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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="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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#general_idea'>General idea</a></li> <li><a href='#history'>History</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#BCobordism'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-Cobordisms</a></li> <li><a href='#HQFTs'>Homotopy Quantum Field Theories</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#11_dimensional_hqfts_with_background_a_'>1+1 dimensional HQFTs with background a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(G,1)</annotation></semantics></math></a></li> <li><a href='#equivariant'>Equivariant</a></li> </ul> <li><a href='#history_2'>History</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general_idea">General idea</h3> <p>What are called a <strong>homotopy quantum field theories</strong> (HQFTs) are <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theories">topological quantum field theories</a> (TQFT) defined on <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> that are equipped with the extra <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> into a given <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>Hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bord_n(B)</annotation></semantics></math> denotes a <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a> suitably equipped with maps into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, then an HQFT is a <a class="existingWikiWord" href="/nlab/show/monoidal+functor">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>Z</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Bord</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><msup><mo stretchy="false">)</mo> <mo lspace="thinmathspace" rspace="thinmathspace">∐</mo></msup><mo>⟶</mo><msup><mi>Vect</mi> <mo>⊗</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z \;\colon\; Bord_n(B)^{\coprod} \longrightarrow Vect^{\otimes} \,. </annotation></semantics></math></div> <h3 id="history">History</h3> <p>HQFTs were first defined (under a different name) by <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a> as early as 1988 in (<a href="#Segal88">Segal 88</a>).</p> <p>Starting from 1991, several papers by <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, including <a href="#FreedQuinn1991">FreedQuinn1991</a>, further developed the notion of a <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a>, where bordisms in the domain category are equipped with a map to a target space (such as the <a class="existingWikiWord" href="/nlab/show/Eilenberg%E2%80%93MacLane+space">Eilenberg–MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(G,1)</annotation></semantics></math> for a finite group <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>). See, in particular, Theorem 1.7 and the preceding discussion on page 6 of <a href="#FreedQuinn1991">FreedQuinn1991</a>.</p> <p>From 1999, HQFTs were studied systematically by <a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a> (<a href="#Turaev99">Turaev 99</a>) for 2-dimensional <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>/<a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> and extended to 3-dimensional ones in (<a href="#Turaev00">Turaev 00</a>). Turaev also introduced the term “homotopy quantum field theory”.</p> <p>At about the same time, (<a href="#BrightwellTurner00">Brightwell-Turner 00</a>) looked at what they called the <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> surface category and its <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s. There are two viewpoints which interact and complement each other. Turaev’s seems to be to see HQFTs as an extension of the tool kit for studying <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> already given by <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>s, whilst in Brightwell and Turner’s, it is the ‘background space’, which is probed by the surfaces in the sense of <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a>.</p> <p>In the proof of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> in (<a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">Lurie 09</a>) the concept of HQFTs was refined to <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> by considering an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bord_n(X)</annotation></semantics></math> with maps to a given <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</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>. For these the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> essentially says (see at <a href="cobordism+hypothesis#StatementForCobordismsInAManifold">For framed cobordisms in a topological space</a>) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mo lspace="thinmathspace" rspace="thinmathspace">∐</mo></msup></mrow><annotation encoding="application/x-tex">Bord_n(X)^\coprod</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+construction">free construction</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+with+duals">(∞,n)-category with duals</a> on the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <p>An HQFT is going to be defined as assigning data to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordisms. We first introduce these</p> <ul> <li><a href="#BCobordism">B-cobordisms</a></li> </ul> <p>and then define</p> <ul> <li><a href="#HQFTs">HQFTs</a></li> </ul> <p>themselves in terms of these.</p> <h3 id="BCobordism"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-Cobordisms</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifold</strong> is a pair <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 closed oriented <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>-<a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> (with a choice of base point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math> in each connected component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>), and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g : X \to B</annotation></semantics></math>, called the <strong>characteristic map</strong>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msub><mi>m</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">g(m_i) = \ast</annotation></semantics></math> for each base point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphism</strong> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifolds, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : ( X, g) \to ( Y, h)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\phi : X \to Y</annotation></semantics></math> of the manifolds, preserving the orientation, taking base points into base points and such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mi>ϕ</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">h\phi = g</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If as is often the case, the manifolds under consideration will be <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and then ‘<a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>’ is interpreted as ‘<a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>’, but equally well we can position the theory in the category of PL-manifolds or triangulable topological manifolds with the obvious changes. In fact for some of the time it is convenient to develop constructions for simplicial complexes rather than manifolds, as it is triangulations that provide the basis for the combinatorial descriptions of the structures that we will be using.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Man</mi></mstyle><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Man}(n,B)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/category">category</a> 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">n</annotation></semantics></math>-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifolds and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphisms. We define a ‘sum’ operation on this category using <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a>. The disjoint union of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifolds is defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⨿</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⨿</mo><mi>Y</mi><mo>,</mo><mi>g</mi><mo>⨿</mo><mi>h</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">( X, g) \amalg ( Y, h) := ( X\amalg Y, g\amalg h),</annotation></semantics></math></div> <p>with the obvious characteristic map, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>⨿</mo><mi>h</mi><mo>:</mo><mi>X</mi><mo>⨿</mo><mi>Y</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g\amalg h : X \amalg Y \to B</annotation></semantics></math>. With this ‘sum’ operation, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Man</mi></mstyle><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Man}(n,B)</annotation></semantics></math> becomes a symmetric monoidal category with the unit being given by the empty <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifold, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math>, with the empty characteristic map. Of course, this is an <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>-manifold by default.</p> </div> <p>These <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifolds are the objects of interest, but they have to be related by the analogue of <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> for this setting.</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordism</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W,F)</annotation></semantics></math>, from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_0,g)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,h)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">W : X_0 \to X_1</annotation></semantics></math> endowed with a <a class="existingWikiWord" href="/nlab/show/homotopy+class">homotopy class</a> relative to the boundary of a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>W</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">F : W \to B</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><mi>F</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><mo>=</mo><mi>g</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>F</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow></msub><mo>=</mo><mi>h</mi></mrow><annotation encoding="application/x-tex"> F|_{X_0} = g \,, \;\;\;\; F|_{X_1} = h </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Generally, unless necessary in this entry, we will not make a notational distinction between the <a class="existingWikiWord" href="/nlab/show/homotopy+class">homotopy class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and any of its representatives.</p> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordisms</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mi>W</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo><msup><mi>F</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi : (W,F) \to (W^\prime, F^\prime)</annotation></semantics></math>, is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>W</mi><mo>→</mo><msup><mi>W</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><annotation encoding="application/x-tex">\psi : W \to W^\prime</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><mi>ψ</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mo>+</mo></msub><mi>W</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∂</mo> <mo>+</mo></msub><msup><mi>W</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo><mspace width="1em"></mspace><mi>ψ</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mo>−</mo></msub><mi>W</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∂</mo> <mo>−</mo></msub><msup><mi>W</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo></mrow><annotation encoding="application/x-tex">\psi (\partial_+W) = \partial_+W^\prime, \quad \psi (\partial_-W) = \partial_-W^\prime,</annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mi>ψ</mi><mo>=</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">F^\prime \psi = F</annotation></semantics></math>, in the obvious sense of homotopy classes relative to the boundary.</p> </div> <p>We can glue <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordisms along their boundaries, or more generally, along a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphism between their boundaries, in the usual way. This gives rise to a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>HCobord</mi></mstyle><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{HCobord}(n,B)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordisms</p> <p>The detailed structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> and the resulting category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>HCobord</mi></mstyle><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{HCobord}(n,B)</annotation></semantics></math> is given in (<a href="#Rodrigues03">Rodrigues 03, appendix</a>), at least in the important case of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifolds. This category is a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with strict <a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a>.</p> <h3 id="HQFTs">Homotopy Quantum Field Theories</h3> <p>The general absract definition of an HQFT is now the following.</p> <p>Fix an integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \geq 0</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/field">field</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. All vector spaces will be tacitly assumed to be finite dimensional. In general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> can be replaced by a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> merely by replacing finite dimensional vector spaces by projective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> of finite type, but we will not do this here.</p> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>A <strong>homotopy quantum field theory</strong> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>HCobord</mi></mstyle><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{HCobord}(n,B)</annotation></semantics></math> to the category, <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, of finite dimensional vector spaces over the <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> </div> <p>This definiting unwinds to the following structure in components</p> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n + 1)</annotation></semantics></math>-dimensional homotopy quantum field theory</strong>, <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>, with background <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> assigns</p> <ul> <li> <p>to any <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>-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifold, <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>, a vector space, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">\tau{(X,g)}</annotation></semantics></math>,</p> </li> <li> <p>to any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphism, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : (X, g) \to ( Y, h)</annotation></semantics></math>, 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">n</annotation></semantics></math>-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-manifolds, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-linear isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>τ</mi><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>→</mo><mi>τ</mi><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">\tau(\phi) : \tau{(X, g)} \to \tau{( Y, h)}</annotation></semantics></math>,</p> </li> </ul> <p>and</p> <ul> <li>to any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordism, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W,F) : (X_0,g_0) \to (X_1,g_1)</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-linear transformation, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>:</mo><mi>τ</mi><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>→</mo><mi>τ</mi><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">\tau(W) : \tau{(X_0,g_0)} \to \tau{(X_1,g_1)}</annotation></semantics></math>.</li> </ul> <p>These assignments are to satisfy the following axioms:</p> <ol> <li> <p><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 functorial in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Man</mi></mstyle><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Man}(n,B)</annotation></semantics></math>, i.e., for two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphisms, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi: (X, g) \to ( Y, h)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : ( Y, h) \to (P,j)</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mi>ψ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\tau(\phi\psi) = \tau(\phi)\tau(\psi),</annotation></semantics></math> and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">1_{(X,g)}</annotation></semantics></math> is the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphism on <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>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mrow><mi>τ</mi><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow></msub></mrow><annotation encoding="application/x-tex">\tau(1_{(X,g)}) = 1_{\tau{(X,g)}}</annotation></semantics></math></p> </li> <li> <p>There are natural isomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mi>τ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⨿</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">c_{(X,g),(Y,h)} : \tau((X,g)\amalg (Y,h)) \cong \tau(X,g)\otimes \tau(Y,h),</annotation></semantics></math></div> <p>and an isomorphism, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">u : \tau(\emptyset) \cong K</annotation></semantics></math>, that satisfy the usual axioms for a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordisms, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W,F) : (X,g) \to (Y,h)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><msup><mi>Y</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo><msup><mi>h</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,G): (Y^\prime, h^\prime) \to (P,j)</annotation></semantics></math> glued along a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mi>Y</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo><msup><mi>h</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi :(Y,h) \to (Y^\prime,h^\prime)</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><msub><mo>⨿</mo> <mi>ψ</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\tau((W,F)\amalg_\psi (V,G))= \tau(V,G)\tau(\psi)\tau(W,F).</annotation></semantics></math></p> </li> <li> <p>For the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordism, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>I</mi><mo>×</mo><mi>X</mi><mo>,</mo><msub><mn>1</mn> <mi>g</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1_{(X,g)} = (I\times X, 1_g)</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">\tau( 1_{(X,g)}) = 1_{\tau(X,g)}.</annotation></semantics></math></p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W,F) : (X,g) \to (Y,h)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo><msup><mi>g</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mi>Y</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>,</mo><msup><mi>h</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,G) : (X^\prime,g^\prime) \to (Y^\prime,h^\prime)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo><mo>:</mo><mi>∅</mi><mo>→</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">(P,J): \emptyset \to \emptyset</annotation></semantics></math>, some fairly obvious diagrams are commutative.</p> </li> </ol> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>These axioms are slightly different from those given in the original paper of Turaev in 1999. The really significant difference is in axiom 4, which is weaker than as originally formulated, where any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cobordism structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">I \times X</annotation></semantics></math> was considered as trivial. The effect of this change is important as it is now the case that the HQFT is determined by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-type of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, cf. (<a href="#Rodrigues03">Rodrigues 03</a>).</p> <p>With the revised version of the axioms, it becomes possible to attempt to classify HQFTs with a given <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. Turaev did this in the original paper with <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">n = 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(G,1)</annotation></semantics></math>. The results of Brightwell and Turner essentially gave the solution for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(A,2)</annotation></semantics></math>.</p> </div> <h2 id="examples">Examples</h2> <h3 id="11_dimensional_hqfts_with_background_a_">1+1 dimensional HQFTs with background a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(G,1)</annotation></semantics></math></h3> <p>If we look at the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n= 1</annotation></semantics></math> and with background an <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+space">Eilenberg-Mac Lane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(G,1)</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</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>, then HQFTs correspond to <a class="existingWikiWord" href="/nlab/show/crossed+G-algebra">crossed G-algebras</a>, in much the same way that commutative <a class="existingWikiWord" href="/nlab/show/Frobenius+algebras">Frobenius algebras</a> correspond to <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a>s. There the correspondence is given by a <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math>, corresponds to the <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(S^1)</annotation></semantics></math>. This is because the circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a>, sometimes called a Frobenius object, in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Bord_2</annotation></semantics></math> of 2d-<a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s between 1-manifolds.</p> <p>In the case of HQFTs, the role of the circle is replaced by the family of circles with characteristic maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. Each one gives, combinatorially, a circle together with a labelling of the boundary by an element of <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>. (It does not seem to be known how to get 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>-graded version of an abstract Frobenius object that will correspond to this situation, although this is probably not too hard to do.)</p> <h3 id="equivariant">Equivariant</h3> <p>In (<a href="#MooreSegal06">Moore-Segal 06</a>) are discussed <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>-<span class="newWikiWord">equivariant TFT<a href="/nlab/new/equivariant+TFT">?</a></span>s and it is shown that they naturally correspond to a simple case of Turaev’s HQFTs. They relate (1+1) equivariant TFTs to Turaev’s <a class="existingWikiWord" href="/nlab/show/crossed+G-algebras">crossed G-algebras</a> (which they call Turaev algebras).</p> <h2 id="history_2">History</h2> <p>The original definition is due to <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, who introduced them under the term <em><span class="newWikiWord">elliptic objects<a href="/nlab/new/elliptic+objects">?</a></span></em>. Specifically, in <a href="#Segal88">Segal 88</a>, we read in §6:</p> <blockquote> <p>6. Speculation about the definition of elliptic cohomology</p> <p>For any space <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> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒫</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{P}_X</annotation></semantics></math> be the category whose objects are the points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and whose morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math> are the paths in <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> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math>, two such paths being identified if they differ only by reparametrization. A functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒫</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{P}_X</annotation></semantics></math> to finite dimensional vector spaces is essentially the same thing as a vector bundle 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> with a connection. (The functor must be continuous in a suitable sense.) It is well known how K-theory is constructed from such objects.</p> <p>I have described elsewhere [23] a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> whose objects are all compact oriented one-dimensional manifolds, and whose morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">S_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S_1</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>Σ</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Sigma,\alpha)</annotation></semantics></math>, where <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> is a Riemann surface with boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\partial\Sigma</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is an isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\partial X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>S</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">S_1-S_0</annotation></semantics></math>. Two pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Sigma,\alpha)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mo>′</mo><mo>,</mo><mi>α</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Sigma',\alpha')</annotation></semantics></math> are identified if they are isomorphic. For any space <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> one can now define a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_X</annotation></semantics></math>. Its objects 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>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S,s)</annotation></semantics></math>, where <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> is an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s \colon S \to X</annotation></semantics></math> is a map. Its morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>s</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S_0,s_0)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S_1,s_1)</annotation></semantics></math> are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Sigma,\alpha,\sigma)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(\Sigma,\alpha)\colon S_0 \to S_1</annotation></semantics></math> is a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\sigma\colon\Sigma\to X</annotation></semantics></math> is a map compatible with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>O</mi></msub><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s_O,s_1)</annotation></semantics></math>. The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_X</annotation></semantics></math> is a natural analogue of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒫</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{P}_X</annotation></semantics></math> which gives rise to vector bundles.</p> <p>It is appropriate to consider functors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> to the category <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">\mathcal{V}</annotation></semantics></math> of topological vector spaces and trace-class maps. If such a functor <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> is holomorphic in the natural sense then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(S^1)</annotation></semantics></math> is a positive energy representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Diff(S^1)</annotation></semantics></math> of finite type. More precisely, as is familiar in the representation theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Diff(S^1)</annotation></semantics></math>, one must consider projective representations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> of some definite positive integral level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. Imposing a further condition — the contraction condition below — on the functor <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> ensures that the character of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(S^1)</annotation></semantics></math> is a modular form of weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>Now let us define an elliptic object of level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> 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> as a projective functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mi>X</mi></msub><mo>→</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">E\colon\mathcal{C}_X\to \mathcal{V}</annotation></semantics></math> of level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> which is holomorphic and satisfies the contraction condition. Such an object consists of an infinite dimensional vector bundle on the loop space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>, equivariant under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Diff(S^1)</annotation></semantics></math>, together with some additional data amounting to a kind of connection. The primary example is the spin bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}X</annotation></semantics></math>, which is defined when <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 spin manifold with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_1=0</annotation></semantics></math>.</p> <p>I have nothing precise to say about elliptic objects, but it seems to me quite likely that the objects of each level lead to an interesting cohomology theory, and that the theories for different levels are related by “Bott maps”. That would fit in well with Theorem (5.3), for just as elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(B G)</annotation></semantics></math> are elated to flat bundles, so elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ell</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ell^*(B G)</annotation></semantics></math> seem to be related to flat elliptic objects, i.e. ones such that the operator associated to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Sigma,\alpha,\sigma)</annotation></semantics></math> depends 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> only up to homotopy, and is therefore a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\pi_1(\Sigma)\to G</annotation></semantics></math>.</p> <p>I should mention that the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> can be modified by equipping the Riemann surfaces <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> with chosen spin structures. That is certainly needed to obtain genuine elliptic cohomology.</p> <p>Finally I return to the “contraction property”. This is motivated by the path-integral point of view. If a surface <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> is a morphism from <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> to itself then the trace of the operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(\Sigma)\colon E(S)\to E(S)</annotation></semantics></math> associated to <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> must depend only on the <em>closed</em> surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Σ</mi><mo stretchy="false">ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\check\Sigma</annotation></semantics></math> obtained by attaching the two boundary pieces of <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> to each other. Thus if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>τ</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_\tau</annotation></semantics></math> is the annulus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mo stretchy="false">|</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>τ</mi></mrow></msup><mo stretchy="false">|</mo><mo>≤</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo>≤</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{z \in \mathbf{C} \mid |e^{i\tau}|\le |z|\le 1\}</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>Σ</mi><mo stretchy="false">ˇ</mo></mover> <mi>τ</mi></msub><mo>≅</mo><msub><mover><mi>Σ</mi><mo stretchy="false">ˇ</mo></mover> <mrow><mi>τ</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\check\Sigma_\tau\cong \check\Sigma_{\tau'}</annotation></semantics></math>, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>′</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau'=-1/\tau</annotation></semantics></math>, and therefore the trace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mi>τ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(\Sigma_\tau)</annotation></semantics></math> is invariant under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau\mapsto -1/\tau</annotation></semantics></math>.</p> <p>Brylinski [9] has proposed a similar approach to elliptic cohomology.</p> <p>Postscript. After giving this talk, I learnt of the work [29] 1 which gives a good account of the Dirac operator on loop space from the path integral point of view.</p> </blockquote> <h2 id="references">References</h2> <p>The original definition is due to <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a> (who introduced them under the name (flat) <em><span class="newWikiWord">elliptic objects<a href="/nlab/new/elliptic+objects">?</a></span></em>), see §6 of</p> <ul> <li id="Segal88"><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Elliptic cohomology (after Landweber–Stong, Ochanine, Witten, and others)</em>, Séminaire Bourbaki, 40e année, 1987–88, No. 695 (<a href="http://www.numdam.org/item/?id=SB_1987-1988__30__187_0">numdam:SB_1987-1988__30__187_0</a>)</li> </ul> <p>These ideas were further developed in</p> <ul> <li id="FreedQuinn1991"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+S.+Freed">Daniel S. Freed</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Quinn">Frank Quinn</a>, <em>Chern-Simons Theory with Finite Gauge Group</em>, <a href="https://arxiv.org/abs/hep-th/9111004">arXiv:hep-th/9111004</a>.</p> </li> <li id="Quinn95"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Quinn">Frank Quinn</a>, <em>Lectures on axiomatic topological quantum field theory</em>, in <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Karen+Uhlenbeck">Karen Uhlenbeck</a> (eds.) <em>Geometry and Quantum Field Theory</em> <strong>1</strong> (1995) [<a href="https://doi.org/10.1090/pcms/001">doi:10.1090/pcms/001</a>]</p> </li> </ul> <p>The theory of HQFTs was developed in</p> <ul> <li id="Turaev99"> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Homotopy field theory in dimension 2 and group-algebras</em> (<a href="http://arxiv.org/abs/math.QA/9910010">arXiv:math.QA/9910010</a>)</p> </li> <li id="Turaev00"> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Homotopy field theory in dimension 3 and crossed group-categories</em> (<a href="http://arxiv.org/abs/math.GT/0005291">arXiv:math.GT/0005291</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a> (with appendices by <a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a> and <a class="existingWikiWord" href="/nlab/show/Alexis+Virelizier">Alexis Virelizier</a>): <em>Homotopy Quantum Field Theory</em>, Tracts in Mathematics <strong>10</strong>, European Mathematical Society (2010) [<a href="https://ems.press/books/etm/79">ems:etm/79</a>]</p> </li> <li id="BrightwellTurner00"> <p>M. Brightwell and P. Turner, <em>Representations of the homotopy surface category of a simply connected space</em>, J. Knot Theory and its Ramifications, 9 (2000), 855–864.</p> </li> <li id="Rodrigues03"> <p>G. Rodrigues, <em>Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1 + 1</em>, J. Knot Theory and its Ramifications, 12 (2003) 287–317 (<a href="http://arxiv.org/abs/math.QA/0105018">arXiv:math.QA/0105018</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Porter">T. Porter</a> and <a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">V. Turaev</a>, <em>Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-algebras</em>, Journal of Homotopy and Related Structures 3(1), 2008, 113–159. (<a href="http://arxiv.org/abs/math.QA/0512032">arXiv:math.QA/0512032</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Formal Homotopy Quantum Field Theories II: Simplicial Formal Maps</em>, Cont. Math. 431, p. 375 - 404 (Streetfest volume: Categories in Algebra, Geometry and Mathematical Physics - edited by A. Davydov, M. Batanin, and M. Johnson, S. Lack, and A. Neeman) (<a href="http://arxiv.org/abs/math.QA/0512034">arXiv:math.QA/0512034</a>)</p> </li> </ul> <p>A treatment of HQFTs that includes some details of the links with TQFTs is given in <a class="existingWikiWord" href="/nlab/files/HQFT-XMenagerie.pdf" title="">HQFTs meet the Menagerie</a>, which is a set of notes prepared by <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a> for a school and workshop in Lisbon, Feb. 2011.</p> <p>Related ideas are discussed in</p> <ul> <li id="MooreSegal06"><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>D-branes and K-theory in 2D topological field theory</em> (<a href="http://arxiv.org/abs/hep-th/0609042">arXiv:hep-th/0609042</a>)</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>, <a class="existingWikiWord" href="/nlab/show/Timothy+Porter">Timothy Porter</a>, <em>A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids</em> [<a href="https://arxiv.org/abs/2301.02491">arXiv:2301.02491</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <a class="existingWikiWord" href="/nlab/show/Lukas+Woike">Lukas Woike</a>, <em>Extended Homotopy Quantum Field Theories and their Orbifoldization</em>, Journal of Pure and Applied Algebra <strong>224</strong> 4 (2020) 106213 [<a href="https://arxiv.org/abs/1802.08512">arXiv:1802.08512</a>, <a href="https://doi.org/10.1016/j.jpaa.2019.106213">doi:10.1016/j.jpaa.2019.106213</a>]</p> </li> </ul> <p>Understanding <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a> of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> as an <a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended</a> <a class="existingWikiWord" href="/nlab/show/homotopy+field+theory">homotopy field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lukas+M%C3%BCller">Lukas Müller</a>, <a class="existingWikiWord" href="/nlab/show/Lukas+Woike">Lukas Woike</a>, <em>Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory</em>, J. Homotopy Relat. Struct. <strong>15</strong> (2020) 113–142 [<a href="https://arxiv.org/abs/1802.10455">arXiv:1802.10455</a>, <a href="https://doi.org/10.1007/s40062-019-00242-3">doi:10.1007/s40062-019-00242-3</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 7, 2025 at 09:24:27. 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