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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4315/#Item_3" title="Discuss this page in its dedicated thread on the nForum" 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> <blockquote> <p>This entry is about polarization of <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> (or of any <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>) into canonical “position” coordinates and <a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a>. Different concepts of a similar name include the <em><a class="existingWikiWord" href="/nlab/show/polarization+identity">polarization identity</a></em> (such as in an <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner product space</a> or a <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a>) or <em><a class="existingWikiWord" href="/nlab/show/wave+polarization">wave polarization</a></em> (such as polarized <a class="existingWikiWord" href="/nlab/show/light">light</a>). On the other hand, the concept of <em><a class="existingWikiWord" href="/nlab/show/polarized+algebraic+variety">polarized algebraic variety</a></em> is closely related.</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="geometric_quantization">Geometric quantization</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong> <strong><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <em><a href="geometry+of+physics#LagrangiansAndActionFunctionals">Lagrangians and Action functionals</a></em> + <em><a href="geometry+of+physics#GeometricQuantization">Geometric Quantization</a></em></p> <h2 id="prerequisites">Prerequisites</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> </li> </ul> </li> </ul> <h2 id="prequantum_field_theory">Prequantum field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a> = <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a></p> <ul> <li> <p>prequantum 1-bundle = <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, regular<a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>,<a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> = lift of <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+0-bundle">prequantum 0-bundle</a> = <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> </li> </ul> </li> </ul> <h2 id="geometric_quantization">Geometric quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+leaf">Bohr-Sommerfeld leaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">geometric quantization by push-forward</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+non-integral+forms">geometric quantization of non-integral forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/coherent+state+%28in+geometric+quantization%29">coherent state</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil+theorem">Borel-Weil theorem</a>, <a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schubert+calculus">Schubert calculus</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/geometric+quantization+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="symplectic_geometry">Symplectic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#OfASymplecticManifold'>Of a symplectic manifold</a></li> <li><a href='#OfAPoissonLieAlgebroid'>Of a Poisson Lie algebroid</a></li> <li><a href='#OfACourantLie2Algebroid'>Of a Courant Lie 2-algebroid</a></li> <li><a href='#OfASymplecticLienAlgebroid'>Of a symplectic Lie <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>-algebroid</a></li> <li><a href='#of_an_plectic_smooth_groupoid'>Of an <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>-plectic smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#real_polarizations'>Real polarizations</a></li> <li><a href='#K&#228;hlerPolarization'>Kähler polarizations</a></li> <li><a href='#convex_polarizations'>Convex polarizations</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <ul> <li><a href='#quantum_states_and_wave_functions'>Quantum states and wave functions</a></li> <li><a href='#bohrsommerfeld_leaves'>Bohr-Sommerfeld leaves</a></li> <li><a href='#liouville_integrability'>Liouville integrability</a></li> <li><a href='#refinement_to_higher_geometric_quantization'>Refinement to higher geometric quantization</a></li> <li><a href='#refinement_to_higher_quantization_by_pushforward'>Refinement to higher quantization by push-forward</a></li> <li><a href='#other'>Other</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>For a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> regarded as the <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> of a <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a>, a choice of <em>polarization</em> is, locally, a choice of decomposition of the <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</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> into “<a class="existingWikiWord" href="/nlab/show/canonical+coordinates">canonical coordinates</a>” and “<a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a>”.</p> <p>The archtypical example is that where <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>Σ</mi></mrow><annotation encoding="application/x-tex">X = T^* \Sigma</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. In this case the canonical <a class="existingWikiWord" href="/nlab/show/canonical+coordinates">canonical coordinates</a> are those parameterizing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> itself, while the canonical <a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a> are coordinates on each <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of the cotangent bundle.</p> <p>But for general symplectic manifolds there is no such canonical choice of coordinates and momenta. Moreover, in general there is not even a global notion of canonical momenta. Instead, a choice of (real) polarization is a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> of phase space by <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifolds">Lagrangian submanifolds</a> and then</p> <ul> <li> <p>the “<a class="existingWikiWord" href="/nlab/show/canonical+coordinates">canonical coordinates</a>” are coordinates on the corresponding <a class="existingWikiWord" href="/nlab/show/leaf+space">leaf space</a> (parameterizing the leaves);</p> </li> <li> <p>the “<a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a>” are coordinates along each <a class="existingWikiWord" href="/nlab/show/leaf">leaf</a>. If there is no <em>typical leaf</em> then this is not a globally defined notion, only the polarization itself is.</p> </li> </ul> <p>Locally this is a choice of <a class="existingWikiWord" href="/nlab/show/coordinate+patch">coordinate patch</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \mathbb{R}^{2n} \to X</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> takes the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>ω</mi><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mi>i</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \phi^* \omega = \sum_{i = 1}^n \mathbf{d} q^i \wedge \mathbf{d}p_i </annotation></semantics></math></div> <p>where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>q</mi> <mi>i</mi></msup><mo>:</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>≃</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{q^i : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}</annotation></semantics></math> are the canonical coordinates on the first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-factor of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>o</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>≃</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{p_o : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\}</annotation></semantics></math> are canonical coordinates on the second <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-factor.</p> <h2 id="definition">Definition</h2> <p>The traditional notion of polarization applies to a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>.</p> <ul> <li><a href="OfASymplecticManifold">Of a symplectic manifold</a></li> </ul> <p>Symplectic manifold are the lowest step in a tower of notions in <a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a> which proceeds with <a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a> for all <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> and manifolds refined to <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoids">smooth infinity-groupoids</a>. The next simplest cases in this tower are <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroids">symplectic Lie n-algebroids</a>, which for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroids">Poisson Lie algebroids</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroids">Courant Lie 2-algebroids</a>:</p> <ul> <li> <p><a href="#OfAPoissonLieAlgebroid">Of a Poisson Lie algebroid</a></p> </li> <li> <p><a href="#OfACourantLie2Algebroid">Of a Courant Lie 2-algebroid</a></p> </li> <li> <p><a href="#OfASymplecticLienAlgebroid">Of a symplectic Lie n-algebroid</a></p> </li> </ul> <h3 id="OfASymplecticManifold">Of a symplectic manifold</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>.</p> <div class="num_defn" id="ByLagrangianFoliation"> <h6 id="definition_2">Definition</h6> <p>A <strong>real polarization</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> by <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifolds">Lagrangian submanifolds</a>.</p> </div> <p>For instance (<a href="#Weinstein">Weinstein, p. 9</a>).</p> <p>More generally</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <strong>polarization</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> is a choice of involutive <a class="existingWikiWord" href="/nlab/show/Lagrangian+subspace">Lagrangian subbundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>↪</mo><msub><mi>T</mi> <mi>ℂ</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{P} \hookrightarrow T_{\mathbb{C}} X</annotation></semantics></math> of of the <a class="existingWikiWord" href="/nlab/show/complexification">complexified</a> <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>For instance (<a href="#BatesWeinstein">Bates-Weinstein, def. 7.4</a>)</p> <h3 id="OfAPoissonLieAlgebroid">Of a Poisson Lie algebroid</h3> <p>A <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</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">\mathfrak{P}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>. Regarding its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> as the algebra of functions on a <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a>, that dg-manifold carries a graded <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>. One can then say</p> <div class="num_defn" id="ForPoissonLieAlgebroidyByLagrangianFoliation"> <h6 id="definition_4">Definition</h6> <p>A dg-<a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a> 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><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{P}, \omega)</annotation></semantics></math> is also called a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>-structure</strong>. (<a href="#Severa">Ševera, section 4</a>).</p> <p>Hence we might say <strong>real polarization</strong> 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><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{P}, \omega)</annotation></semantics></math> is a foliation by dg-Lagrangian submanifolds.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \pi)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> underlying a <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔓</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{P}, \omega)</annotation></semantics></math>, a dg-Lagrangian submanifold 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><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{P}, \omega)</annotation></semantics></math> corresponds to a <a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \pi)</annotation></semantics></math>.</p> </div> <p>(<a href="#Severa">Ševera, section 4</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The dg-Lagrangian submanifolds also correspond to <a class="existingWikiWord" href="/nlab/show/branes">branes</a> in the <a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a> (see there) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔓</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{P}, \omega)</annotation></semantics></math>.</p> </div> <h3 id="OfACourantLie2Algebroid">Of a Courant Lie 2-algebroid</h3> <p>A <a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a> <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">\mathfrak{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math>. Regarding its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> as the algebra of functions on a <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a>, that dg-manifold carries a graded <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>. One can then say</p> <div class="num_defn" id="ForPoissonLieAlgebroidyByLagrangianFoliation"> <h6 id="definition_5">Definition</h6> <p>A dg-<a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℭ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{C}, \omega)</annotation></semantics></math> is also called a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>-structure</strong>. (<a href="#Severa">Ševera, section 4</a>).</p> <p>Hence we might say <strong>real polarization</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℭ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{C}, \omega)</annotation></semantics></math> is a foliation by dg-Lagrangian submanifolds.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℭ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{C}, \omega)</annotation></semantics></math> correspond to <a class="existingWikiWord" href="/nlab/show/Dirac+structures">Dirac structures</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℭ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{C}, \omega)</annotation></semantics></math>.</p> </div> <p>(<a href="#Severa">Ševera, section 4</a>)</p> <h3 id="OfASymplecticLienAlgebroid">Of a symplectic Lie <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>-algebroid</h3> <p>??</p> <h3 id="of_an_plectic_smooth_groupoid">Of an <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>-plectic smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</h3> <p>A simple notion of a real polarization for <a class="existingWikiWord" href="/nlab/show/2-plectic+manifolds">2-plectic manifolds</a> is considered within the context of higher geometric quantization in <a href="#Rogers">Rogers, Chap. 7</a>.</p> <h2 id="examples">Examples</h2> <h3 id="real_polarizations">Real polarizations</h3> <p>(…)</p> <h3 id="K&#228;hlerPolarization">Kähler polarizations</h3> <p>If the <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> lifts to the structure of a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>J</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, J, g)</annotation></semantics></math>, hence with <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(-,-) = \omega(-,I(-))</annotation></semantics></math>, then the holomorphic/antiholomorphic decomposition induced by the <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> structure is a polarization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>. Polarizations of this form are therefore called <strong><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarizations">Kähler polarizations</a></strong>.</p> <h3 id="convex_polarizations">Convex polarizations</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/p-convex+polarization">p-convex polarization</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <h3 id="quantum_states_and_wave_functions">Quantum states and wave functions</h3> <p>Upon (<a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric</a>) <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the physical system described by the <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> is supposed to be a function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> – or rather a <a class="existingWikiWord" href="/nlab/show/section">section</a> of a <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> which is a “wave-function that only depends on the canonical coordinates”, not on the canonical momenta. In terms of polarizations this is formalized by saying that a <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">quantum state</a> is a section which is <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant constant</a> along the <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> of the polarization (along the “momentum direction”).</p> <h3 id="bohrsommerfeld_leaves">Bohr-Sommerfeld leaves</h3> <p>After a choice of <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> lifting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, a <strong><a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+leaf">Bohr-Sommerfeld leaf</a></strong> of a (real) polarization is a <a class="existingWikiWord" href="/nlab/show/leaf">leaf</a> on which the prequantum line bundle is not just flat, but also trivializable as a <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a> with connection.</p> <h3 id="liouville_integrability">Liouville integrability</h3> <p>If a polarization on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2n</annotation></semantics></math>-dimensional symplectic manifold is generated from <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 class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a> whose <a class="existingWikiWord" href="/nlab/show/Hamiltonians">Hamiltonians</a> commute with each other under the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> (and one of them is regarded as that generating time evolution of a mechanical system) then one speaks of a <a class="existingWikiWord" href="/nlab/show/Liouville+integrable+system">Liouville integrable system</a>.</p> <h3 id="refinement_to_higher_geometric_quantization">Refinement to higher geometric quantization</h3> <div> <p><strong><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> from binary and non-degenerate <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_1"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></th><th><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrated</a> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> = <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_2"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_3"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-</a><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>/ <a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</a> <a class="existingWikiWord" href="/nlab/show/leaf">leaf</a></th><th>= <a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th><a class="existingWikiWord" href="/nlab/show/n-module">(n+1)-module</a> of <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_4"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math></th><th>discussed in:</th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a></td><td style="text-align: left;">–</td><td style="text-align: left;">ordinary <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></em></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></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/Poisson+sigma-model">Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a> (of underlying <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>)</td><td style="text-align: left;"><a href="Poisson+sigma-model#Branes">brane of Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a> = <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over <a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantiized</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/extended+geometric+quantization+of+2d+Chern-Simons+theory">extended geometric quantization of 2d Chern-Simons theory</a></em></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+2-groupoid">symplectic 2-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-plectic+geometry">3-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+structure">Dirac structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> in <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_5"><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/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">(n+1)-plectic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_6"><semantics><mrow><mi>d</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d = n+1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(adapted from <a class="existingWikiWord" href="/nlab/show/Some+title+containing+the+words+%22homotopy%22+and+%22symplectic%22%2C+e.g.+this+one">Ševera 00</a>)</p></div> <h3 id="refinement_to_higher_quantization_by_pushforward">Refinement to higher quantization by push-forward</h3> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a></th><th><a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> / <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></th><th><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">generalized orientation</a>/<a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></th><th><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></th><th>incarnation as <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> in <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">third integral SW class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">W_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin%E1%B6%9C+structure">spinᶜ structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">K-theoretic geometric quantization</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly">Freed-Witten anomaly</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/EO%28n%29">EO(n)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">w_4</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+Morava+K-theory">integral Morava K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K(2)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">seventh integral SW class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">W_7</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Diaconescu-Moore-Witten+anomaly">Diaconescu-Moore-Witten anomaly</a> in <a href="Diaconescu-Moore-Witten+anomaly#ReferencesInterpretationInSecondMoravaKTheory">Kriz-Sati interpretation</a></td></tr> </tbody></table> </div> <h3 id="other">Other</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+transformation">canonical transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polarized+algebraic+variety">polarized algebraic variety</a></p> </li> </ul> <h2 id="references">References</h2> <p>Lecture notes include</p> <ul id="Weinstein"> <li><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>Lectures on Symplectic manifolds</em> Lecture 2 <em>Lagrangian splittings, real and complex polarizations, Kähler manifolds</em>, CBMS Regional Conference Series in Mathematics, AMS (1977)</li> </ul> <ul id="BatesWeinstein"> <li>Sean Bates, <a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>Lectures on the geometry of quantization</em>, <a href="http://www.math.berkeley.edu/~alanw/GofQ.pdf">pdf</a></li> </ul> <ul> <li>Kristin Shaw, <em>An introduction to polarizations</em> (<a href="http://www.math.toronto.edu/karshon/grad/2006-07/polarizations.pdf">pdf</a>)</li> </ul> <p>and section 4 and 5 of</p> <ul id="Blau"> <li><a class="existingWikiWord" href="/nlab/show/Matthias+Blau">Matthias Blau</a>, <em>Symplectic Geometry and Geometric Quantization</em> (<a class="existingWikiWord" href="/nlab/files/BlauGeometricQuantization.pdf" title="pdf">pdf</a>)</li> </ul> <p>or section 5 of</p> <ul> <li>A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, <em>Mathematical Foundations of Geometric Quantization</em> Extracta Math. 13 (1998) 135-238 (<a href="http://arxiv.org/abs/math-ph/9904008">arXiv:math-ph/9904008</a>)</li> </ul> <p>or</p> <ul> <li>Yuichi Nohara, <em>Independence of Polarization in Geometric Quantization</em> (<a href="http://geoquant2007.mi.ras.ru/nohara.pdf">pdf</a>)</li> </ul> <p>Lagrangian submanifolds of <a class="existingWikiWord" href="/nlab/show/L-infinity+algebroids">L-infinity algebroids</a> are considered in</p> <ul> <li id="Severa"><a class="existingWikiWord" href="/nlab/show/Pavol+%C5%A0evera">Pavol Ševera</a>, <em>Some title containing the words “homotopy” and “symplectic”, e.g. this one</em> (<a href="http://arxiv.org/abs/math/0105080">arXiv:0105080</a>)</li> </ul> <p>In the case that the polarization integrates to the <a class="existingWikiWord" href="/nlab/show/action">action</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> one may think of passing to polarized sections as equivlent to passing 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">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+equivalence">gauge equivalence classes</a>. This point of view is highlighted in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gabriel+Catren">Gabriel Catren</a>, <em>On the Relation Between Gauge and Phase Symmetries</em>, Foundations of Physics 44 (12):1317-1335 (2014) (<a href="http://philpapers.org/rec/CATOTR">web</a>)</li> </ul> <p>More on <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> for <a class="existingWikiWord" href="/nlab/show/real+polarizations">real polarizations</a>:</p> <ul> <li>Pau Mir, <a class="existingWikiWord" href="/nlab/show/Eva+Miranda">Eva Miranda</a>: <em>Geometric quantization via cotangent models</em>, Anal. Math. Phys. <strong>11</strong> (2021) 118 [<a href="https://doi.org/10.1007/s13324-021-00559-4">doi:10.1007/s13324-021-00559-4</a>, <a href="https://arxiv.org/abs/2102.02699">arXiv:2102.02699</a>]</li> </ul> <p>A candidate for polarizations for higher geometric quantization in <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>-plectic geometry is discussed in Chapter 7 of</p> <ul> <li id="Rogers"><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Higher symplectic geometry</em> PhD thesis (<a href="http://arxiv.org/abs/1106.4068">arXiv:1106.4068</a>).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 2, 2025 at 15:19:02. See the <a href="/nlab/history/polarization" 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/polarization" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4315/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/polarization/24" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/polarization" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/polarization" accesskey="S" class="navlink" id="history" rel="nofollow">History (24 revisions)</a> <a href="/nlab/show/polarization/cite" style="color: black">Cite</a> <a href="/nlab/print/polarization" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/polarization" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>