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<span class="vector-toc-numb">2</span> <span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Algebro-geometric:_as_a_sheaf" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebro-geometric:_as_a_sheaf"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Algebro-geometric: as a sheaf</span> </div> </a> <ul id="toc-Algebro-geometric:_as_a_sheaf-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Concrete:_as_a_smooth_manifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Concrete:_as_a_smooth_manifold"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Concrete: as a smooth manifold</span> </div> </a> <ul id="toc-Concrete:_as_a_smooth_manifold-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Batchelor's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Batchelor's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Batchelor's theorem</span> </div> </a> <ul id="toc-Batchelor's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odd_symplectic_structures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Odd_symplectic_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Odd symplectic structures</span> </div> </a> <button aria-controls="toc-Odd_symplectic_structures-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Odd symplectic structures subsection</span> </button> <ul id="toc-Odd_symplectic_structures-sublist" class="vector-toc-list"> <li id="toc-Odd_symplectic_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Odd_symplectic_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Odd symplectic form</span> </div> </a> <ul id="toc-Odd_symplectic_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Antibracket" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Antibracket"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Antibracket</span> </div> </a> <ul id="toc-Antibracket-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-P_and_SP-manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#P_and_SP-manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>P and SP-manifolds</span> </div> </a> <ul id="toc-P_and_SP-manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laplacian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplacian"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Laplacian</span> </div> </a> <ul id="toc-Laplacian-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-SUSY" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#SUSY"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>SUSY</span> </div> </a> <ul id="toc-SUSY-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none"> Supersymmetric generalization of manifolds</div> <p>In <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>supermanifolds</b> are generalizations of the <a href="/wiki/Manifold" title="Manifold">manifold</a> concept based on ideas coming from <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a>. Several definitions are in use, some of which are described below. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Informal_definition">Informal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=1" title="Edit section: Informal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An informal definition is commonly used in physics textbooks and introductory lectures. It defines a <b>supermanifold</b> as a <a href="/wiki/Manifold" title="Manifold">manifold</a> with both <a href="/wiki/Boson" title="Boson">bosonic</a> and <a href="/wiki/Fermion" title="Fermion">fermionic</a> coordinates. Locally, it is composed of <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">coordinate charts</a> that make it look like a "flat", "Euclidean" <a href="/wiki/Superspace" title="Superspace">superspace</a>. These local coordinates are often denoted by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,\theta ,{\bar {\theta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,\theta ,{\bar {\theta }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4227a24e03b129f1edf9fc1efca4a6075cef43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.653ex; height:3.176ex;" alt="{\displaystyle (x,\theta ,{\bar {\theta }})}"></span></dd></dl> <p>where <i>x</i> is the (<a href="/wiki/Real_number" title="Real number">real-number</a>-valued) <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> coordinate, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228647b7d4a18b6c8c0c390b439a61da8fafec76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \theta \,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c601696fb36006323b35d998005eee37a587d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.676ex;" alt="{\displaystyle {\bar {\theta }}}"></span> are <a href="/wiki/Grassmann_number" title="Grassmann number">Grassmann-valued</a> spatial "directions". </p><p>The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. This includes, among other things a compact definition of <a href="/wiki/Functional_integral" class="mw-redirect" title="Functional integral">functional integrals</a>, the proper treatment of ghosts in <a href="/wiki/BRST_quantization" title="BRST quantization">BRST quantization</a>, the cancellation of infinities in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, Witten's work on the <a href="/wiki/Atiyah-Singer_index_theorem" class="mw-redirect" title="Atiyah-Singer index theorem">Atiyah-Singer index theorem</a>, and more recent applications to <a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">mirror symmetry</a>. </p><p>The use of Grassmann-valued coordinates has spawned the field of <a href="/wiki/Supermathematics" title="Supermathematics">supermathematics</a>, wherein large portions of geometry can be generalized to super-equivalents, including much of <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a> and most of the theory of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> (such as <a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebras</a>, <i>etc.</i>) However, issues remain, including the proper extension of <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> to supermanifolds. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Three different definitions of supermanifolds are in use. One definition is as a sheaf over a <a href="/wiki/Ringed_space" title="Ringed space">ringed space</a>; this is sometimes called the "<a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebro-geometric</a> approach".<sup id="cite_ref-rogers_1-0" class="reference"><a href="#cite_note-rogers-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This approach has a mathematical elegance, but can be problematic in various calculations and intuitive understanding. A second approach can be called a "concrete approach",<sup id="cite_ref-rogers_1-1" class="reference"><a href="#cite_note-rogers-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> as it is capable of simply and naturally generalizing a broad class of concepts from ordinary mathematics. It requires the use of an infinite number of supersymmetric generators in its definition; however, all but a finite number of these generators carry no content, as the concrete approach requires the use of a <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">coarse</a> <a href="/wiki/Topological_space" title="Topological space">topology</a> that renders almost all of them equivalent. Surprisingly, these two definitions, one with a finite number of supersymmetric generators, and one with an infinite number of generators, are equivalent.<sup id="cite_ref-rogers_1-2" class="reference"><a href="#cite_note-rogers-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-rogers-8_2-0" class="reference"><a href="#cite_note-rogers-8-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>A third approach describes a supermanifold as a <a href="/w/index.php?title=Base_topos&action=edit&redlink=1" class="new" title="Base topos (page does not exist)">base topos</a> of a <a href="/w/index.php?title=Superpoint&action=edit&redlink=1" class="new" title="Superpoint (page does not exist)">superpoint</a>. This approach remains the topic of active research.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Algebro-geometric:_as_a_sheaf">Algebro-geometric: as a sheaf</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=3" title="Edit section: Algebro-geometric: as a sheaf"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although supermanifolds are special cases of <a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">noncommutative manifolds</a>, their local structure makes them better suited to study with the tools of standard <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="/wiki/Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed spaces</a>. </p><p>A supermanifold <b>M</b> of dimension (<i>p</i>,<i>q</i>) is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>M</i> with a <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of <a href="/wiki/Superalgebra" title="Superalgebra">superalgebras</a>, usually denoted <i>O<sub><b>M</b></sub></i> or C<sup>∞</sup>(<b>M</b>), that is locally isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} ^{p})\otimes \Lambda ^{\bullet }(\xi _{1},\dots \xi _{q})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⊗</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} ^{p})\otimes \Lambda ^{\bullet }(\xi _{1},\dots \xi _{q})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c6c80bc3c8e9dbbbb4cee64dc8a3f2c55237a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.761ex; height:3.009ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} ^{p})\otimes \Lambda ^{\bullet }(\xi _{1},\dots \xi _{q})}"></span>, where the latter is a Grassmann (Exterior) algebra on <i>q</i> generators. </p><p>A supermanifold <b>M</b> of dimension (1,1) is sometimes called a <a href="/w/index.php?title=Super-Riemann_surface&action=edit&redlink=1" class="new" title="Super-Riemann surface (page does not exist)">super-Riemann surface</a>. </p><p>Historically, this approach is associated with <a href="/wiki/Felix_Berezin" title="Felix Berezin">Felix Berezin</a>, <a href="/w/index.php?title=Dimitry_Leites&action=edit&redlink=1" class="new" title="Dimitry Leites (page does not exist)">Dimitry Leites</a>, and <a href="/wiki/Bertram_Kostant" title="Bertram Kostant">Bertram Kostant</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Concrete:_as_a_smooth_manifold">Concrete: as a smooth manifold</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=4" title="Edit section: Concrete: as a smooth manifold"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A different definition describes a supermanifold in a fashion that is similar to that of a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>, except that the model space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a670215fd4556c78acd92bdc55d472548b7a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.737ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{p}}"></span> has been replaced by the <i>model superspace</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}^{p}\times \mathbb {R} _{a}^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>×</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}^{p}\times \mathbb {R} _{a}^{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4c0df3e5ef86b123d3d0b7e3eb05bafe5febdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.358ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{c}^{p}\times \mathbb {R} _{a}^{q}}"></span>. </p><p>To correctly define this, it is necessary to explain what <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e6edcd7d1b849c15c6cf0412815395b16290cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.622ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{c}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6bfcc68246042e86b9941d519d56facf522178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.78ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{a}}"></span> are. These are given as the even and odd real subspaces of the one-dimensional space of <a href="/wiki/Grassmann_number" title="Grassmann number">Grassmann numbers</a>, which, by convention, are generated by a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of anti-commuting variables: i.e. the one-dimensional space is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \otimes \Lambda (V),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>⊗</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \otimes \Lambda (V),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3deee8e20990a7101f1c7274b909ae1323e2067d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.375ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \otimes \Lambda (V),}"></span> where <i>V</i> is infinite-dimensional. An element <i>z</i> is termed <i>real</i> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d2f0b0d7dfabc9e4cebf14c91771f4de60bd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.331ex; height:2.343ex;" alt="{\displaystyle z=z^{*}}"></span>; real elements consisting of only an even number of Grassmann generators form the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e6edcd7d1b849c15c6cf0412815395b16290cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.622ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{c}}"></span> of <i>c-numbers</i>, while real elements consisting of only an odd number of Grassmann generators form the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6bfcc68246042e86b9941d519d56facf522178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.78ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{a}}"></span> of <i>a-numbers</i>. Note that <i>c</i>-numbers commute, while <i>a</i>-numbers anti-commute. The spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f29576e43219a95e594006b6ffacd60e78a39d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{c}^{p}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{a}^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{a}^{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffb787784ba70a67a10acf0f7fa666216db57d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.78ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{a}^{q}}"></span> are then defined as the <i>p</i>-fold and <i>q</i>-fold Cartesian products of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e6edcd7d1b849c15c6cf0412815395b16290cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.622ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{c}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6bfcc68246042e86b9941d519d56facf522178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.78ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{a}}"></span>.<sup id="cite_ref-dewitt_4-0" class="reference"><a href="#cite_note-dewitt-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Just as in the case of an ordinary manifold, the supermanifold is then defined as a collection of <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">charts</a> glued together with differentiable transition functions.<sup id="cite_ref-dewitt_4-1" class="reference"><a href="#cite_note-dewitt-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This definition in terms of charts requires that the transition functions have a <a href="/wiki/Smooth_structure" title="Smooth structure">smooth structure</a> and a non-vanishing <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>. This can only be accomplished if the individual charts use a topology that is <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">considerably coarser</a> than the vector-space topology on the Grassmann algebra. This topology is obtained by projecting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f29576e43219a95e594006b6ffacd60e78a39d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{c}^{p}}"></span> down to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a670215fd4556c78acd92bdc55d472548b7a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.737ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{p}}"></span> and then using the natural topology on that. The resulting topology is <i>not</i> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, but may be termed "projectively Hausdorff".<sup id="cite_ref-dewitt_4-2" class="reference"><a href="#cite_note-dewitt-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>That this definition is equivalent to the first one is not at all obvious; however, it is the use of the coarse topology that makes it so, by rendering most of the "points" identical. That is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{c}^{p}\times \mathbb {R} _{a}^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>×</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{c}^{p}\times \mathbb {R} _{a}^{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4c0df3e5ef86b123d3d0b7e3eb05bafe5febdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.358ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{c}^{p}\times \mathbb {R} _{a}^{q}}"></span> with the coarse topology is essentially isomorphic<sup id="cite_ref-rogers_1-3" class="reference"><a href="#cite_note-rogers-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-rogers-8_2-1" class="reference"><a href="#cite_note-rogers-8-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{p}\otimes \Lambda ^{\bullet }(\xi _{1},\dots \xi _{q})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>⊗</mo> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{p}\otimes \Lambda ^{\bullet }(\xi _{1},\dots \xi _{q})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3ed57d7db034ab466e3ed6b0798fbdc39f39c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.278ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{p}\otimes \Lambda ^{\bullet }(\xi _{1},\dots \xi _{q})}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold <b>M</b> is contained in its sheaf <i>O<sub><b>M</b></sub></i> of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves. </p><p>An alternative approach to the dual point of view is to use the <a href="/wiki/Functor_of_points" class="mw-redirect" title="Functor of points">functor of points</a>. </p><p>If <b>M</b> is a supermanifold of dimension (<i>p</i>,<i>q</i>), then the underlying space <i>M</i> inherits the structure of a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> whose sheaf of smooth functions is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{M}/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{M}/I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde3139c13d358584eb854daa5791faf10708671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.067ex; height:2.843ex;" alt="{\displaystyle O_{M}/I}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> is the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by all odd functions. Thus <i>M</i> is called the underlying space, or the body, of <b>M</b>. The quotient map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{M}\to O_{M}/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{M}\to O_{M}/I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c7cab8c1831bc1be786bc787c518c70b114970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.413ex; height:2.843ex;" alt="{\displaystyle O_{M}\to O_{M}/I}"></span> corresponds to an injective map <i>M</i> → <b>M</b>; thus <i>M</i> is a submanifold of <b>M</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Let <i>M</i> be a manifold. The <i>odd tangent bundle</i> ΠT<i>M</i> is a supermanifold given by the sheaf Ω(<i>M</i>) of differential forms on <i>M</i>.</li> <li>More generally, let <i>E</i> → <i>M</i> be a <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a>. Then Π<i>E</i> is a supermanifold given by the sheaf Γ(ΛE<sup>*</sup>). In fact, Π is a <a href="/wiki/Functor" title="Functor">functor</a> from the <a href="/w/index.php?title=Category_of_vector_bundles&action=edit&redlink=1" class="new" title="Category of vector bundles (page does not exist)">category of vector bundles</a> to the <a href="/w/index.php?title=Category_of_supermanifolds&action=edit&redlink=1" class="new" title="Category of supermanifolds (page does not exist)">category of supermanifolds</a>.</li> <li><a href="/wiki/Supergroup_(physics)" title="Supergroup (physics)">Lie supergroups</a> are examples of supermanifolds.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Batchelor's_theorem"><span id="Batchelor.27s_theorem"></span>Batchelor's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=7" title="Edit section: Batchelor's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form Π<i>E</i>. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalence of categories</a>. It was published by <a href="/wiki/Marjorie_Batchelor" title="Marjorie Batchelor">Marjorie Batchelor</a> in 1979.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> of Batchelor's theorem relies in an essential way on the existence of a <a href="/wiki/Partition_of_unity" title="Partition of unity">partition of unity</a>, so it does not hold for complex or real-analytic supermanifolds. </p> <div class="mw-heading mw-heading2"><h2 id="Odd_symplectic_structures">Odd symplectic structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=8" title="Edit section: Odd symplectic structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Odd_symplectic_form">Odd symplectic form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=9" title="Edit section: Odd symplectic form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-odd <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic structure</a>. All natural geometric objects on a supermanifold are graded. In particular, the bundle of two-forms is equipped with a grading. An odd symplectic form ω on a supermanifold is a closed, odd form, inducing a non-degenerate pairing on <i>TM</i>. Such a supermanifold is called a <a href="/w/index.php?title=P-manifold&action=edit&redlink=1" class="new" title="P-manifold (page does not exist)">P-manifold</a>. Its graded dimension is necessarily (<i>n</i>,<i>n</i>), because the odd symplectic form induces a pairing of odd and even variables. There is a version of the Darboux theorem for P-manifolds, which allows one to equip a P-manifold locally with a set of coordinates where the odd symplectic form ω is written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\sum _{i}d\xi _{i}\wedge dx_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>d</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\sum _{i}d\xi _{i}\wedge dx_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ad05adc5c6df29286a27068131000c4ed667bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.895ex; height:5.509ex;" alt="{\displaystyle \omega =\sum _{i}d\xi _{i}\wedge dx_{i},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> are even coordinates, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf3bf8407299b66e55ad17ab166e08a93d3466c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.818ex; height:2.509ex;" alt="{\displaystyle \xi _{i}}"></span> odd coordinates. (An odd symplectic form should not be confused with a Grassmann-even <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic form</a> on a supermanifold. In contrast, the Darboux version of an even symplectic form is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}dp_{i}\wedge dq_{i}+\sum _{j}{\frac {\varepsilon _{j}}{2}}(d\xi _{j})^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}dp_{i}\wedge dq_{i}+\sum _{j}{\frac {\varepsilon _{j}}{2}}(d\xi _{j})^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f3278e15f9a091f7ad97354fe46da8650fb768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.628ex; height:6.509ex;" alt="{\displaystyle \sum _{i}dp_{i}\wedge dq_{i}+\sum _{j}{\frac {\varepsilon _{j}}{2}}(d\xi _{j})^{2},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i},q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i},q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dddaebf666d2fcc572fdb85f93a6d998a28e4146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.929ex; height:2.009ex;" alt="{\displaystyle p_{i},q_{i}}"></span> are even coordinates, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf3bf8407299b66e55ad17ab166e08a93d3466c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.818ex; height:2.509ex;" alt="{\displaystyle \xi _{i}}"></span> odd coordinates and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f968845121a91c222089c0edebd9db9250fa2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.993ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{j}}"></span> are either +1 or −1.) </p> <div class="mw-heading mw-heading3"><h3 id="Antibracket">Antibracket</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=10" title="Edit section: Antibracket"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an odd symplectic 2-form ω one may define a <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a> known as the <b>antibracket</b> of any two functions <i>F</i> and <i>G</i> on a supermanifold by </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{F,G\}={\frac {\partial _{r}F}{\partial z^{i}}}\omega ^{ij}(z){\frac {\partial _{l}G}{\partial z^{j}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{F,G\}={\frac {\partial _{r}F}{\partial z^{i}}}\omega ^{ij}(z){\frac {\partial _{l}G}{\partial z^{j}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0718cbf340c9a2e0e211c1395e63d8309927af7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.897ex; height:5.676ex;" alt="{\displaystyle \{F,G\}={\frac {\partial _{r}F}{\partial z^{i}}}\omega ^{ij}(z){\frac {\partial _{l}G}{\partial z^{j}}}.}"></span></dd></dl></dd></dl> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2f68e4448ffa436adfb338fec4afabf2c21bf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.208ex; height:2.509ex;" alt="{\displaystyle \partial _{r}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ddebd8ba3ca5cfeae7246db89c1c62c2d376346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.957ex; height:2.509ex;" alt="{\displaystyle \partial _{l}}"></span> are the right and left <a href="/wiki/Derivative" title="Derivative">derivatives</a> respectively and <i>z</i> are the coordinates of the supermanifold. Equipped with this bracket, the algebra of functions on a supermanifold becomes an <a href="/wiki/Antibracket_algebra" class="mw-redirect" title="Antibracket algebra">antibracket algebra</a>. </p><p>A <a href="/wiki/Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">coordinate transformation</a> that preserves the antibracket is called a <a href="/w/index.php?title=P-transformation&action=edit&redlink=1" class="new" title="P-transformation (page does not exist)">P-transformation</a>. If the <a href="/wiki/Berezinian" title="Berezinian">Berezinian</a> of a P-transformation is equal to one then it is called an <a href="/w/index.php?title=SP-transformation&action=edit&redlink=1" class="new" title="SP-transformation (page does not exist)">SP-transformation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="P_and_SP-manifolds">P and SP-manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=11" title="Edit section: P and SP-manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the <a href="/wiki/Darboux_theorem" class="mw-redirect" title="Darboux theorem">Darboux theorem</a> for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}^{n|n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}^{n|n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c87666ef13f720b5d8c755e01300994a6a0e475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.633ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}^{n|n}}"></span> glued together by P-transformations. A manifold is said to be an <a href="/w/index.php?title=SP-manifold&action=edit&redlink=1" class="new" title="SP-manifold (page does not exist)">SP-manifold</a> if these transition functions can be chosen to be SP-transformations. Equivalently one may define an SP-manifold as a supermanifold with a nondegenerate odd 2-form ω and a <a href="/wiki/Density_function" class="mw-redirect" title="Density function">density function</a> ρ such that on each <a href="/wiki/Coordinate_patch" class="mw-redirect" title="Coordinate patch">coordinate patch</a> there exist <a href="/wiki/Darboux_coordinates" class="mw-redirect" title="Darboux coordinates">Darboux coordinates</a> in which ρ is identically equal to one. </p> <div class="mw-heading mw-heading3"><h3 id="Laplacian">Laplacian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=12" title="Edit section: Laplacian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may define a <a href="/wiki/Laplacian_operator" class="mw-redirect" title="Laplacian operator">Laplacian operator</a> Δ on an SP-manifold as the operator which takes a function <i>H</i> to one half of the <a href="/wiki/Divergence" title="Divergence">divergence</a> of the corresponding <a href="/wiki/Hamiltonian_vector_field" title="Hamiltonian vector field">Hamiltonian vector field</a>. Explicitly one defines </p> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H={\frac {1}{2\rho }}{\frac {\partial _{r}}{\partial z^{a}}}\left(\rho \omega ^{ij}(z){\frac {\partial _{l}H}{\partial z^{j}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta H={\frac {1}{2\rho }}{\frac {\partial _{r}}{\partial z^{a}}}\left(\rho \omega ^{ij}(z){\frac {\partial _{l}H}{\partial z^{j}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52fabe093b31eb0c8cd0de881ef902b059dcdc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.366ex; height:6.176ex;" alt="{\displaystyle \Delta H={\frac {1}{2\rho }}{\frac {\partial _{r}}{\partial z^{a}}}\left(\rho \omega ^{ij}(z){\frac {\partial _{l}H}{\partial z^{j}}}\right).}"></span></dd></dl></dd></dl></dd></dl> <p>In Darboux coordinates this definition reduces to </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ={\frac {\partial _{r}}{\partial x^{a}}}{\frac {\partial _{l}}{\partial \theta _{a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {\partial _{r}}{\partial x^{a}}}{\frac {\partial _{l}}{\partial \theta _{a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0564363b7f7c84734126c3c911c070ae02528f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.967ex; height:5.843ex;" alt="{\displaystyle \Delta ={\frac {\partial _{r}}{\partial x^{a}}}{\frac {\partial _{l}}{\partial \theta _{a}}}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>where <i>x</i><sup><i>a</i></sup> and <i>θ</i><sub><i>a</i></sub> are even and odd coordinates such that </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =dx^{a}\wedge d\theta _{a}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>∧</mo> <mi>d</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =dx^{a}\wedge d\theta _{a}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5303c396f9d236c677a129160c43bc944124c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.829ex; height:2.676ex;" alt="{\displaystyle \omega =dx^{a}\wedge d\theta _{a}.}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>The Laplacian is odd and nilpotent </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{2}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8e876ea085ad001bb97f30ed46de939e1a62b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.898ex; height:2.676ex;" alt="{\displaystyle \Delta ^{2}=0.}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>One may define the <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> of functions <i>H</i> with respect to the Laplacian. In <a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9205088">Geometry of Batalin-Vilkovisky quantization</a>, <a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Albert Schwarz</a> has proven that the integral of a function <i>H</i> over a <a href="/wiki/Lagrangian_submanifold" class="mw-redirect" title="Lagrangian submanifold">Lagrangian submanifold</a> <i>L</i> depends only on the cohomology class of <i>H</i> and on the <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> class of the body of <i>L</i> in the body of the ambient supermanifold. </p> <div class="mw-heading mw-heading2"><h2 id="SUSY">SUSY</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=13" title="Edit section: SUSY"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A pre-SUSY-structure on a supermanifold of dimension (<i>n</i>,<i>m</i>) is an odd <i>m</i>-dimensional distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\subset TM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>⊂</mo> <mi>T</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\subset TM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2d1ac672577ba6ca252a29bdbca5acfc991cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.922ex; height:2.176ex;" alt="{\displaystyle P\subset TM}"></span>. With such a distribution one associates its Frobenius tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}P\mapsto TM/P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>P</mi> <mo stretchy="false">↦</mo> <mi>T</mi> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}P\mapsto TM/P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30506e577785b1d9a5feb1d37759fcd869b12e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.922ex; height:3.176ex;" alt="{\displaystyle S^{2}P\mapsto TM/P}"></span> (since <i>P</i> is odd, the skew-symmetric Frobenius tensor is a symmetric operation). If this tensor is non-degenerate, e.g. lies in an open orbit of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(P)\times GL(TM/P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL(P)\times GL(TM/P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54f6a9e5ea33cabd2910f48e2b23d14da6572a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.01ex; height:2.843ex;" alt="{\displaystyle GL(P)\times GL(TM/P)}"></span>, <i>M</i> is called <i>a SUSY-manifold</i>. SUSY-structure in dimension (1, <i>k</i>) is the same as odd <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">contact structure</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Supergeometry" title="Supergeometry">Supergeometry</a></li> <li><a href="/wiki/Graded_manifold" title="Graded manifold">Graded manifold</a></li> <li><a href="/wiki/Batalin%E2%80%93Vilkovisky_formalism" title="Batalin–Vilkovisky formalism">Batalin–Vilkovisky formalism</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-rogers-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-rogers_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rogers_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-rogers_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-rogers_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Alice_Rogers" title="Alice Rogers">Alice Rogers</a>, <i>Supermanifolds: Theory and Applications</i>, World Scientific, (2007) <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-3203-21-1" title="Special:BookSources/978-981-3203-21-1">978-981-3203-21-1</a> <i>(See <a rel="nofollow" class="external text" href="http://www.worldscientific.com/doi/suppl/10.1142/1878/suppl_file/1878_chap01.pdf">Chapter 1</a>)</i></span> </li> <li id="cite_note-rogers-8-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-rogers-8_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rogers-8_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Rogers, <i>Op. Cit.</i> <i>(See Chapter 8.)</i></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/supermanifold">supermanifold</a> at the <a href="/wiki/NLab" title="NLab"><i>n</i>Lab</a></span> </li> <li id="cite_note-dewitt-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-dewitt_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dewitt_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-dewitt_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">Bryce DeWitt</a>, <i>Supermanifolds</i>, (1984) Cambridge University Press <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521%2B42377%2B5" title="Special:BookSources/0521+42377+5">0521 42377 5</a> <i>(See chapter 2.)</i></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatchelor1979" class="citation cs2">Batchelor, Marjorie (1979), "The structure of supermanifolds", <i>Transactions of the American Mathematical Society</i>, <b>253</b>: 329–338, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1998201">10.2307/1998201</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1998201">1998201</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0536951">0536951</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=The+structure+of+supermanifolds&rft.volume=253&rft.pages=329-338&rft.date=1979&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D536951%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1998201%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1998201&rft.aulast=Batchelor&rft.aufirst=Marjorie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASupermanifold" class="Z3988"></span></span> </li> </ol></div></div> <ul><li>Joseph Bernstein, "<a rel="nofollow" class="external text" href="http://www.math.ias.edu/QFT/fall/">Lectures on Supersymmetry (notes by Dennis Gaitsgory)</a>", <i>Quantum Field Theory program at IAS: Fall Term</i></li> <li>A. Schwarz, "<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9205088">Geometry of Batalin-Vilkovisky quantization</a>", ArXiv hep-th/9205088</li> <li>C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, <i>The Geometry of Supermanifolds</i> (Kluwer, 1991) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-1440-9" title="Special:BookSources/0-7923-1440-9">0-7923-1440-9</a></li> <li>L. Mangiarotti, <a href="/wiki/Gennadi_Sardanashvily" title="Gennadi Sardanashvily">G. Sardanashvily</a>, <i>Connections in Classical and Quantum Field Theory</i> (World Scientific, 2000) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-02-2013-8" title="Special:BookSources/981-02-2013-8">981-02-2013-8</a> (<a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0910.0092">0910.0092</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Supermanifold&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" 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title="Supersymmetry">Supersymmetry</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">General topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Supersymmetric_gauge_theory" title="Supersymmetric gauge theory">Supersymmetric gauge theory</a></li> <li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a></li> <li><a href="/wiki/Super_vector_space" title="Super vector space">Super vector space</a></li> <li><a href="/wiki/Supergeometry" title="Supergeometry">Supergeometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supermathematics" title="Supermathematics">Supermathematics</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Superalgebra" title="Superalgebra">Superalgebra</a></li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li> <li><a href="/wiki/Super-Poincar%C3%A9_algebra" title="Super-Poincaré algebra">Super-Poincaré algebra</a></li> <li><a href="/wiki/Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li> <li><a href="/wiki/Supergroup_(physics)" title="Supergroup (physics)">Supergroup</a></li> <li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Harmonic_superspace" title="Harmonic superspace">Harmonic superspace</a></li> <li><a href="/wiki/Super_Minkowski_space" title="Super Minkowski space">Super Minkowski space</a></li> <li><a class="mw-selflink selflink">Supermanifold</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supercharge" title="Supercharge">Supercharge</a></li> <li><a href="/wiki/R-symmetry" title="R-symmetry">R-symmetry</a></li> <li><a href="/wiki/Supermultiplet" title="Supermultiplet">Supermultiplet</a></li> <li><a href="/wiki/Short_supermultiplet" title="Short supermultiplet">Short supermultiplet</a></li> <li><a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_state" title="Bogomol'nyi–Prasad–Sommerfield state">BPS state</a></li> <li><a href="/wiki/Superpotential" title="Superpotential">Superpotential</a></li> <li><a href="/wiki/D-term" title="D-term">D-term</a></li> <li><a href="/wiki/Fayet%E2%80%93Iliopoulos_D-term" title="Fayet–Iliopoulos D-term"> FI D-term</a></li> <li><a href="/wiki/F-term" title="F-term">F-term</a></li> <li><a href="/wiki/Moduli_(physics)" title="Moduli (physics)">Moduli space</a></li> <li><a href="/wiki/Supersymmetry_breaking" title="Supersymmetry breaking">Supersymmetry breaking</a></li> <li><a href="/wiki/Konishi_anomaly" title="Konishi anomaly">Konishi anomaly</a></li> <li><a href="/wiki/Seiberg_duality" title="Seiberg duality">Seiberg duality</a></li> <li><a href="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten theory</a></li> <li><a href="/wiki/Witten_index" title="Witten index">Witten index</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino_gauge" title="Wess–Zumino gauge">Wess–Zumino gauge</a></li> <li><a href="/wiki/Supersymmetric_localization" title="Supersymmetric localization">Localization</a></li> <li><a href="/wiki/Mu_problem" title="Mu problem">Mu problem</a></li> <li><a href="/wiki/Little_hierarchy_problem" title="Little hierarchy problem">Little hierarchy problem</a></li> <li><a href="/wiki/Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Electric–magnetic duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coleman%E2%80%93Mandula_theorem" title="Coleman–Mandula theorem">Coleman–Mandula</a></li> <li><a href="/wiki/Haag%E2%80%93%C5%81opusza%C5%84ski%E2%80%93Sohnius_theorem" title="Haag–Łopuszański–Sohnius theorem">Haag–Łopuszański–Sohnius</a></li> <li><a href="/wiki/Supersymmetry_nonrenormalization_theorems" title="Supersymmetry nonrenormalization theorems">Nonrenormalization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Field theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li> <li><a href="/wiki/N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li> <li><a href="/wiki/4D_N_%3D_1_global_supersymmetry" title="4D N = 1 global supersymmetry"> 4D N = 1</a></li> <li><a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li> <li><a href="/wiki/Super_QCD" title="Super QCD">Super QCD</a></li> <li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0) superconformal</a></li> <li><a href="/wiki/ABJM_superconformal_field_theory" title="ABJM superconformal field theory">ABJM superconformal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pure_4D_N_%3D_1_supergravity" title="Pure 4D N = 1 supergravity">Pure 4D N = 1 supergravity</a></li> <li><a href="/wiki/4D_N_%3D_1_supergravity" title="4D N = 1 supergravity">4D N = 1 supergravity</a></li> <li><a href="/wiki/N_%3D_8_supergravity" title="N = 8 supergravity">N = 8 supergravity</a></li> <li><a href="/wiki/Higher_dimensional_supergravity" class="mw-redirect" title="Higher dimensional supergravity">Higher dimensional</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D supergravity</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li> <li><a href="/wiki/Gauged_supergravity" title="Gauged supergravity">Gauged supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Superpartner" title="Superpartner">Superpartners</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axino" title="Axino">Axino</a></li> <li><a href="/wiki/Chargino" title="Chargino">Chargino</a></li> <li><a href="/wiki/Gaugino" title="Gaugino">Gaugino</a></li> <li><a href="/wiki/Goldstino" title="Goldstino">Goldstino</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviscalar" title="Graviscalar">Graviscalar</a></li> <li><a href="/wiki/Higgsino" title="Higgsino">Higgsino</a></li> <li><a href="/wiki/Lightest_supersymmetric_particle" title="Lightest supersymmetric particle">LSP</a></li> <li><a href="/wiki/Neutralino" title="Neutralino">Neutralino</a></li> <li><a href="/wiki/R-hadron" title="R-hadron">R-hadron</a></li> <li><a href="/wiki/Sfermion" title="Sfermion">Sfermion</a></li> <li><a href="/wiki/Sgoldstino" title="Sgoldstino">Sgoldstino</a></li> <li><a href="/wiki/Stop_squark" title="Stop squark">Stop squark</a></li> <li><a href="/wiki/Superghost" title="Superghost">Superghost</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Researchers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ian_Affleck" title="Ian Affleck">Affleck</a></li> <li><a href="/wiki/Jonathan_Bagger" title="Jonathan Bagger">Bagger</a></li> <li><a href="/wiki/Marjorie_Batchelor" title="Marjorie Batchelor">Batchelor</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Michael_Dine" title="Michael Dine">Dine</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Jim_Gates" class="mw-redirect" title="Jim Gates">Gates</a></li> <li><a href="/wiki/Yuri_Golfand" title="Yuri Golfand">Golfand</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Claus_Montonen" title="Claus Montonen">Montonen</a></li> <li><a href="/wiki/David_Olive" title="David Olive">Olive</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Warren_Siegel" title="Warren Siegel">Siegel</a></li> <li><a href="/wiki/Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li> <li><a href="/wiki/Alice_Rogers" title="Alice Rogers">Rogers</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Supermanifold&oldid=1250683544">https://en.wikipedia.org/w/index.php?title=Supermanifold&oldid=1250683544</a>"</div></div> <div id="catlinks" class="catlinks" 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