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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#darboux_coordinates'>Darboux coordinates</a></li> <li><a href='#SymplecticStructure'>Relation to almost symplectic structure</a></li> <li><a href='#relation_to_almost_hermitian_and_khler_structure'>Relation to almost Hermitian and Kähler structure</a></li> <li><a href='#symplectomorphisms'>Symplectomorphisms</a></li> <li><a href='#poisson_structure'>Poisson structure</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>symplectic manifold</strong> is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of even <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>dim</mi><mi>X</mi><mo>=</mo><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">dim X = 2 n</annotation></semantics></math>;</p> </li> <li> <p>equipped with a <strong>symplectic form</strong>:</p> <ul> <li> <p>a closed smooth <a class="existingWikiWord" href="/nlab/show/differential+form">2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^2_{cl}(X)</annotation></semantics></math>;</p> </li> <li> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is <strong>non-degenerate</strong>, which means equivalently that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ω</mi> <mrow><mo>∧</mo><mi>n</mi></mrow></msup><mo>=</mo><mi>ω</mi><mo>∧</mo><mi>ω</mi><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega^{\wedge n}=\omega\wedge\omega\wedge\cdots\wedge\omega</annotation></semantics></math> has the maximal <a class="existingWikiWord" href="/nlab/show/rank">rank</a> at every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p\in X</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mn>2</mn></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>X</mi><mo>,</mo><msub><mi>ω</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^2 T^*_p X,\omega_p)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p\in X</annotation></semantics></math>.</p> </li> </ul> </li> </ul> </li> </ul> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A real <strong>symplectic manifold</strong> <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</p> <ul> <li> <p>a <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 <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p>equipped with a <strong>symplectic atlas</strong>:</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> consisting of smooth <a class="existingWikiWord" href="/nlab/show/chart">chart</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi_i:U_i\to X</annotation></semantics></math> as usual,</p> </li> <li> <p>such that the transition functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>:</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∩</mo><msub><mi>ϕ</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∩</mo><msub><mi>ϕ</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_j^{-1}\circ\phi_i:\phi_i^{-1}(\phi_i(U_i)\cap\phi_j(U_j))\to \phi_j^{-1}(\phi_i(U_i)\cap\phi_j(U_j))</annotation></semantics></math> preserve the standard symplectic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>0</mn></msub><mo>=</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><msub><mi>dx</mi> <mi>i</mi></msub><mo>∧</mo><msub><mi>dp</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\omega_0=\sum_{i=1}^n dx_i\wedge dp_i</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math> with the basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>p</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1,\ldots,x_n,p_1,\ldots,p_n)</annotation></semantics></math>.</p> </li> </ul> </li> </ul> </div> <div class="num_note" id="TangentCotangentIsomorphism"> <h6 id="remark">Remark</h6> <p>The non-degeneracy of the symplectic form implies that it defines an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</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><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>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega(-,-) : \Gamma(T X) \to \Gamma(T^* X) </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/section">section</a>s of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> – <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>s – and <a class="existingWikiWord" href="/nlab/show/section">section</a>s of the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> – <a class="existingWikiWord" href="/nlab/show/differential+form">differential 1-form</a>s – 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> by the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>∈</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∈</mo><msubsup><mi>T</mi> <mi>x</mi> <mo>*</mo></msubsup><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (v \in T_x X) \mapsto (\omega(v,-) \in T^*_x X) \,. </annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>The vector fields in the image of the exact 1-forms under the isomorphism, remark <a class="maruku-ref" href="#TangentCotangentIsomorphism"></a>, are called <strong><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></strong>.</p> </div> <p>This means that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in C^\infty(X)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>H</mi></mrow><annotation encoding="application/x-tex">d H</annotation></semantics></math> its differential 1-form, the corresponding <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>H</mi></msub><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v_H \in \Gamma(T X)</annotation></semantics></math> is the unique vector field such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>H</mi><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>H</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> d H = \omega(v_H, -) \, </annotation></semantics></math></div> <p>Equivalently, for <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> a <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>ω</mi><mo>=</mo><msub><mi>ω</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>d</mi><msup><mi>x</mi> <mi>i</mi></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">\phi^*\omega = \omega_{i j} d x^i \wedge d x^j</annotation></semantics></math> the symplectic form on this patch, the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">v_H</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>H</mi></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>H</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mo stretchy="false">(</mo><msup><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mo>∂</mo> <mi>j</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> v_H = \frac{\partial H}{\partial x^i} (\omega^{-1})^{i j} \partial_j \,. </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <h3 id="darboux_coordinates">Darboux coordinates</h3> <p>By <a class="existingWikiWord" href="/nlab/show/Darboux%27s+theorem">Darboux's theorem</a> every symplectic manifold has an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> by <a class="existingWikiWord" href="/nlab/show/coordinate+charts">coordinate charts</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><mo>≃</mo><mi>U</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n} \simeq U \hookrightarrow X</annotation></semantics></math> on which the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> takes the canonical form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>=</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mi>d</mi><msup><mi>x</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\omega|_U = \sum_{k = 1}^n d x^{2k} \wedge d x^{2 k+1}</annotation></semantics></math>.</p> <h3 id="SymplecticStructure">Relation to almost symplectic structure</h3> <p>The existence of a 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^2(X)</annotation></semantics></math> which is non-degenerate (but not necessarily closed) is equivalent to the existence of a <a class="existingWikiWord" href="/nlab/show/G-structure">Sp-structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> along the inclusion of the <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> into the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sp(2n) \hookrightarrow GL(2n) \,. </annotation></semantics></math></div> <p>Such an <em>Sp(2n)-structure</em> is also called an <em>almost symplectic structure</em> 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>. Adding the extra condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \omega = 0</annotation></semantics></math> – the condition for <a class="existingWikiWord" href="/nlab/show/integrability+of+G-structures">integrability of G-structures</a> – makes it a genuine symplectic structure. See at <em><a href="integrability+of+G-structures#ExampleSymplecticStructure">integrability of G-structures – Examples – Symplectic structure</a></em>.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a></em> on a symplectic or almost symplectic manifold is in turn <a class="existingWikiWord" href="/nlab/show/lift+of+the+structure+group">lift of the structure group</a> to the <a class="existingWikiWord" href="/nlab/show/metaplectic+group">metaplectic group</a>.</p> <h3 id="relation_to_almost_hermitian_and_khler_structure">Relation to almost Hermitian and Kähler structure</h3> <p>By the <a href="#SymplecticStructure">above</a>, a symplectic manifold structure is an <a class="existingWikiWord" href="/nlab/show/integrable+G-structure">integrable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})</annotation></semantics></math>-structure. Further <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> along the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> inclusion of the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n) \hookrightarrow Sp(2n,\mathbb{R})</annotation></semantics></math> yields is an <a class="existingWikiWord" href="/nlab/show/almost+Hermitian+structure">almost Hermitian structure</a>. If that is again <a class="existingWikiWord" href="/nlab/show/integrable+G-structure">first order integrable</a> then it is <em><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+structure">Kähler structure</a></em>.</p> <p>Such a refinement from symplectic to Kähler structure is also called a choice of <em><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a></em>.</p> <h3 id="symplectomorphisms">Symplectomorphisms</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math> a symplectic manifold, the <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma(T X)</annotation></semantics></math> that generate <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>s that preserve the symplectic structure are precisely the locally <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The condition in question is that the <a class="existingWikiWord" href="/nlab/show/Lie+derivative">Lie derivative</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>v</mi></msub><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> L_v \omega = 0 </annotation></semantics></math></div> <p>vanishes. By <a class="existingWikiWord" href="/nlab/show/Cartan%27s+magic+formula">Cartan's magic formula</a> and using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \omega = 0</annotation></semantics></math> this is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \iota_v \omega = 0 \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> it follows that there is locally a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>H</mi><mo>=</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi></mrow><annotation encoding="application/x-tex">d H = \iota_v \omega</annotation></semantics></math>.</p> </div> <h3 id="poisson_structure">Poisson structure</h3> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> a symplectic manifold, define a bilinear skew-symmetric map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo>:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \{-,-\} : C^\infty(X) \otimes C^\infty(X) \to C^\infty(X) </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo stretchy="false">}</mo><mo>:</mo><mo>=</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mi>F</mi></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mi>H</mi></msub></mrow></msub><mi>ω</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{F,H\} := \iota_{v_F} \iota_{v_H} \omega \,. </annotation></semantics></math></div></div> <p>In a <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a> this says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><mi>F</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><mi>H</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>j</mi></msup></mrow></mfrac><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{F,H\} = (\frac{\partial F}{\partial x^i}) (\omega^{-1})^{i j} (\frac{\partial H}{\partial x^j}) \,. </annotation></semantics></math></div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-,-\}</annotation></semantics></math> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>.</p> </div> <h2 id="examples">Examples</h2> <ul> <li> <p>every <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> is canonically also a symplectic manifold;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a> over any <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a> becomes a symplectic manifold after dividing out symmetries: the reduced <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a>, <a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a>, <a class="existingWikiWord" href="/nlab/show/sesquilinear+form">sesquilinear form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+form">Kähler form</a>, <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+orbifold">symplectic orbifold</a></p> </li> </ul> <p>The notion of symplectic manifold is equivalent to that of <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>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>. (See there.)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+singularity">symplectic singularity</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+resolution">symplectic resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+connection">symplectic connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/canonical+momentum">canonical momentum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holomorphic+symplectic+manifold">holomorphic symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a></p> </li> </ul> <div> <table><thead><tr><th>type of <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_f62603fb534695b8e21ede1fcde98df3436f73ef_1"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner product space</a></th><th>condition on <a class="existingWikiWord" href="/nlab/show/orthogonal">orthogonal</a> space <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_f62603fb534695b8e21ede1fcde98df3436f73ef_2"><semantics><mrow><msup><mi>W</mi> <mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">W^\perp</annotation></semantics></math></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/isotropic+subspace">isotropic subspace</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_f62603fb534695b8e21ede1fcde98df3436f73ef_3"><semantics><mrow><mi>W</mi><mo>⊂</mo><msup><mi>W</mi> <mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">W \subset W^\perp </annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coisotropic+subspace">coisotropic subspace</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_f62603fb534695b8e21ede1fcde98df3436f73ef_4"><semantics><mrow><msup><mi>W</mi> <mo>⊥</mo></msup><mo>⊂</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">W^\perp \subset W</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian+subspace">Lagrangian subspace</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_f62603fb534695b8e21ede1fcde98df3436f73ef_5"><semantics><mrow><mi>W</mi><mo>=</mo><msup><mi>W</mi> <mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">W = W^\perp</annotation></semantics></math></td><td style="text-align: left;">(for <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic space</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_f62603fb534695b8e21ede1fcde98df3436f73ef_6"><semantics><mrow><mi>W</mi><mo>∩</mo><msup><mi>W</mi> <mo>⊥</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W \cap W^\perp = \{0\}</annotation></semantics></math></td><td style="text-align: left;">(for <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>)</td></tr> </tbody></table> </div><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> <h2 id="references">References</h2> <p>See the references at <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>.</p> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion-invariants</a> of almost symplectic structures includes</p> <ul> <li id="AlbuquerquePicken11">Rui Albuquerque, Roger Picken, <em>On invariants of almost symplectic connections</em> (<a href="http://arxiv.org/abs/1107.1860">arXiv:1107.1860</a>)</li> </ul> <p>The generalization of the notion of symplectic manifolds to <a class="existingWikiWord" href="/nlab/show/dg-manifolds">dg-manifolds</a> is sometimes known as <em><a class="existingWikiWord" href="/nlab/show/NQ-supermanifolds">PQ-supermanifolds</a></em> , due to</p> <ul> <li>M. Alexandrov, <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">M. Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">A. Schwarz</a>, O. Zaboronsky, <em>The geometry of the master equation and topological quantum field theory</em>, Int. J. Modern Phys. A 12(7):1405–1429, 1997 (<a href="http://arxiv.org/abs/hep-th/9502010">arXiv:hep-th/9502010</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/symplectic+orbifolds">symplectic orbifolds</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Misha+Verbitsky">Misha Verbitsky</a>, <em>Holomorphic symplectic geometry and orbifold singularities</em>, Asian J. Math. 4 (2000), no. 3, 553-563 (<a href="https://arxiv.org/abs/math/9903175">arXiv:math/9903175</a>)</li> </ul> <p>On a proposal for <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of symplectic manifolds:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vardan+Oganesyan">Vardan Oganesyan</a>, <em>The first step towards symplectic homotopy theory</em> [<a href="https://arxiv.org/abs/2304.10529">arXiv:2304.10529</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 12:55:56. See the <a href="/nlab/history/symplectic+manifold" 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/symplectic+manifold" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2524/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/symplectic+manifold/38" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/symplectic+manifold" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/symplectic+manifold" accesskey="S" class="navlink" id="history" rel="nofollow">History (38 revisions)</a> <a href="/nlab/show/symplectic+manifold/cite" style="color: black">Cite</a> <a href="/nlab/print/symplectic+manifold" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/symplectic+manifold" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>