elementary step in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>/<a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">higher topos theory</a>, but not common in traditional differential geometry) turns out to be useful and drives all of the refinements, generalizations and theorems that we discuss in the following: we will see that higher <a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is essentially the geometry insice <a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-topos">higher slice categories</a> of <a class="existingWikiWord" href="/nlab/show/infinity-stack">higher stacks</a> over <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">higher moduli stacks</a> of <a class="existingWikiWord" href="/nlab/show/principal+infinity-connection">higher principal connections</a>.</p> <p>Before we get there, notice the following…</p> <h3 id="TheNeedForHigherPrequantumBundles">The need for higher prequantum bundles</h3> <p>The tools of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> mainly apply to <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> and only partially to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. In particular in the context of <em><a class="existingWikiWord" href="/nlab/show/extended+prequantum+field+theory">extended prequantum field theory</a></em> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <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> a <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</a> over the (<a class="existingWikiWord" href="/nlab/show/phase+space">phase</a>-)space of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> is to be refined (de-<a class="existingWikiWord" href="/nlab/show/transgression">transgressed</a>) to a <em><a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum n-bundle</a></em> over the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>. Therefore in order to apply <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> to <a class="existingWikiWord" href="/nlab/show/extended+prequantum+field+theory">extended prequantum field theory</a> to obtain <a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a> we first need extended/higher prequantum geometry.</p> <p>For instance the <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum 3-bundle</a> for standard <a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d</a> <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is modulated by the differential <a class="existingWikiWord" href="/nlab/show/smooth+first+fractional+Pontryagin+class">smooth first fractional Pontryagin class</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><msub><mi>Spin</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>forget</mi><mspace width="thickmathspace"></mspace><mi>connections</mi></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><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><mo stretchy="false">↓</mo></mtd> <mtd><mi>geometric</mi><mspace width="thickmathspace"></mspace><mi>realization</mi></mtd></mtr> <mtr><mtd><mi>B</mi><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow && \downarrow & forget \; connections \\ \mathbf{B}Spin &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ \downarrow && \downarrow & geometric\;realization \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& K(\mathbb{Z},4) } \,, </annotation></semantics></math></div> <p>modulating/classifying the universal <em><a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle+with+connection">Chern-Simons circle 3-bundle with connection</a></em> (also known as a <em><a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></em>) over the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</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>-Chern-Simons theory, which is the moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><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}G_{conn}</annotation></semantics></math> 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>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a>.</p> <p>Similarly the <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum 7-bundle</a> for <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> on <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <a class="existingWikiWord" href="/nlab/show/principal+infinity-connections">principal 2-connections</a> is given by the differential <a class="existingWikiWord" href="/nlab/show/smooth+second+fractional+Pontryagin+class">smooth second fractional Pontryagin class</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><msub><mi>String</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>forget</mi><mspace width="thickmathspace"></mspace><mi>connections</mi></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><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><mo stretchy="false">↓</mo></mtd> <mtd><mi>geometric</mi><mspace width="thickmathspace"></mspace><mi>realization</mi></mtd></mtr> <mtr><mtd><mi>B</mi><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>8</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}String_{conn} &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow && \downarrow & forget\; connections \\ \mathbf{B}String &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 U(1) \\ \downarrow && \downarrow & geometric\; realization \\ B String &\stackrel{\frac{1}{6}p_2}{\to}& K(\mathbb{Z},8) } \,, </annotation></semantics></math></div> <p>modulating/classifying the universal <em><a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle+with+connection">Chern-Simons circle 7-bundle with connection</a></em> over the moduli 2-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}String_{conn}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <a class="existingWikiWord" href="/nlab/show/principal+infinity-connection">principal 2-connections</a>.</p> <p>Therefore we want to lift the <a href="#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices">above</a> table of traditional notions to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>…</p> <h3 id="brief_recollection_higher_geometry">Brief recollection: Higher geometry</h3> <p>In order to say this, clearly we need some basics of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</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><mi>Groupoids</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>nerve</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Categories</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Kan</mi><mi>complexes</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>Categories</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Groupoids \\ & \swarrow && \searrow^{\mathrlap{nerve}} \\ Categories && && Kan complexes \\ & \searrow && \swarrow \\ && (\infty,1)-Categories } \,. </annotation></semantics></math></div> <p>Important construction principle for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>: <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a> with some subset of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>↪</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \hookrightarrow Mor(\mathcal{C})</annotation></semantics></math> declared to be “<a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>”, the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></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>W</mi></msub><mi>𝒞</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> L_W \mathcal{C} \in (\infty,1)Cat </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/universal+construction">universal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> by universally turning each weak equivalence into an actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> in the sense of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>In particular let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a>, assumed for simplicity to have <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>. Declare then that in the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>KanCplx</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(C^{op}, KanCplx)</annotation></semantics></math>, hence in <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-valued presheaves, the weak equivalences are the <a class="existingWikiWord" href="/nlab/show/stalk">stalkwise</a> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> of Kan complexes. Then</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>H</mi></mstyle><mo>≔</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>L</mi> <mi>W</mi></msub><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>KanCplx</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \coloneqq Sh_{\infty}(C) \coloneqq L_{W} Func(C^{op}, KanCplx) </annotation></semantics></math></div> <p>is called the <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></em> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>An <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a>-object <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> in such an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(G)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group">group</a> is called an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> “with geometric structure as encoded by the test spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>”. The canonical source of <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>-groups are the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+products">homotopy fiber products</a> of point inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">* \to X</annotation></semantics></math> of any object X, 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><mi>Ω</mi><mi>X</mi><mo>≔</mo><mo>*</mo><munder><mo>×</mo><mi>X</mi></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega X \coloneqq {*} \underset{X}{\times} {*} \,. </annotation></semantics></math></div> <p>In fact this are <em>all</em> the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> that there are, up to equivalence: forimg <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mover><munderover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>≃</mo></munderover><mover><mo>←</mo><mi>Ω</mi></mover></mover><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}} \mathbf{H}^{*/}_{\geq 1} </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> and <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected+object+in+an+%28%E2%88%9E%2C1%29-topos">connected</a> objects. The inverse equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></em> operation.</p> <p>We say that such an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <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> is <em><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a></em> if it is equipped with an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a> (co)/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monads">(∞,1)-monads</a></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a></th><th></th><th><a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a></th><th></th><th><a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a></th></tr></thead><tbody><tr><td style="text-align: left;">idemp. monad</td><td style="text-align: left;"></td><td style="text-align: left;">idemp. comonad</td><td style="text-align: left;"></td><td style="text-align: left;">idemp. monad</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>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</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><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</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><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</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><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</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><mo>♯</mo></mrow><annotation encoding="application/x-tex">\sharp</annotation></semantics></math></td></tr> </tbody></table> <p>This implies (strictly speaking we need <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> for that, coming from another adjoint triple of (co)monads) that for every <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \in Grp(\mathbf{H})</annotation></semantics></math> there is a canonical object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><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> which modulats <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-connections">principal ∞-connections</a>.</p> <h3 id="higher_atiyah_groupoids">Higher Atiyah groupoids</h3> <p>Looking at the <a href="#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices">above</a> table and noticing the <a href="#TheNeedForHigherPrequantumBundles">above</a> need for higher prequantum bundles, we should try to find an analogous table of concepts in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>, something like this:</p> <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> <p>(…)</p> <p>The way all these notions and theorems work is by considering <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-groups">automorphism ∞-groups</a> of the classifying (or rather: modulating) maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/prequantum+%E2%88%9E-bundle">prequantum ∞-bundle</a> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over the domain. For instance</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><mo>∇</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>∇</mo></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{QuantMorph}(\nabla) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \,. </annotation></semantics></math></div> <p>The others are obtained by succesively forgetting connection data. For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">BiSect</mo><mo stretchy="false">(</mo><mi>Cou</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></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><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \BiSect(Cou(\nabla)) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla_1}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_1}} \\ && \mathbf{B}(\mathbf{B}\mathbb{G}_{conn}) } \right\} \,. </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><mo lspace="0em" rspace="thinmathspace">BiSect</mo><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mn>0</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mn>0</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \BiSect(At(\nabla)) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla_0}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_0}} \\ && \mathbf{B}\mathbb{G} } \right\} \,. </annotation></semantics></math></div> <p>The extension sequence is then schematically simply the following</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><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>∇</mo></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \array{ && X \\ & \swarrow & & \searrow \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \; \to \; \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \; \to \; \left\{ \array{ X && \stackrel{\simeq}{\to} && X } \right\} </annotation></semantics></math></div> <p>in this generality this now includes various other notions, too:</p> <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> <h3 id="the_central_theorem_quantomorphism_group_extensions">The central theorem: Quantomorphism <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>-group extensions</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>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 <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> a higher prequantum geometry with respect to <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> there is a long <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</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><mi>Ω</mi><mi>𝔾</mi><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mrow><mo>(</mo><mo>∇</mo><mo>)</mo></mrow><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo>∇</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mrow><mo>(</mo><mrow><mo>(</mo><mi>Ω</mi><mi>𝔾</mi><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mrow><mo>(</mo><mo>∇</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \to \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}\left(\left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \right) \,. </annotation></semantics></math></div> <p>Similarly there is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+infinity-group">Heisenberg infinity-group</a> extension</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>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{Heis}(\nabla) \to G </annotation></semantics></math></div></div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> sequence of prop. <a class="maruku-ref" href="#QuantomorphismExtension"></a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</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>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝔓𝔬𝔦𝔰𝔰𝔬𝔫</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>𝒳</mi> <mi>Ham</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mi>ω</mi></mrow></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \to \mathfrak{Poisson}(X,\omega) \to \mathcal{X}_{Ham}(X,\omega) \stackrel{\iota_{(-)\omega}}{\to} \mathbf{B}\mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \,, </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><mi>𝔓𝔬𝔦𝔰𝔰𝔬𝔫</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Poisson}(X,\omega)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a> as defined in (<a href="#Rogers11">Rogers 11</a>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒳</mi> <mi>Ham</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{X}_{Ham}</annotation></semantics></math> is the Lie algebra of <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a> restricted to the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \flat (\mathbf{B}^{n-1})\mathbb{R})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> for flat <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> in the given degree, regarded as an abelian <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>.</p> </li> </ul> </div> <p>The following table shows what this sequence reduces to when one chooses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = \mathbf{B}^{n-1}U(1)</annotation></semantics></math>.</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> <h3 id="examples__and__as_heisenberg_groups">Examples: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math> as Heisenberg <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>-groups</h3> <div class="num_example"> <h6 id="example">Example</h6> <p>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">G</annotation></semantics></math> a simply connected semisimple compact Lie group such as the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, let</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>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></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 \coloneqq \exp\left(2 \pi i \int_{S^1} [S^1, \tfrac{1}{2}\hat \mathbf{p}_1]\right) \;\colon\; G \to \mathbf{B}^2 U(1)_{conn} </annotation></semantics></math></div> <p>be the canonical <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> over it. Then the <a class="existingWikiWord" href="/nlab/show/Heisenberg+infinity-group">Heisenberg 2-group</a> <a class="existingWikiWord" href="/nlab/show/infinity-group+extension">extension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> U(1)\mathbf{FlatConn}(G) \to \mathbf{Heis}(\nabla) \to G </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> extension</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>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>String</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1) \to String(G) \to G \,. </annotation></semantics></math></div></div> <p>(by classification of extensions by cohomology… by Lie 2-algebra computation…)</p> <p>(and analogously for <a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a>…)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>6</mn></msup><mi>U</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>→</mo><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mrow><mo>(</mo><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mrow><mo>[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>]</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>→</mo><mi>String</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}^6 U\left(1\right) \to \mathbf{Heis}\left(\exp\left(2 \pi i \int_{S^1} \left[S^1, \tfrac{1}{2}\hat \mathbf{p}_2\right] \right)\right) \to String </annotation></semantics></math></div> <h2 id="constructions_in_higher_prequantum_geometry">Constructions in higher prequantum geometry</h2> <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><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="related_entries">Related entries</h2> <ul> <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/definite+parameterization+of+WZW+terms">definite parameterization of WZW terms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definite+globalization+of+WZW+terms">definite globalization of WZW terms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+structures">higher structures</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/basic+bundle+gerbe">basic bundle gerbe</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></em>.</p> </blockquote> <ul> <li id="FRS13a"> <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 <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>-gerbe connections in geometric prequantization</a></em>, Rev. Math. Phys. <strong>28</strong> 06 1650012 (2016) [<a href="http://arxiv.org/abs/1304.0236">arXiv:1304.0236</a>, <a href="https://doi.org/10.1142/S0129055X16500124">doi:10.1142/S0129055X16500124</a>]</p> </li> <li id="FRS13b"> <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> <li id="Schreiber13"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em> (<a href="http://arxiv.org/abs/1310.7930">arXiv:1310.7930</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 23, 2023 at 17:09:12. See the <a href="/nlab/history/higher+prequantum+geometry" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/higher+prequantum+geometry" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/higher+prequantum+geometry/12" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/higher+prequantum+geometry" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/higher+prequantum+geometry" accesskey="S" class="navlink" id="history" rel="nofollow">History (12 revisions)</a> <a href="/nlab/show/higher+prequantum+geometry/cite" style="color: black">Cite</a> <a href="/nlab/print/higher+prequantum+geometry" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/higher+prequantum+geometry" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>