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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="supergeometry">Supergeometry</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="complex_geometry">Complex geometry</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#NonProjectedSuperRiemannSurfaceModuli'>Non-projected super-moduli of super Riemann surfaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesSuperModuliOfSuperRiemannSurfaces'>Super-Moduli space of super Riemann surfaces</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>super Riemann surface</em> is the analog of a <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>.</p> <h2 id="examples">Examples</h2> <p>A <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> with <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> that is also a <a class="existingWikiWord" href="/nlab/show/branched+cover">branched cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^{(1 \vert 1)}</annotation></semantics></math> is canonically endowed with the structure of a super Riemann surface with <a class="existingWikiWord" href="/nlab/show/Ramond+punctures">Ramond punctures</a> (<a href="#DonagiWitten15">Donagi-Witten 15</a>). This implies that <a class="existingWikiWord" href="/nlab/show/adinkras">adinkras</a> encode certain very special super Riemann surfaces <a href="#DoranIgaLandweberMendez-Diez13">Doran & Iga & Landweber & Mendez-Diez 13</a>.</p> <h2 id="properties">Properties</h2> <div> <h3 id="NonProjectedSuperRiemannSurfaceModuli">Non-projected super-moduli of super Riemann surfaces</h3> <p>A <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> is called <em>projected</em> if it <a class="existingWikiWord" href="/nlab/show/retraction">retracts</a> onto its bosonic <a class="existingWikiWord" href="/nlab/show/body">body</a>. (That’s not the wording used in the literature, though.)</p> <p>Since the computation of <a class="existingWikiWord" href="/nlab/show/superstring+scattering+amplitudes">superstring scattering amplitudes</a> involves a <a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a> over the <a class="existingWikiWord" href="/nlab/show/supergeometry">super</a> <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of the given type of super Riemann surfaces, it is of interested to know when this <a class="existingWikiWord" href="/nlab/show/moduli+space+of+super+Riemann+surfaces">moduli space of super Riemann surfaces</a> is projected, as that allows to separate the bosonic from the fermionic sectors of this “<a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>”.</p> <p>However, it turns out that the super-<a class="existingWikiWord" href="/nlab/show/moduli+space+of+super+Riemann+surfaces">moduli space of super Riemann surfaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔐</mi> <mrow><mi>g</mi><mo>,</mo><msub><mi>n</mi> <mi>S</mi></msub><mo>,</mo><msub><mi>n</mi> <mi>R</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{M}_{g, n_S, n_R}</annotation></semantics></math> is generically <em>not</em> projected beyond low <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> (the string’s <a class="existingWikiWord" href="/nlab/show/loop+order">loop order</a>), depending on</p> <ul> <li> <p>the number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">n_S</annotation></semantics></math> of Neveu-punctures</p> </li> <li> <p>the number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">n_R</annotation></semantics></math> of Ramond-punctures.</p> </li> </ul> <p>Specifically:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔐</mi> <mrow><mi>g</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{M}_{g, 0, 0}</annotation></semantics></math> is not projected for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">g \geq 5</annotation></semantics></math>,</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔐</mi> <mrow><mi>g</mi><mo>,</mo><msub><mi>n</mi> <mi>S</mi></msub><mo>≥</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{M}_{g, n_S \geq 1, 0}</annotation></semantics></math> is not projected for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>≥</mo><msub><mi>n</mi> <mi>S</mi></msub><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g \geq n_S + 1</annotation></semantics></math></p> <p>&lbrack;<a href="#DonagiWitten15">Donagi & Witten 2015</a>&rbrack;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔐</mi> <mrow><mi>g</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{M}_{g, 0, 2r \geq 2}</annotation></semantics></math> is not projected for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>5</mn><mi>r</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g \geq 5r + 1</annotation></semantics></math></p> <p>&lbrack;<a href="#DonagiOtt23">Donagi & Ott 2023</a>&rbrack;.</p> </li> </ul> <p>On the other hand, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">g = 2</annotation></semantics></math> (stringy <a class="existingWikiWord" href="/nlab/show/2-loop">2-loop</a>) remains the highest order for which integration over the moduli space has actually been considered/performed, see <a href="string+scattering+amplitude#DHokerPhong02">D’Hoker & Phong 2002</a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+scattering+amplitude">string scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+changing+operator">picture changing operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The concept of super Riemann surfaces originates with the following articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Friedan">Daniel Friedan</a>, <em>Notes On String Theory and 2-Dimensional Conformal Field Theory</em>. 1986</p> </li> <li> <p>M.A. Baranov, <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, <em>On the Multiloop Contribution to the String Theory</em> Int.J.Mod.Phys., A2:1773, 1987</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Critical Dimensions of String Theories and the Dualizing Sheaf on the Moduli Space of (Super) Curves</em>, Funct.Anal.Appl., 20:244-245,</p> <p>1987</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steve+Giddings">Steve Giddings</a>, P. Nelson, <em>The Geometry of super Riemann surfaces</em>, Commun. Math. Phys., 116, (1988), 607</p> </li> </ul> <p>See also</p> <ul> <li id="DHokerPhong05"><a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, <a class="existingWikiWord" href="/nlab/show/Duong+Phong">Duong Phong</a>, <em>Complex geometry and supergeometry</em>, Current Developments in Mathematics 2005 (2007): 1-40 (<a href="https://projecteuclid.org/euclid.cdm/1223654523">euclid:1223654523</a>)</li> </ul> <p>Discussion of super Riemann surfaces induced by <a class="existingWikiWord" href="/nlab/show/supermultiplets">supermultiplets</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>-extended <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d = 1</annotation></semantics></math> supersymmetry – via <a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a> symbols:</p> <ul> <li id="DoranIgaLandweberMendez-Diez13"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Doran">Charles Doran</a>, <a class="existingWikiWord" href="/nlab/show/Kevin+Iga">Kevin Iga</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Landweber">Greg Landweber</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+M%C3%A9ndez-Diez">Stefan Méndez-Diez</a>, <em>Geometrization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi></mrow><annotation encoding="application/x-tex">\mathcal{N}</annotation></semantics></math>-Extended 1-Dimensional Supersymmetry Algebras</em>, Adv. Theor. Math. Phys. <strong>19</strong> 5 (2015) 1043-1113 [<a href="https://arxiv.org/abs/1311.3736">arXiv:1311.3736</a>, <a href="https://dx.doi.org/10.4310/ATMP.2015.v19.n5.a4">doi:10.4310/ATMP.2015.v19.n5.a4</a>, <a href="https://www.charlesdoran.net/uploads/6/7/5/1/6751141/22_geometrization_of_n_extended_1_dimensional_supersymmetry_algebras_i_2015.pdf">pdf</a>]</p> </li> <li id="DoranIgaKostiukMendes-Diez16"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Doran">Charles Doran</a>, <a class="existingWikiWord" href="/nlab/show/Kevin+Iga">Kevin Iga</a>, Jordan Kostiuk, <a class="existingWikiWord" href="/nlab/show/Stefan+M%C3%A9ndez-Diez">Stefan Méndez-Diez</a>, <em>Geometrization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi></mrow><annotation encoding="application/x-tex">\mathcal{N}</annotation></semantics></math>-Extended 1-Dimensional Supersymmetry Algebras II</em>, Adv. Theor. Math> Physics <strong>22</strong> 3 (2018) [<a href="https://arxiv.org/abs/1610.09983">arXiv:1610.09983</a>]</p> </li> </ul> <div> <h3 id="ReferencesSuperModuliOfSuperRiemannSurfaces">Super-Moduli space of super Riemann surfaces</h3> <p>On the <a class="existingWikiWord" href="/nlab/show/moduli+space+of+super+Riemann+surfaces">moduli space of</a> <a class="existingWikiWord" href="/nlab/show/super+Riemann+surfaces">super Riemann surfaces</a> (the <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometric</a> analog of the <a class="existingWikiWord" href="/nlab/show/moduli+space+of+Riemann+surfaces">moduli space of Riemann surfaces</a>):</p> <ul> <li id="Rabin87"> <p><a class="existingWikiWord" href="/nlab/show/Jeffrey+Rabin">Jeffrey Rabin</a>, <em>Supermanifolds and Super Riemann Surfaces</em>, In: H.C. Lee et. al (eds.) <em>Super Field Theories</em>, NATO Science Series (Series B: Physics), vol 160. Springer (1987) (<a href="https://doi.org/10.1007/978-1-4613-0913-0_34">doi:10.1007/978-1-4613-0913-0_34</a>, <a href="https://www.osti.gov/servlets/purl/6687531">pdf</a>)</p> </li> <li id="LeBrunRothstein88"> <p>Claude LeBrun, Mitchell Rothstein, <em>Moduli of super Riemann surfaces</em>, Comm. Math. Phys. Volume 117, Number 1 (1988), 159-176 (<a href="https://projecteuclid.org/euclid.cmp/1104161598">euclid:cmp/1104161598</a>)</p> </li> <li id="Witten12"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Notes On Super Riemann Surfaces And Their Moduli</em>, Pure and Applied Mathematics Quarterly Volume 15 (2019) Number 1 Special Issue on Super Riemann Surfaces and String Theory [<a href="https://dx.doi.org/10.4310/PAMQ.2019.v15.n1.a2">doi:10.4310/PAMQ.2019.v15.n1.a2</a><a href="http://arxiv.org/abs/1209.2459">arXiv:1209.2459</a>]</p> </li> <li id="DonagiWitten15"> <p><a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>: <em>Supermoduli Space Is Not Projected</em>, Proc. Symp. Pure Math. <strong>90</strong> (2015) 19-72 [<a href="https://inspirehep.net/literature/1231519">spire:1231519</a>, <a href="http://arxiv.org/abs/1304.7798">arXiv:1304.7798</a>]</p> </li> <li> <p><a href="http://scgp.stonybrook.edu/archives/10356">Supermoduli Workshop: May 18 – 22, 2015</a>, videos of lecture courses by <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, <a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a> and <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a></p> </li> <li id="CodogniViviani17"> <p>Giulio Codogni, Filippo Viviani, <em>Moduli and Periods of Supersymmetric Curves</em>, Adv. Theor. Math. Phys. 23 (2019) 2, 345-402 (<a href="https://arxiv.org/abs/1706.04910">arXiv:1706.04910</a>, <a href="https://dx.doi.org/10.4310/ATMP.2019.v23.n2.a2">doi:10.4310/ATMP.2019.v23.n2.a2</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ugo+Bruzzo">Ugo Bruzzo</a>, Daniel Hernández Ruipérez, <em>The supermoduli of SUSY curves with Ramond punctures</em>, RACSAM <strong>115</strong> 144 (2021) [<a href="https://doi.org/10.1007/s13398-021-01078-4">doi:10.1007/s13398-021-01078-4</a>, <a href="https://arxiv.org/abs/1910.12236">arXiv:1910.12236</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nadia+Ott">Nadia Ott</a><em>The Supermoduli Space of Genus Zero Susy Curves with Ramond Punctures</em>, PhD thesis, University of Minnesota (2020) [<a href="https://www.proquest.com/openview/7b7eaefa3b66fdc1eb16eab554d54153/1/advanced">proquest:28094035</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nadia+Ott">Nadia Ott</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+A.+Voronov">Alexander A. Voronov</a>: <em>The supermoduli space of genus zero SUSY curves with Ramond punctures</em>, Journal of Geometry and Physics <strong>185</strong> (2023) 104726 [<a href="https://arxiv.org/abs/1910.05655">arXiv:1910.05655</a>, <a href="https://doi.org/10.1016/j.geomphys.2022.104726">doi:10.1016/j.geomphys.2022.104726</a>]</p> </li> <li> <p>Dimitri Skliros: <em>Moving NS Punctures on Super Spheres</em>, SIGMA <strong>20</strong> (2024) 090 [<a href="https://doi.org/10.3842/SIGMA.2024.090">doi:10.3842/SIGMA.2024.090</a>, <a href="https://arxiv.org/abs/2307.06355">arXiv:2307.06355</a>]</p> </li> <li id="DonagiOtt23"> <p><a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, <a class="existingWikiWord" href="/nlab/show/Nadia+Ott">Nadia Ott</a>, <em>Supermoduli Space with Ramond punctures is not projected</em> [<a href="https://inspirehep.net/literature/2688635">spire:2688635</a>, <a href="https://arxiv.org/abs/2308.07957">arXiv:2308.07957</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dimitry+Leites">Dimitry Leites</a>, Alexander S. Tikhomirov: <em>Non-split superstrings of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">|</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1|2)</annotation></semantics></math></em> [<a href="https://arxiv.org/abs/2503.02416">arXiv:2503.02416</a>]</p> </li> </ul> <p>Further discussion of <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometric</a> <a class="existingWikiWord" href="/nlab/show/Teichm%C3%BCller+space">Teichmüller space</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Penner">Robert Penner</a>, <a class="existingWikiWord" href="/nlab/show/Anton+Zeitlin">Anton Zeitlin</a>, <em>Decorated Super-Teichmüller Space</em> (<a href="https://arxiv.org/abs/1509.06302">arXiv:1509.06302</a>)</p> </li> <li> <p>Ivan C.H. Ip, <a class="existingWikiWord" href="/nlab/show/Robert+Penner">Robert Penner</a>, <a class="existingWikiWord" href="/nlab/show/Anton+Zeitlin">Anton Zeitlin</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathcal{N}=2</annotation></semantics></math> Super-Teichmüller Theory</em>, Advances in Mathematics 336 (2018) 409-454 (<a href="https://arxiv.org/abs/1605.08094">arXiv:1605.08094</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anton+Zeitlin">Anton Zeitlin</a>, <em>Super-Teichmüller spaces and related structures</em> (<a href="https://arxiv.org/abs/1811.09939">arXiv:1811.09939</a>)</p> </li> </ul> <p>In relation to <a class="existingWikiWord" href="/nlab/show/fat+graphs">fat graphs</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Albert+S.+Schwarz">Albert S. Schwarz</a>, <a class="existingWikiWord" href="/nlab/show/Anton+M.+Zeitlin">Anton M. Zeitlin</a>, <em>Super Riemann surfaces and fatgraphs</em>, Universe <strong>9</strong> 9 (2023) 384 [<a href="https://doi.org/10.3390/universe9090384">doi:10.3390/universe9090384</a>, <a href="https://arxiv.org/abs/2307.02706">arXiv:2307.02706</a>]</li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on March 5, 2025 at 07:20:28. 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