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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="supergeometry">Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="integration_theory">Integration theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong></p> <table><thead><tr><th>analytic integration</th><th>cohomological integration</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/measure">measure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/virtual+fundamental+class">virtual</a>) <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+integration">Riemann</a>/<a class="existingWikiWord" href="/nlab/show/Lebesgue+integration">Lebesgue integration</a> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">of differential forms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward in generalized cohomology</a>/<a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">in differential cohomology</a></td></tr> </tbody></table> <h2 id="analytic_integration">Analytic integration</h2> <p><a class="existingWikiWord" href="/nlab/show/integral+calculus">integral calculus</a></p> <p><a class="existingWikiWord" href="/nlab/show/Riemann+integration">Riemann integration</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+integration">Lebesgue integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/line+integral">line integral</a>/<a class="existingWikiWord" href="/nlab/show/contour+integration">contour integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a>, <a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a>, <a class="existingWikiWord" href="/nlab/show/fermionic+path+integral">fermionic path integral</a></p> <p><a class="existingWikiWord" href="/nlab/show/Kontsevich+integral">Kontsevich integral</a>, <a class="existingWikiWord" href="/nlab/show/Selberg+integral">Selberg integral</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+Selberg+integral">elliptic Selberg integral</a></p> <h2 id="cohomological_integration">Cohomological integration</h2> <p><a class="existingWikiWord" href="/nlab/show/integration+in+ordinary+differential+cohomology">integration in ordinary differential cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+in+differential+K-theory">integration in differential K-theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a></p> <h2 id="variants">Variants</h2> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a></p> <p><a class="existingWikiWord" href="/nlab/show/Batalin-Vilkovisky+integral">Batalin-Vilkovisky integral</a></p></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#no_topdegree_forms_in_supergeometry'>No top-degree forms in supergeometry</a></li> <li><a href='#solution'>Solution</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>There are different ways to define a <a class="existingWikiWord" href="/nlab/show/volume+form">differential volume element</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. Some of these definitions can be carried over to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, others cannot. The possibly most familiar way of talking about differential volume elements, in terms of top-degree <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s, does <em>not</em> carry over to <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>s.</p> <h3 id="no_topdegree_forms_in_supergeometry">No top-degree forms in supergeometry</h3> <p>In supergeometry the notion of <em>top-degree form</em> does not in general make sense, since there are no top-degree wedge powers of “<a class="existingWikiWord" href="/nlab/show/superdifferential+form">odd 1-forms</a>”: if for instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\theta_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\theta_2</annotation></semantics></math> are odd functions on some <a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p|q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>θ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d \theta_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>θ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d \theta_2</annotation></semantics></math> are their <a class="existingWikiWord" href="/nlab/show/differential+1-forms">differential 1-forms</a>, then the wedge product of these is symmetric in 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><msub><mi>θ</mi> <mn>1</mn></msub><mo>∧</mo><mi>d</mi><msub><mi>θ</mi> <mn>2</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>d</mi><msub><mi>θ</mi> <mn>2</mn></msub><mo>∧</mo><mi>d</mi><msub><mi>θ</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d\theta_1 \wedge d \theta_2 = + d\theta_2 \wedge d \theta_1 \,. </annotation></semantics></math></div> <p>Notice the plus sign on the right, which is the product of one minus sign for interchanging <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\theta_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\theta_2</annotation></semantics></math>, and another minus sign for interchanging the two differentials. See at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em> for more on this.</p> <p>Accordingly, the wedge product of the differential of an odd function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> with itself does not in general vanish:</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>d</mi><mi>θ</mi><mo>∧</mo><mi>d</mi><mi>θ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>θ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (d \theta \wedge d\theta = 0) \Leftrightarrow (\theta = 0) \,. </annotation></semantics></math></div> <p>On the cartesian supermanifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n|m}</annotation></semantics></math> with canonical even coordinate functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi> <mi>i</mi></msup><msubsup><mo stretchy="false">}</mo> <mn>1</mn> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{x^i\}_1^n</annotation></semantics></math> and canonical odd coordinate functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>θ</mi> <mi>j</mi></msup><msubsup><mo stretchy="false">}</mo> <mn>1</mn> <mi>m</mi></msubsup></mrow><annotation encoding="application/x-tex">\{\theta^j\}_1^m</annotation></semantics></math> the differential form which one would want to regard as the canonical <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> is</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>:</mo><mo>=</mo><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mi>n</mi></msup><mo>∧</mo><mi>d</mi><msup><mi>θ</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>θ</mi> <mi>m</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega := d x^1 \wedge \cdots \wedge d x^n \wedge d\theta^1 \wedge \cdots \wedge d\theta^m \,. </annotation></semantics></math></div> <p>Due to the above, this is not a <em>top</em> form, since for instance</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>∧</mo><mi>d</mi><msup><mi>θ</mi> <mn>1</mn></msup><mo>≠</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega \wedge d\theta^1 \neq 0 \,. </annotation></semantics></math></div> <p>But this example also indicates the solution: apparently for integration it is not really essential that a form is a top power, what is rather essential is that it is, locally, the wedge product of a <em>basis</em> of 1-forms. This perspective then does lead to a sensible definition of volume forms (and more generally “integrable forms”) on supermanifolds, described below.</p> <h3 id="solution">Solution</h3> <p>Therefore the naïve identification of differential volume measures with top degree forms has to be refined. The idea is to characterize a volume form by other means, in particular as an equivalence class of choices of <a class="existingWikiWord" href="/nlab/show/bases">bases</a> for the space of 1-forms, and then to define <strong>integrable forms</strong> to be pairs consisting of such a generalized volume form and a <strong>multivector</strong>: this pair is supposed to represent the differential form one would obtain could one contract the multivector with the volume form, as in ordinary differential geometry.</p> <p>The definition of integration of integrable forms in supergeometry in terms of multivector fields leads, in the case that the supermanifold in an <a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a> to the <a class="existingWikiWord" href="/nlab/show/BV+theory">BV formalism</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/integration">integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="DeligneMorgan99"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, Ch 3 in: <em>Notes on Supersymmetry (following <a class="existingWikiWord" href="/nlab/show/Joseph+Bernstein">Joseph Bernstein</a>)</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, <strong>1</strong>, Amer. Math. Soc. Providence (1999) 41-97 [<a href="https://bookstore.ams.org/qft-1-2-s">ISBN:978-0-8218-2014-8</a>, <a href="http://www.math.ias.edu/qft">web version</a>, <a class="existingWikiWord" href="/nlab/files/DeligneMorgan-NotesOnSusy.pdf" title="pdf">pdf</a>]</p> </li> <li id="Mirković04"> <p><a class="existingWikiWord" href="/nlab/show/Ivan+Mirkovi%C4%87">Ivan Mirković</a>, Sec 4-6 in: <em>Notes on Super Math</em>, in <em><a href="https://people.math.umass.edu/~mirkovic/0.SEMINARS/1.QFT/Fall.04.html">Quantum Field Theory Seminar</a></em>, lecture notes (2004) [<a href="https://people.math.umass.edu/~mirkovic/0.SEMINARS/1.QFT/1.SuperMath/8.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Mirkovic-NotesOnSupermathematics.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Exposition:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="http://www.math.uni-hamburg.de/home/schreiber/sin.pdf">Integration over supermanifolds</a>&</em></li> </ul> <p>A general abstract discussion in terms of <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a> theory:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+Paugam">Frédéric Paugam</a>, <em>Homotopical Poisson Reduction of gauge theories</em> (<a href="http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf">pdf</a>)</li> </ul> <p>Geometric discussion of <a class="existingWikiWord" href="/nlab/show/picture+number">picture number</a> appearing in the context of <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a> (and originally seen in the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the <a class="existingWikiWord" href="/nlab/show/NSR+superstring">NSR superstring</a>, crucial in <a class="existingWikiWord" href="/nlab/show/superstring+field+theory">superstring field theory</a>) is due to</p> <ul> <li id="Belopolsky97b"><a class="existingWikiWord" href="/nlab/show/Alexander+Belopolsky">Alexander Belopolsky</a>, <em>Picture changing operators in supergeometry and superstring theory</em> (<a href="https://arxiv.org/abs/hep-th/9706033">arXiv:hep-th/9706033</a>)</li> </ul> <p>and further amplified in</p> <ul> <li id="Witten12"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, appendix D of <em>Notes On Super Riemann Surfaces And Their Moduli</em> (<a href="http://arxiv.org/abs/1209.2459">arXiv:1209.2459</a>)</li> </ul> <p>In this perspective picture number is an extra grading on <a class="existingWikiWord" href="/nlab/show/differential+forms+on+supermanifolds">differential forms on supermanifolds</a> induced from a choice of <em>integral top-form</em> needed to define <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a>:</p> <ul> <li id="CatenacciGrassiNoja18"> <p><a class="existingWikiWord" href="/nlab/show/Roberto+Catenacci">Roberto Catenacci</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Grassi">Pietro Grassi</a>, <a class="existingWikiWord" href="/nlab/show/Simone+Noja">Simone Noja</a>, <em>Superstring Field Theory, Superforms and Supergeometry</em>, Journal of Geometry and Physics Volume 148, February 2020, 103559 (<a href="https://arxiv.org/abs/1807.09563">arXiv:1807.09563</a>)</p> <p>(with an eye towards <a class="existingWikiWord" href="/nlab/show/superstring+field+theory">superstring field theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C.+A.+Cremonini">C. A. Cremonini</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Grassi">Pietro Grassi</a>, <em>Pictures from Super Chern-Simons Theory</em> (<a href="https://arxiv.org/abs/1907.07152">arXiv:1907.07152</a>)</p> <p>(with an eye towards <a class="existingWikiWord" href="/nlab/show/super+Chern-Simons+theory">super Chern-Simons theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C.+A.+Cremonini">C. A. Cremonini</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Grassi">Pietro Grassi</a>, S. Penati, <em>Supersymmetric Wilson Loops via Integral Forms</em> (<a href="https://arxiv.org/abs/2003.01729">arXiv:2003.01729</a>)</p> <p>(for super-<a class="existingWikiWord" href="/nlab/show/Wilson+lines">Wilson lines</a>)</p> </li> </ul> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pietro+Grassi">Pietro Grassi</a>, <em>Integral Forms and Applications</em>, Sestri Levante 2015 (<a href="https://agenda.infn.it/event/8823/attachments/55101/64989/Grassi-SL.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GrassiIntegralForms2015.pdf" title="pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li>Sergio L. Cacciatori, Simone Noja, Riccardo Re, <em>The Unifying Double Complex on Supermanifolds</em> (<a href="https://arxiv.org/abs/2004.10906">arXiv:2004.10906</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 9, 2024 at 03:21:15. 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