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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> (2|1)-dimensional Euclidean field theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/16178/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</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/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> <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> </div> </div> <div class="standout"> <p>This is a sub-entry of <a class="existingWikiWord" href="/nlab/show/geometric+models+for+elliptic+cohomology">geometric models for elliptic cohomology</a> and <a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology">A Survey of Elliptic Cohomology</a></p> <p>See there for background and context.</p> <p>This entry here is about the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2|1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/super-cobordism">super-cobordism</a> categories where cobordisms are <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>s, and about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>the</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">the (2|1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>s given by functors on these.</p> </div> <p>Previous:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Axiomatic+field+theories+and+their+motivation+from+topology">Axiomatic field theories and their motivation from topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf">(2,1)-dimensional Euclidean field theories and tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">bordism categories following Stolz-Teichner</a></p> </li> </ul> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#explicit_description_of_'>explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mn>1</mn> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord_{1}^{fam}</annotation></semantics></math></a></li> <li><a href='#explicit_description_of__2'>explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>Bord</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E Bord_{2}</annotation></semantics></math></a></li> <li><a href='#explicit_description_of__3'>explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mn>2</mn> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord_{2}^{fam}</annotation></semantics></math></a></li> <li><a href='#references'>References</a></li> </ul> </div> <h1 id="idea">Idea</h1> <p>Previously we had defined smooth categories of <a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian cobordism</a>s. Now we pass from <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a>s to <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>s and define the corresponding smooth <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a>. Then we define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d|\delta)</annotation></semantics></math>-dimensional Euclidean field theories to be smooth representations of these categories.</p> <p>As described at <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf">(2,1)-dimensional Euclidean field theories and tmf</a>, the idea is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2|1)</annotation></semantics></math>-dimensional Euclidean field theories are a geometric model for <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a>. While there is no complete proof of this so far, in the next and final session</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+forms+from+partition+functions">modular forms from partition functions</a></li> </ul> <p>it will be shows that the claim is true at least for the <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> over the point: the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2|1)</annotation></semantics></math>-dimensional EFT is a modular form. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2|1)</annotation></semantics></math>-dimensional EFTs do yield the correct <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> of <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> over the point.</p> <h1 id="details">Details</h1> <p>Let <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>s.</p> <p>We will define a <a class="existingWikiWord" href="/nlab/show/stack">stack</a>/<a class="existingWikiWord" href="/nlab/show/fibered+category">fibered category</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SDiff</mi></mrow><annotation encoding="application/x-tex">SDiff</annotation></semantics></math> called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>Bord</mi> <mrow><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">E Bord_{2|1}</annotation></semantics></math> whose morphisms are smooth families of (2|1)-dimensional <a class="existingWikiWord" href="/nlab/show/super-cobordism">super-cobordism</a>s, and a <a class="existingWikiWord" href="/nlab/show/stack">stack</a>/<a class="existingWikiWord" href="/nlab/show/fibered+category">fibered category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>sTV</mi> <mi>fam</mi></msup></mrow><annotation encoding="application/x-tex">sTV^{fam}</annotation></semantics></math> of topological super vector bundles.</p> <p>So recall</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/supergeometry+-+contents">supergeometry</a>.</li> </ul> <p><strong>question</strong>: What is the right notion of Riemannian or Euclidean structure on <a class="existingWikiWord" href="/nlab/show/super-cobordism">super-cobordism</a>s?</p> <p><strong>strategy</strong>: From the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> perspective we need some structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> such that the “space” of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps(\Sigma,X)</annotation></semantics></math> naturally carries some measure that allows to perform a <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>.</p> <p>This perspective suggests certain generalizations of the notion of <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> to <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>s which may be a little different than what one might have thought of naively.</p> <p>We want to define <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>s as a generalization of <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> with <em>flat</em> Riemannian metric.</p> <p>notice that there is a canonical bijection between</p> <ul> <li> <p>flat <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>s on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> </li> <li> <p>a maximal <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> consisting of charts such that all transition functions belong the the <strong>Euclidean group</strong> or <strong>Galileo group</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Eucl</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>⋊</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Eucl(\mathbb{R}^d) := \mathbb{R}^d \rtimes O(\mathbb{R}^d) </annotation></semantics></math></div> <p>(rigid translations and rotations)</p> </li> </ul> <p>In analogy to that we define:</p> <p>Similarly a <strong>Euclidean structure</strong> on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d|\delta)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> is defined using the Euclidean <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> given by the <a class="existingWikiWord" href="/nlab/show/semidirect+product">semidirect product</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Eucl</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi></mrow></msup><mo>⋊</mo><mi>Spin</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Eucl(\mathbb{R}^{d|\delta}) := \mathbb{R}^{d|\delta} \rtimes Spin(\mathbb{R}^d) </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/Spin">Spin</a> group acts on the translations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d|\delta}</annotation></semantics></math> in a way to be specified.</p> <p>first recall the notion of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></li> </ul> <p><strong>goal</strong> replace the standard Euclidean group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>,</mo><mi>Eucl</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>d</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{R}^d, Eucl(\mathbb{R}^d))</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,G)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a suitable <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a suitable <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>.</p> <p>This leads to the notion of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>.</li> </ul> <p>The morphisms of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>Bord</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E Bord_{(2|1)}</annotation></semantics></math> will be <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s that are <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>s.</p> <p><strong>goal</strong> define the <a class="existingWikiWord" href="/nlab/show/fibered+category">fibered category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi><msubsup><mi>Bord</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi></mrow> <mi>sfam</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>cSDiff</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ E Bord_{d|\delta}^{sfam} \\ \downarrow \\ cSDiff } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cSDiff</mi></mrow><annotation encoding="application/x-tex">cSDiff</annotation></semantics></math> is the category of <a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a>s.</p> <p>The objects of this fibered category are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Y</mi> <mo>±</mo></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>←</mo></mtd> <mtd><msup><mi>Y</mi> <mi>c</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>S</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y^{\pm} &\to& Y &\leftarrow& Y^c \\ & \searrow & \downarrow & \swarrow \\ && S } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y \to S</annotation></semantics></math> is a family of <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>s of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d|\delta)</annotation></semantics></math>.</p> <p>For the non-super, non-family version of <strong>Euclidean bordism</strong> we require that the core <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Y^c</annotation></semantics></math> is totally geodesic in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <p>now for the superversion we require that there exist charts (in the open atlas) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y \to S</annotation></semantics></math> covering all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Y^c</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>Y</mi><msub><mo>⊃</mo> <mi>open</mi></msub><mi>U</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd></mtd> <mtd><mi>V</mi><mo>⊂</mo><mi>S</mi><mo>×</mo><msubsup><mi>ℝ</mi> <mi>cs</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi></mrow></msubsup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mo>⊃</mo></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>⊃</mo></msup></mtd></mtr> <mtr><mtd><msup><mi>Y</mi> <mi>c</mi></msup><mo>⊃</mo><mi>U</mi><mo>∩</mo><msup><mi>Y</mi> <mi>c</mi></msup></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>V</mi><mo>∩</mo><mi>S</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo stretchy="false">|</mo><mi>δ</mi></mrow></msup><mo>⊂</mo><mi>S</mi><mo>×</mo><msubsup><mi>ℝ</mi> <mi>cs</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo stretchy="false">|</mo><mi>δ</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && S \\ & \nearrow && \nwarrow \\ Y \supset_{open} U &&\stackrel{\phi}{\to}&& V \subset S \times \mathbb{R}^{d|\delta}_{cs} \\ \downarrow^{\supset} &&&& \downarrow^{\supset} \\ Y^c \supset U \cap Y^c &&\stackrel{\simeq}{\to}&& V \cap S \times \mathbb{R}^{d-1|\delta} \subset S \times \mathbb{R}^{d-1|\delta}_{cs} } </annotation></semantics></math></div> <p>next, a <strong>Euclidean superbordism</strong> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>0</mn></msub><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y_0 \to S</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>1</mn></msub><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y_1 \to S</annotation></semantics></math> is a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>Σ</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>S</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y_1 &\stackrel{i_1}{\to}& \Sigma &\stackrel{i_0}{\leftarrow}& Y_1 \\ & \searrow & \downarrow & \swarrow \\ && S } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i_0, i_1</annotation></semantics></math> are morphisms (of families of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,G)</annotation></semantics></math>-spaces) satisfying the (+)-condition and the (c)-condition described at <a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">bordism categories following Stolz-Teichner</a>.</p> <p>Now a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi></mrow> <mi>sfam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord^{sfam}_{d|\delta}</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>S</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">Y_0 \to S_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Y_1 \to S_1</annotation></semantics></math> is a bordism fitting into a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>Y</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msup></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>S</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>S</mi><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma &\stackrel{i_1}{\leftarrow}& f^* Y_1 &\to& Y_1 \\ \uparrow^{i_0} &\searrow& \downarrow && \downarrow \\ Y_1 &\to & S_0 &\stackrel{f}{\to}& S1 } </annotation></semantics></math></div> <p>and we identify bordisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>,</mo><mi>Σ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\Sigma, \Sigma'</annotation></semantics></math> if they are isometric – namely isomorphic in the category of <a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a>s – “rel boundary”.</p> <p><strong>definition</strong> A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d|\delta)</annotation></semantics></math>-dimensional Euclidean field theory</strong> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msubsup><mi>Fun</mi> <mi>csDiff</mi> <mo>⊗</mo></msubsup><mo stretchy="false">(</mo><mi>E</mi><msubsup><mi>Bord</mi> <mrow><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">|</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow> <mi>sfam</mi></msubsup><mo>,</mo><msup><mi>TV</mi> <mi>sfam</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E \in Fun_{csDiff}^\otimes(E Bord_{(d|\delta)}^{sfam}, TV^{sfam}) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/fibered+category">fibered cateories</a> (i. <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a> as well as <a class="existingWikiWord" href="/nlab/show/cartesian+functor">cartesian functor</a> ) over the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cSDiff</mi></mrow><annotation encoding="application/x-tex">cSDiff</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a>s.</p> <p><strong>Definition</strong> (roughly) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>TV</mi> <mi>sfam</mi></msup></mrow><annotation encoding="application/x-tex">TV^{sfam}</annotation></semantics></math> is the category of families of topological vector spaces parameterized by <a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a>s.</p> <p>Recall that the ordinary category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TV</mi></mrow><annotation encoding="application/x-tex">TV</annotation></semantics></math> is the category of complete <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a>, locally convex <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>.</p> <p>define the <a class="existingWikiWord" href="/nlab/show/projective+tensor+product">projective tensor product</a> of two such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>,</mo><mi>W</mi><mo>∈</mo><mi>TV</mi></mrow><annotation encoding="application/x-tex">V, W \in TV</annotation></semantics></math>. This is a certain completion of the algebraic tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><msub><mo>⊗</mo> <mi>alg</mi></msub><mi>W</mi></mrow><annotation encoding="application/x-tex">V \otimes_{alg} W</annotation></semantics></math> with respect to the projective topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><msub><mo>⊗</mo> <mi>alg</mi></msub><mi>W</mi></mrow><annotation encoding="application/x-tex">V \otimes_{alg} W</annotation></semantics></math>.</p> <p>This will be the coarsest <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (the one with the least open sets) making the following maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi><msub><mo>⊗</mo> <mi>alg</mi></msub><mi>W</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><mi>f</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>V</mi><mo>×</mo><mi>W</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V \otimes_{alg} W &\to& Z \\ & \nearrow_{f'} \\ V \times W } </annotation></semantics></math></div> <p>continuous.</p> <p><strong>Remark</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>N</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>alg</mi></msub><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mo>≃</mo></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>⊂</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(M \times N) &\leftarrow& C^\infty(M) \otimes_{alg} C^\infty(N) \\ & {}_{\simeq}\nwarrow & \downarrow^{\subset} \\ && C^\infty(M) \otimes C^\infty(N) } </annotation></semantics></math></div> <p><strong>Definition</strong> the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>TV</mi> <mi>sfam</mi></msup></mrow><annotation encoding="application/x-tex">TV^{sfam}</annotation></semantics></math> are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S,V)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of locally complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-graded <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a> with the structure of a sheaf of modules of the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">O_S</annotation></semantics></math>.</p> <p><strong>goal</strong> define the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2|1)</annotation></semantics></math>-dimensional Euclidean field theory.</p> <p><strong>definition</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> be an EFT as above.</p> <p>We may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}_+ \times h</annotation></semantics></math> (positive axis times upper half plane) as moduli space of Euclidean tori, where for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℓ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">(\ell, \tau) \in \mathbb{R}_+ \times h</annotation></semantics></math> we get a torus (regarded as a <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>) denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{\ell,\tau}</annotation></semantics></math>. This is the torus given by the lattice spanned by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mi>Re</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ell Re(\tau) + Im(\tau)</annotation></semantics></math> in the upper half plane. Then for the ordinary EFT we would define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>E</mi></msub><mo>:</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>h</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> Z_E : \mathbb{R}_+ \times h \to \mathbb {C} </annotation></semantics></math></div><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>ℓ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\ell,\tau) \mapsto E(T_{\ell,\tau}) </annotation></semantics></math></div> <p>For the superversion we put</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>E</mi></msub><mo>:</mo><mo>=</mo><msub><mi>Z</mi> <mrow><msub><mi>E</mi> <mi>red</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> Z_{E} := Z_{E_{red}} </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi><msubsup><mi>Bord</mi> <mrow><mn>2</mn><mo stretchy="false">|</mo><mn>1</mn></mrow> <mi>sfam</mi></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>ρ</mi></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mi>E</mi></msup></mtd></mtr> <mtr><mtd><mi>E</mi><msubsup><mi>Bord</mi> <mrow><mn>2</mn><mo>,</mo><mi>Spin</mi></mrow> <mi>fam</mi></msubsup></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>E</mi> <mi>red</mi></msub></mrow></mover></mtd> <mtd><msup><mi>TV</mi> <mi>fam</mi></msup></mtd> <mtd><mo>↪</mo></mtd> <mtd><msup><mi>TV</mi> <mi>sfam</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && E Bord_{2|1}^{sfam} \\ & {}^{\rho}\nearrow && \searrow^{E} \\ E Bord_{2, Spin}^{fam} &\stackrel{E_{red}}{\to}& TV^{fam} & \hookrightarrow & TV^{sfam} } </annotation></semantics></math></div> <h1 id="examples">Examples</h1> <h2 id="explicit_description_of_">explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mn>1</mn> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord_{1}^{fam}</annotation></semantics></math></h2> <p>See <a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">bordism categories following Stolz-Teichner</a>.</p> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mn>1</mn> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord_1^{fam}</annotation></semantics></math> is generated from</p> <ul> <li> <p>the <em>family of right elbows_</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>1</mn><mo>−</mo><mi>E</mi><msubsup><mi>Bord</mi> <mrow><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow> <mi>fam</mi></msubsup><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>∋</mo></mtd> <mtd><mi>R</mi></mtd> <mtd><mo>:</mo><mo>=</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>ℝ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>S</mi><mo>:</mo><mo>=</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 1-E Bord_{\mathbb{R}_+}^{fam}(\emptyset, pt \coprod pt) & \ni& R & := \mathbb{R}_+ \times \mathbb{R} \\ && \downarrow \\ && S := \mathbb{R}_+ } </annotation></semantics></math></div></li> <li> <p>the point-family of the left elbox</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>L</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>S</mi><mo>:</mo><mo>=</mo><mi>pt</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ L_0 \\ \downarrow \\ S := pt } </annotation></semantics></math></div></li> <li> <p>the family of intervals in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow> <mi>fam</mi></msubsup><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>pt</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E Bord^{fam}_{\mathbb{R}^+}(pt,pt)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>I</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ I \\ \downarrow \\ \mathbb{R}_{\geq 0} } </annotation></semantics></math></div></li> </ul> <p>Because:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msubsup><mi>Fun</mi> <mi>Diff</mi> <mo>⊗</mo></msubsup><mo stretchy="false">(</mo><mi>E</mi><msup><mi>Bord</mi> <mi>fam</mi></msup><mo>,</mo><msup><mi>TV</mi> <mi>fam</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E \in Fun^{\otimes}_{Diff}(E Bord^{fam}, TV^{fam}) </annotation></semantics></math></p> <p>is determined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>pt</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><mi>V</mi><mo>∈</mo><mi>TV</mi></mrow><annotation encoding="application/x-tex"> E(pt) =: V \in TV </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><mi>λ</mi><mo>:</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> E(L_0) =: \lambda : V \otimes V \to \mathbb{R} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><mi>ρ</mi><mo>∈</mo><msub><mi>TV</mi> <mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo>,</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>,</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E(R) =: \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><mi>H</mi></mrow></msup><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>ℝ</mi> <mo>≥</mo></msub><mo>,</mo><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V)) </annotation></semantics></math></div> <p>forms a <em>smooth</em> semigroup under composition generated by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H \in End(V) </annotation></semantics></math></div> <p>(the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi><mo>⊗</mo><mi>V</mi></mtd> <mtd></mtd> <mtd><mover><mo>↪</mo><mi>λ</mi></mover></mtd> <mtd></mtd> <mtd><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mi>ρ</mi></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><mi>H</mi></mrow></msup></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>ℝ</mi> <mo>+</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V \otimes V &&\stackrel{\lambda}{\hookrightarrow}&& End(V) \\ & {}_{\rho}\nwarrow && \nearrow_{e^{- t H}} \\ && \mathbb{R}_+ } </annotation></semantics></math></div> <p>so due to smoothness the data collapses to the infinitesimal data</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (V, \lambda, H) </annotation></semantics></math></div> <p><strong>example – ordinary quantum mechanics</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a>. Then set</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>:</mo><mo>=</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">H:= \Delta</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/Laplace+operator">Laplace operator</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>:</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V := C^\infty(M) \subset L^2(M)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> is the restriction of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(M)</annotation></semantics></math> inner product to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></p> </li> </ul> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><mi>H</mi></mrow></msup></mrow><annotation encoding="application/x-tex">e^{-t H}</annotation></semantics></math> is “trace class” in the non-standard sense described above in that it makes the above diagram commute.</p> <p>So everything as known from standard <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> textbooks, except that we don’t use the full <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of states, but just the <a class="existingWikiWord" href="/nlab/show/Frechet+space">Frechet space</a> of smooth functions.</p> <h2 id="explicit_description_of__2">explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>Bord</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E Bord_{2}</annotation></semantics></math></h2> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mi>Bord</mi> <mn>2</mn></msub><msub><mo></mo><mi>oriented</mi></msub><msup><mo></mo><mi>fam</mi></msup></mrow><annotation encoding="application/x-tex">E Bord_{2}_{oriented}^{fam}</annotation></semantics></math> has the following generators:</p> <p>objects are generated from</p> <ul> <li> <p>the <strong>circle</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>ℓ</mi></msub><mo>:</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mi>ℤ</mi><mo>⋅</mo><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">K_\ell := \mathbb{R}^2/\mathbb{Z}\cdot \ell</annotation></semantics></math> of length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\ell \gt 0</annotation></semantics></math> (with collars!! that’s why it looks like a cylinder of circumference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math>)</p> <p>notice that we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell </annotation></semantics></math> as parameteriing translation by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>⋊</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Eucl</mi> <mi>or</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^2 \rtimes SO(2) = Eucl_{or}(\mathbb{R}^2)</annotation></semantics></math></p> <p>and the circle with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(+)</annotation></semantics></math>/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)</annotation></semantics></math>-collars reversed</p> </li> </ul> <p>morphisms are generated from</p> <ul> <li> <p><strong>cylinders</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{\ell,\tau}</annotation></semantics></math> which as a manifold is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mi>ℤ</mi><mo>⋅</mo><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\simeq \mathbb{R}^2/\mathbb{Z}\cdot \ell</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> parameterizes the embedding of the outgoing circle: the incoming circle is embedded in the canonical way (the identity map on the cylinder, really), while the outgoing circle is embedded by translating the cylinder by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo>⋅</mo><mi>Re</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ell \cdot Re(\tau)</annotation></semantics></math> upwards and rotated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mi>⋯</mi><mi>Im</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ell \cdots Im(\tau)</annotation></semantics></math>.</p> </li> <li> <p><strong>right elbows</strong> which are the same as the cylinder, except that now the second circle is embedded after reflection so that it becomes an ingoing circle.</p> </li> <li> <p>the <strong>thin left elbow</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ℓ</mi></msub></mrow><annotation encoding="application/x-tex">L_\ell</annotation></semantics></math>, similar to the above, with arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\tau = 0</annotation></semantics></math></p> </li> <li> <p>the <strong>torus</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>τ</mi></msub></mrow><annotation encoding="application/x-tex">T_\tau</annotation></semantics></math> obtained from the cylinder by gluing incoming and outgoing</p> </li> </ul> <p><strong>notice</strong> the <strong>pair of pants</strong> is not a morphism in the category at all! since, recall, we require all bordisms to be <em>flat</em> and all boundaries to be <em>geodesics</em> . There is no way to put such a flat metric on the trinion.</p> <p>satisfying the relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>ℓ</mi></msub><mo>∘</mo><msub><mi>R</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub><mo>=</mo><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> L_\ell \circ R_{\ell, \tau} = T_{\ell, \tau} </annotation></semantics></math></div> <p>as for the non-family version, but now also with the new relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo>′</mo><mo>,</mo><mi>τ</mi><mo>′</mo></mrow></msub><mo>=</mo><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> T_{\ell', \tau'} = T_{\ell, \tau} </annotation></semantics></math></div> <p>whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mo>′</mo><mo>=</mo><mi>ℓ</mi><mo>⋅</mo><mo stretchy="false">|</mo><mi>c</mi><mi>τ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\ell' = \ell \cdot|c \tau + d|</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>′</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mi>τ</mi><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mi>τ</mi><mo>+</mo><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\tau' = \frac{a \tau + b }{c \tau + d}</annotation></semantics></math></p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd><mi>c</mi></mtd> <mtd><mi>d</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>∈</mo><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\left(\array{a & b \\ c & d }\right) \in SL_2(\mathbb{Z})</annotation></semantics></math>.</p> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SL_2(\mathbb{Z})</annotation></semantics></math> is generated by</p> <ul> <li> <p>translation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℓ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>ℓ</mi><mo>,</mo><mi>τ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ell, \tau) \mapsto (\ell, \tau + 1)</annotation></semantics></math></p> </li> <li> <p>S-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℓ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>ℓ</mi><mo>⋅</mo><mo stretchy="false">|</mo><mi>τ</mi><mo stretchy="false">|</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mi>τ</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ell, \tau) \mapsto (\ell \cdot |\tau|, - \frac{1}{\tau})</annotation></semantics></math></p> </li> </ul> <p>and now there is <strong>one more relation</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo>,</mo><mi>τ</mi></mrow></msub><mo>=</mo><msub><mi>T</mi> <mrow><mi>ℓ</mi><mo stretchy="false">|</mo><mi>τ</mi><mo stretchy="false">|</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mi>τ</mi></mfrac></mrow></msub></mrow><annotation encoding="application/x-tex"> T_{\ell, \tau} = T_{\ell |\tau|, - \frac{1}{\tau}} </annotation></semantics></math></div> <p>as usual write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mo>=</mo><msup><mi>e</mi> <mrow><mn>2</mn><mi>π</mi><mi>i</mi><mi>τ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">q := e^{2 \pi i \tau}</annotation></semantics></math> which is on the pointed unit disk since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> is half plane since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></p> <h2 id="explicit_description_of__3">explicit description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mn>2</mn> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord_{2}^{fam}</annotation></semantics></math></h2> <p>thwe category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Bord</mi> <mrow><mn>2</mn><mo>,</mo><mi>oriented</mi></mrow> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">E Bord_{2, oriented}^{fam}</annotation></semantics></math> is generated from</p> <p>objects:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>S</mi><mo>=</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{K \\ \downarrow \\ S = \mathbb{R}_+}</annotation></semantics></math></p> </li> <li> <p>morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>L</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>ℝ</mi> <mo>+</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ L \\ \downarrow \\ \mathbb{R}_+ } </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>R</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>h</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z} } </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mo stretchy="false">(</mo><mi>h</mi><mo>∪</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C \\ \downarrow \\ \mathbb{R}_+ \times (h \cup \mathbb{R})/\mathbb{Z} } </annotation></semantics></math></div></li> </ul> <p>subject to the relations</p> <p>… as before (homework 3, problem 4).. and the furhter one:</p> <p>for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>T</mi></mtd> <mtd></mtd> <mtd><msup><mi>α</mi> <mo>*</mo></msup><mi>T</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>h</mi></mtd> <mtd><mover><mo>←</mo><mi>α</mi></mover></mtd> <mtd><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>×</mo><mi>h</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>ℓ</mi><mo>⋅</mo><mo stretchy="false">|</mo><mi>τ</mi><mo stretchy="false">|</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mi>τ</mi></mfrac><mo stretchy="false">)</mo></mtd> <mtd><mover><mrow><mo><</mo><mo>←</mo></mrow><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>ℓ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ T && \alpha^* T \\ \downarrow && \downarrow \\ \mathbb{R}_+ \times h &\stackrel{\alpha}{\leftarrow}& \mathbb{R}_+ \times h \\ \\ (\ell\cdot |\tau|, -\frac{1}{\tau}) &\stackrel{}{\lt\leftarrow}& (\ell, \tau) } </annotation></semantics></math></div> <p>the relation is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>α</mi> <mo>*</mo></msup><mi>T</mi><mo>=</mo><mi>T</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \alpha^* T = T \,. </annotation></semantics></math></div> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em><a class="existingWikiWord" href="/nlab/show/What+is+an+elliptic+object%3F">What is an elliptic object?</a></em> in: <em>Topology, geometry and quantum field theory</em>, London Math. Soc. LNS <strong>308</strong>, Cambridge Univ. Press (2004) 247-343 [<a href="https://math.berkeley.edu/~teichner/Papers/Oxford.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Stolz-Teichner_EllipticObject.pdf" title="pdf">pdf</a>]</p> </li> <li id="Stolz"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Stolz">Stefan Stolz</a> (notes by Arlo Caine), <em>Supersymmetric Euclidean field theories and generalized cohomology</em> Lecture notes (2009) (<a href="http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Stolz">Stefan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>Supersymmetric field theories and generalized cohomology</em> , in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.), <em><a class="existingWikiWord" href="/nlab/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Symposia in Pure Mathematics (2011) [<a href="http://arxiv.org/abs/1108.0189">arXiv:1108.0189</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <em>How do field theories detect the torsion in topological modular forms?</em> [<a href="https://arxiv.org/abs/2303.09138">arXiv:2303.09138</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <em>How do field theories detect the torsion in topological modular forms?</em>, talk at <em><a href="https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html">QFT and Cobordism</a></em>, <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Mar 2023) [<a href="Center+for+Quantum+and+Topological+Systems#BerwickEvansMar23">web</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 17, 2023 at 12:28:22. 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