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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body><div class="rightHandSide toc"> <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 class="standout"> <p>This is a sub-entry of <a class="existingWikiWord" href="/nlab/show/geometric+models+for+elliptic+cohomology">geometric models for elliptic cohomology</a> and <a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology">A Survey of Elliptic Cohomology</a></p> <p>See there for background and context.</p> <p>This entry here is about the definition of <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism categories</a> for <a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian cobordism</a>s.</p> </div> <blockquote> <p><strong>raw material</strong>: this are notes taken more or less verbatim in a seminar – needs polishing</p> </blockquote> <p>Previous:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Axiomatic+field+theories+and+their+motivation+from+topology">Axiomatic field theories and their motivation from topology</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf">(2,1)-dimensional Euclidean field theories and tmf</a></p> </li> </ul> <p>Next:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theories">(2,1)-dimensional Euclidean field theories</a></li> </ul> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian 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/%28infinity%2Cn%29-category+of+cobordisms">(infinity,n)-category of cobordisms</a></p> </li> </ul> <hr /> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#onedim'>description for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d=1</annotation></semantics></math></a></li> <li><a href='#smoothversion'>smooth version / families version</a></li> <li><a href='#fieldtheories'>Riemannian field theories</a></li> <li><a href='#invertible_field_theories'>Invertible Field Theories</a></li> </ul> </div> <h1 id="Idea">Idea</h1> <p>The goal here is to define a category of <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s that carry the structure of <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a>s. Where a functor on an ordinary <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a> defines a <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>, the assignments of a functor on a category of <a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian cobordism</a>s do not only depend on the topology of a given cobordism, but also on its Riemannian structure. In physics terms such a functor is a <em>Euclidean quantum field theory</em> .</p> <blockquote> <p>Notice however that the physicist’s use of the word “Euclidean” is different from the way Stolz-Teichner use it: for a physicist it means that the Riemannian structure is not <em>pseudo</em>-Riemannian. For Stolz-Teichner it means (later on) that the Riemannian metric is <em>flat</em> .</p> </blockquote> <p>One central technical difference between plain topological cobordisms and those with Riemannian structure is that we want the functors on these to smoothly depend on variations of the Riemannian structure. This requires refining the bordism category to a <em>smooth</em> category. By the logic of <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a>, one way to do this is to realize it as a <a class="existingWikiWord" href="/nlab/show/stack">stack</a> on <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> with values in categories. This realization will be described here.</p> <h1 id="topological">Part 1 (topological) bordism category</h1> <p><strong>definition sketch</strong></p> <p>the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">Bord_d</annotation></semantics></math> has</p> <ul> <li> <p>as <a class="existingWikiWord" href="/nlab/show/object">object</a>s closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d-1)</annotation></semantics></math>-dimensional <em>smooth</em> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>s</p> </li> <li> <p>and the <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s are <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional smooth <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>s with boundary, modulo <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> “rel boundaries” (i.e. those that restrict to the identy on the boundary)</p> </li> </ul> <p>The composition of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s is given by gluing of manifolds along their boundary</p> <h1 id="Riemannian">Part 2 Riemannian bordism category</h1> <p>in all of the following</p> <ul> <li> <p>the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> denotes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> <p>without boundary.</p> </li> <li> <p><strong>note on boundaries</strong> technically it is convenient to never ever work with manifolds with Riemannian or other structure with boundary. Instead, we always just mention manifolds without boundary and encoded the way in which they are still to be thouhgt of as <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s by injecting <em><a class="existingWikiWord" href="/nlab/show/collars">collars</a></em> into them. The manifolds with boundary could be obtained by cutting of at the <em>core</em> of these collars (see the definition below) but, while this is morally the idea, in the construction this is never explicitly considered.</p> <p>Also, later when we generalize <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>s to <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>s it will be very convenient not to have to talk about boundaries</p> </li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">R Bord_d</annotation></semantics></math> is defined using bicollars from the beginning</p> <p>an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">R Bord_d</annotation></semantics></math> is a quintuple</p> <p>consisting of</p> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>;</p> </li> <li> <p>a <strong>core</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d-1)</annotation></semantics></math>-manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Y^c</annotation></semantics></math> sitting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>c</mi></msup><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y^c \hookrightarrow Y</annotation></semantics></math> in a thickening <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">Y^d</annotation></semantics></math> – being a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-manifold –</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mo>+</mo></msup><mo>,</mo><msup><mi>Y</mi> <mo>−</mo></msup><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y^+, Y^- \hookrightarrow Y</annotation></semantics></math> two disjointly embedded open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional manifolds such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Y^c</annotation></semantics></math> is in the closure of both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">Y^+</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">Y^-</annotation></semantics></math></p> </li> <li> <p>that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mo>+</mo></msup><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msup><mi>Y</mi> <mo>−</mo></msup><mo>=</mo><mi>Y</mi><mo>\</mo><msup><mi>Y</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">Y^+ \coprod Y^- = Y \backslash Y^C</annotation></semantics></math></p> </li> </ul> </li> </ul> <blockquote> <p>so the <em>picture</em> of an object, which is missing in this writeup here for the moment, is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d-1</annotation></semantics></math>-dimensional Riemannian manifold that is thickened a bit in one further othogonal direction</p> </blockquote> <p><strong>definition</strong> A <strong>Riemannian bordism</strong> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>0</mn></msub><mo>,</mo><msubsup><mi>Y</mi> <mn>0</mn> <mi>c</mi></msubsup><mo>,</mo><msubsup><mi>Y</mi> <mn>0</mn> <mo>±</mo></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y_0,Y_0^c, Y_0^{\pm})</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>Y</mi> <mn>1</mn></msub><mo>,</mo><msubsup><mi>Y</mi> <mn>1</mn> <mi>c</mi></msubsup><mo>,</mo><msubsup><mi>Y</mi> <mn>1</mn> <mo>±</mo></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y_1,Y_1^c, Y_1^{\pm})</annotation></semantics></math> is a triple <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><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Sigma, i_0, i_1)</annotation></semantics></math> where</p> <ul> <li> <p><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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> without boundary</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i=0,1</annotation></semantics></math> an open neighbourhood of the core <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Y</mi> <mi>i</mi> <mi>c</mi></msubsup><mo>↪</mo><msub><mi>W</mi> <mi>i</mi></msub><mover><mo>↪</mo><mi>open</mi></mover><msub><mi>Y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Y_i^c \hookrightarrow W_i \stackrel{open}{\hookrightarrow} Y_i</annotation></semantics></math></p> <p>this defines the intersections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>W</mi> <mi>k</mi> <mo>±</mo></msubsup><mo>:</mo><mo>=</mo><msub><mi>W</mi> <mi>k</mi></msub><mo>∩</mo><msubsup><mi>Y</mi> <mi>k</mi> <mo>±</mo></msubsup></mrow><annotation encoding="application/x-tex">W^\pm_k := W_k \cap Y^\pm_k</annotation></semantics></math> with the two collars for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k = 0,1</annotation></semantics></math>.</p> </li> <li> <p>a smooth map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>k</mi></msub><mo>:</mo><msub><mi>W</mi> <mi>k</mi></msub><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">i_k : W_k \to \Sigma</annotation></semantics></math></p> <p>such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>k</mi></msub><mo>:</mo><msubsup><mi>W</mi> <mi>k</mi> <mo>+</mo></msubsup><mo>∪</mo><msubsup><mi>Y</mi> <mi>k</mi> <mi>c</mi></msubsup><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">i_k : W^+_k \cup Y_k^c \to Z</annotation></semantics></math> is a proper map;</p> </li> <li> <p>(+) for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>i</mi> <mi>k</mi> <mo>+</mo></msubsup><mo>:</mo><mo>=</mo><msub><mi>i</mi> <mi>k</mi></msub><mo stretchy="false">/</mo><msubsup><mi>W</mi> <mi>k</mi> <mo>+</mo></msubsup></mrow><annotation encoding="application/x-tex">i^+_k := i_k/W^+_k</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isometry">isometric</a> embeddings into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>\</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msubsup><mi>W</mi> <mn>1</mn> <mo>−</mo></msubsup><mo>∪</mo><msubsup><mi>Y</mi> <mn>1</mn> <mi>c</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma \backslash i_1(W^-_1 \cup Y^c_1)</annotation></semantics></math></p> <p>i.e. restricted to the (+)-collar the embedding of the thickened object into the would-be cobordisms is isomertric</p> </li> <li> <p>the core <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>c</mi></msup><mo>:</mo><mo>=</mo><mi>Σ</mi><mo>\</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msubsup><mi>W</mi> <mn>0</mn> <mo>+</mo></msubsup><mo stretchy="false">)</mo><mo>∪</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msubsup><mi>W</mi> <mn>1</mn> <mo>−</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma^c := \Sigma \backslash (i_0(W^+_0) \cup i_1(W^-_1))</annotation></semantics></math> is compact</p> <p>i.e. cutting of the (+)-collar of the incoming object and the (-)-collar of the outgoing object yields a compact manifold</p> </li> </ul> </li> </ul> <p><strong>Remark</strong>. Notice that this builds in an asymmetry: the (+)-side is preferred. This is intentionally: also the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TV</mi></mrow><annotation encoding="application/x-tex">TV</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>s will have a similar asymmetry (from the fact that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-dimensional vector spaces there is an evaluation map but not necessarily a coevaluation/unit for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V \otimes V^*</annotation></semantics></math>), similarly, with the above asymmetric definition we have a cobordims <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msup><mi>Y</mi> <mo>*</mo></msup><mo>→</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">Y \coprod Y^* \to \emptyset</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">Y^*</annotation></semantics></math> is obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> by reversing orientation) but <em>not</em> one going the other way round.</p> <p>A big difference between <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>s and the Riemannian QFTs is that for <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>s the vector spaces assigned to objects are necessarily finite-dimensional. So this issue here with infinite-dimensional vector spaces and the asymmetry that this introduces is crucial for Riemannian QFTs.</p> <p><strong>example</strong> Given any <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msub><mi>W</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>W</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \phi : W_0 \to W_1 </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> preserves the decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>W</mi> <mi>k</mi> <mo>±</mo></msubsup><mo>,</mo><msubsup><mi>Y</mi> <mi>k</mi> <mi>c</mi></msubsup></mrow><annotation encoding="application/x-tex">W_k^\pm, Y_k^c</annotation></semantics></math> we get a <a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian cobordism</a> using</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>:</mo><mo>=</mo><msub><mi>W</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \Sigma := W_1 </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>Id</mi> <mrow><msub><mi>W</mi> <mn>1</mn></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>i</mi> <mn>0</mn></msub><mo>=</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex"> i_1 = Id_{W_1}\,,\;\;\;\;\; i_0 = \phi </annotation></semantics></math></div> <p><strong>definition (morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">R Bord_d</annotation></semantics></math>)</strong> morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">Y_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">R Bord_d</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>−</mo><mi>RB</mi></mrow><annotation encoding="application/x-tex">d-RB</annotation></semantics></math> or whatever the notation is) are <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a> classes <em>rel. boundary</em> (see below) of <a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian cobordism</a>s from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">Y_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Y_1</annotation></semantics></math>.</p> <p>We require the commutativity of the following diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>V</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msub><mi>V</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>V</mi><msub><mo>′</mo> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>i</mi><msub><mo>′</mo> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>←</mo><mrow><mi>i</mi><msub><mo>′</mo> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>V</mi><msub><mo>′</mo> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V_1 &\stackrel{i_1}{\to}& X &\stackrel{i_0}{\leftarrow}& V_0 \\ \downarrow^{f_1} && \downarrow && \downarrow^{f_0} \\ V'_1 &\stackrel{i'_1}{\to}& X' &\stackrel{i'_0}{\leftarrow}& V'_0 } </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><msub><mi>f</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F,f_0, f_1)</annotation></semantics></math> is “rel. boundary” if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub><mo>=</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex">f_0 = Id</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>=</mo><mi>Id</mi></mrow><annotation encoding="application/x-tex">f_1 = Id</annotation></semantics></math></p> <p>so an isomorphism “rel boundary” in the sense here (more “rel collars”, really) is an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</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> sitting in a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>V</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msub><mi>V</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>Id</mi></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>Id</mi></msup></mtd></mtr> <mtr><mtd><mi>V</mi><msub><mo>′</mo> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>i</mi><msub><mo>′</mo> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>′</mo></mtd> <mtd><mover><mo>←</mo><mrow><mi>i</mi><msub><mo>′</mo> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>V</mi><msub><mo>′</mo> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V_1 &\stackrel{i_1}{\to}& X &\stackrel{i_0}{\leftarrow}& V_0 \\ \downarrow^{Id} && \downarrow && \downarrow^{Id} \\ V'_1 &\stackrel{i'_1}{\to}& X' &\stackrel{i'_0}{\leftarrow}& V'_0 } </annotation></semantics></math></div> <h2 id="onedim">description for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d=1</annotation></semantics></math></h2> <p>we decribe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R Bord_1</annotation></semantics></math> explicitly</p> <p>it has at least the object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msup><mi>pt</mi> <mo>−</mo></msup></mtd> <mtd><msup><mi>pt</mi> <mi>c</mi></msup></mtd> <mtd><msup><mi>pt</mi> <mo>+</mo></msup></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo></mtd> <mtd><mo>•</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>ℝ</mi><mo>,</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>,</mo><msub><mi>ℝ</mi> <mo>±</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> pt = \left( \array{ pt^- & pt^c & pt^+ \\ -- & \bullet & -- } \right) = (\mathbb{R}, \{0\}, \mathbb{R}_\pm) </annotation></semantics></math></div> <p>which is a point with collar all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>.</p> <p><strong>Lemma</strong> every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R Bord_1</annotation></semantics></math> which is <em>connected</em> and not the empty set is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math></p> <p>now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}_+</annotation></semantics></math> consider the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>t</mi></msub><mo>∈</mo><mi>R</mi><msub><mi>Bord</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>pt</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I_t \in R Bord_1(pt,pt) </annotation></semantics></math></div> <p>defined as the triple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℝ</mi><mo>,</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{R}, i_0, i_1)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">i_0 : \mathbb{R} \to \mathbb{R}</annotation></semantics></math> is the identity map, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">i_1 : \mathbb{R} \to \mathbb{R}</annotation></semantics></math> is translation by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>.</p> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">i_0</annotation></semantics></math> takes the core of in the incoming point to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{R}</annotation></semantics></math> while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i_1</annotation></semantics></math> takes the core of the outgoing point to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}</annotation></semantics></math>. Everything in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> outside of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> is hence “collar” and this describes what naively one would think of as just the <a class="existingWikiWord" href="/nlab/show/interval">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/Riemannian+cobordism">Riemannian cobordism</a>.</p> <p><strong>Lemma</strong> The composition of these cobordisms is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>t</mi></msub><mo>∘</mo><msub><mi>I</mi> <mrow><mi>t</mi><mo>′</mo></mrow></msub><mo>=</mo><msub><mi>I</mi> <mrow><mi>t</mi><mo>+</mo><mi>t</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> I_t \circ I_{t'} = I_{t+t'} </annotation></semantics></math></div> <p>There are also morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mo>+</mo></msub><mo>:</mo><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi><mo>→</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex"> L_+ : pt \coprod pt \to \emptyset </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mo>+</mo></msub><mo>:</mo><mi>∅</mi><mo>→</mo><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex"> R_+ : \emptyset \to pt \coprod pt </annotation></semantics></math></div> <p>which describe morally the same cobordisms as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">I_t</annotation></semantics></math> does, but where both boundary components are regarded as incoming or noth as outgoing, respectively.</p> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">L_t</annotation></semantics></math> is formall given exactly as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">I_t</annotation></semantics></math> only that the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">i_0 : \mathbb{R} \to \mathbb{R}</annotation></semantics></math> is not the identity, but reflection at the origin. This encodes the orientation reversal at that end.</p> <p>This is defined for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t \gt 0</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t= 0</annotation></semantics></math> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">L_0</annotation></semantics></math> is still defined, but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">R_0</annotation></semantics></math> is not!! Exercise: check carefully with the above definition, keeping the asymmetry mentioned there in mind, to show that the obvious definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">R_0</annotation></semantics></math> does not satisfy the axioms above.</p> <p>So this means that we have a cobordism of length 0 going <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">\emptyset \to pt \coprod pt</annotation></semantics></math>, but all cobordisms going the other way round <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi><mo>→</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">pt \coprod pt \to \emptyset</annotation></semantics></math> will have to have non-vanishing length.</p> <p>Another morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R Bord_1</annotation></semantics></math> is the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi><mo>→</mo><mi>pt</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex"> \sigma : pt \coprod pt \to pt \coprod pt </annotation></semantics></math></div> <p>which just interchanges the two points, without having any length.</p> <p><strong>Lemma</strong> We have the following composition laws:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>t</mi></msub><mo>∘</mo><mi>σ</mi><mo>=</mo><msub><mi>L</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">L_t \circ \sigma = L_t</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>t</mi></msub><mo>=</mo><mi>σ</mi><mo>∘</mo><msub><mi>R</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">R_t = \sigma \circ R_t</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>t</mi></msub><msub><mo>∘</mo> <mrow><msub><mi>L</mi> <mn>0</mn></msub></mrow></msub><msub><mi>R</mi> <mrow><mi>t</mi><mo>′</mo></mrow></msub><mo>=</mo><msub><mi>R</mi> <mrow><mi>t</mi><mo>+</mo><mi>t</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">R_t \circ_{L_0} R_{t'} = R_{t+t'}</annotation></semantics></math></p> </li> </ul> <p>where in the last line we have the composition that is obvious once you draw the corresponding picture, which in full beuaty is</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><msub><mi>Id</mi> <mi>pt</mi></msub><mo>⊗</mo><msub><mi>L</mi> <mn>0</mn></msub><mo>⊗</mo><msub><mi>Id</mi> <mi>pt</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>R</mi> <mi>t</mi></msub><mo>⊗</mo><msub><mi>R</mi> <mrow><mi>t</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Id_{pt} \otimes L_0 \otimes Id_{pt}) \circ (R_t \otimes R_{t'}) </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> is given by disjoint union.</p> <p><strong>theorem</strong> the <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><mi>R</mi><msub><mi>Bord</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">R Bord_1</annotation></semantics></math> is <em>generated</em> as a symmetric monoidal category by</p> <ul> <li> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math></p> </li> <li> <p>the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">L_0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>R</mi> <mi>t</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\{R_t\}_{t \gt 0}</annotation></semantics></math></p> </li> </ul> <p>subject to the relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>0</mn></msub><mo>∘</mo><mi>σ</mi><mo>=</mo><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> L_0 \circ \sigma = L_0 </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∘</mo><msub><mi>R</mi> <mi>t</mi></msub><mo>=</mo><msub><mi>R</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex"> \sigma \circ R_t = R_t </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>′</mo><mo>></mo><mn>0</mn><mo>:</mo><msub><mi>R</mi> <mi>t</mi></msub><msub><mo>∘</mo> <mrow><msub><mi>L</mi> <mn>0</mn></msub></mrow></msub><msub><mi>R</mi> <mrow><mi>t</mi><mo>′</mo></mrow></msub><mo>=</mo><msub><mi>R</mi> <mrow><mi>t</mi><mo>+</mo><mi>t</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \forall t,t' \gt 0 : R_t \circ_{L_0} R_{t'} = R_{t + t'} </annotation></semantics></math></div> <p><strong>corollary</strong> symmetric monoidal functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msup><mi>Fun</mi> <mo>⊗</mo></msup><mo stretchy="false">(</mo><mi>R</mi><msub><mi>Bord</mi> <mn>1</mn></msub><mo>,</mo><mi>TV</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E \in Fun^\otimes(R Bord_1, TV) </annotation></semantics></math></div> <p>to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>TV</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">TV_\mathbb{R}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>s are specified by their imagges of these generators. We have</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>:</mo><mi>pt</mi><mo>↦</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">E : pt \mapsto V</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>:</mo><msub><mi>L</mi> <mn>0</mn></msub><mo>↦</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>:</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E : L_0 \mapsto (\lambda : V\otimes V \to \mathbb{R})</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>:</mo><msub><mi>R</mi> <mi>t</mi></msub><mo>↦</mo><msub><mi>ρ</mi> <mi>t</mi></msub><mo>∈</mo><mi>V</mi><mo>⊗</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">E : R_t \mapsto \rho_t \in V \otimes V</annotation></semantics></math></p> </li> </ul> <p>The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\lambda : V \otimes V \to \mathbb{R}</annotation></semantics></math> is necessarily a <em>nondegenerate</em> and <em>symmetric</em> bilinear form and thus may be used to produce and fix an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>≃</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V \simeq V^*</annotation></semantics></math>.</p> <p>This isomorphism is used to get an embedding</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊗</mo><mi>V</mi><mi>to</mi><mi>V</mi><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup><mo>↪</mo><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V \otimes V to V \otimes V^* \hookrightarrow End(V) \,. </annotation></semantics></math></div> <p>The image of this embedding is the set of what in this context will be called “trace class” operators.</p> <p>With respect to this identification the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is to be understood. For varying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\rho_t</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a> (for instance a typical example would be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V = \Gamma(E)</annotation></semantics></math> a space of sections of a vector bundle and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>t</mi></msub><mo>=</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><mi>Δ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\rho_t = e^{-t \Delta}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Laplace+operator">Laplace operator</a> on <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>).</p> <p><strong>note</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\lambda : V \otimes V \to \mathbb{R}</annotation></semantics></math> to be continuous, one cannot use the Hilbert tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_H</annotation></semantics></math></p> <p>the reason is that we have the folloing possible mpas out of the following possible tensor products</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mover><mo>←</mo><mi>λ</mi></mover><mi>V</mi><msub><mo>⊗</mo> <mi>algebraic</mi></msub><mi>V</mi><mover><mo>↪</mo><mrow><mi>finite</mi><mi>rank</mi></mrow></mover><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes_{algebraic} V \stackrel{finite rank}{\hookrightarrow} End(V) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mover><mo>←</mo><mi>λ</mi></mover><mi>V</mi><mo>⊗</mo><mi>V</mi><mover><mo>↪</mo><mrow><mi>trace</mi><mi>class</mi></mrow></mover><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes V \stackrel{trace class}{\hookrightarrow} End(V) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mover><mo>←</mo><mi>λ</mi></mover><mi>V</mi><msub><mo>⊗</mo> <mi>H</mi></msub><mi>V</mi><mover><mo>↪</mo><mrow><mi>Hilbert</mi><mi>Schmitdt</mi></mrow></mover><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes_H V \stackrel{Hilbert Schmitdt}{\hookrightarrow} End(V) </annotation></semantics></math></div> <p>(so here the middle is the projective tensor product, the one that we are actually using)</p> <h2 id="smoothversion">smooth version / families version</h2> <p>We now refine the definition of the categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mi>Bord</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">R Bord_d</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TV</mi></mrow><annotation encoding="application/x-tex">TV</annotation></semantics></math> such that they remember smooth stucture.</p> <blockquote> <p>Effectively, what the following implicitly does is to refine these categories to <a class="existingWikiWord" href="/nlab/show/stack">stack</a>s with values in categories over <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>. The <a class="existingWikiWord" href="/nlab/show/fibered+category">fibred categories</a> that appear in the following, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup><mo>→</mo><mi>Diff</mi></mrow><annotation encoding="application/x-tex">R Bord_d^{fam} \to Diff</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>TV</mi> <mi>fam</mi></msup><mo>→</mo><mi>Diff</mi></mrow><annotation encoding="application/x-tex">TV^{fam} \to Diff</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> of these stacks.</p> </blockquote> <p><strong>definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>TV</mi> <mi>fam</mi></msup></mrow><annotation encoding="application/x-tex">TV^{fam}</annotation></semantics></math></strong></p> <p>recall that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TV</mi></mrow><annotation encoding="application/x-tex">TV</annotation></semantics></math> denotes the category of locally convex Hausdorff <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <p>now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>TV</mi> <mi>fam</mi></msup></mrow><annotation encoding="application/x-tex">TV^{fam}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/fibred+category">fibred category</a> over <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> whose fiber over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Diff</mi></mrow><annotation encoding="application/x-tex">X \in Diff</annotation></semantics></math> is the category of topological <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s over <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>. This has as objects <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s of <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>s, and the morphisms are fiberwise linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-morphisms of bundles in the following sense:</p> <p>let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>,</mo><mi>W</mi><mo>∈</mo><mi>TV</mi></mrow><annotation encoding="application/x-tex">V, W \in TV</annotation></semantics></math></p> <p>Then a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">F : V \to W</annotation></semantics></math> is – for any inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow V</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>W</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mo>⊂</mo></msup></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V &\stackrel{f}{\to}& W \\ \uparrow^\subset & \nearrow \\ U } </annotation></semantics></math></div> <p>– called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">C^1</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">u \in U</annotation></semantics></math> in the direction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \in V</annotation></semantics></math> if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>t</mi><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mi>t</mi></mfrac></mrow><annotation encoding="application/x-tex"> \lim_{t \to 0} \frac{F(u+t v) - F(u)}{t} </annotation></semantics></math></div> <p>exists in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex"> U \times V \to W </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>d</mi><msub><mi>F</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (u,v) \mapsto d F_u(v) </annotation></semantics></math></div> <p>is continuous.</p> <p>Iteratively one defines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">C^n</annotation></semantics></math> and then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>. The morphsims of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TV</mi></mrow><annotation encoding="application/x-tex">TV</annotation></semantics></math>-bundles are supposed to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math> maps in this sense (linear in the fibers, of course)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mi>V</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>S</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>S</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V' &\stackrel{\tilde f}{\to}& V \\ \downarrow &&\downarrow \\ S' &\stackrel{f}{\to}& S } </annotation></semantics></math></div> <p><strong>definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">R Bord_d^{fam}</annotation></semantics></math></strong></p> <p>Similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">R Bord_d^{fam}</annotation></semantics></math> has as objects submersions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y \to S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>c</mi></msup><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y^c \to S</annotation></semantics></math> (not necessarily surjective) with a smooth rank-2 tensor on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> that fiberwise induces the structure of a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> (so these are <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>-families of <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a>s) such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd><msup><mo>←</mo> <mo>⊂</mo></msup></mtd> <mtd><msup><mi>Y</mi> <mi>c</mi></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>submersion</mi></msup></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>proper</mi><mi>subm</mi><mo>.</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>S</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y &\leftarrow^\subset& Y^c \\ \downarrow^{submersion} & \swarrow_{proper subm.} \\ S } </annotation></semantics></math></div> <p>recall that a map is a <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a> if inverse images of compact sets are compact.</p> <p><strong>remark</strong> Notice that if we fix the topology of the fibers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Y \to S</annotation></semantics></math>, then what varies as we vary the fibers is the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> on the fibers, so here each <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> can be thought of as a (subspace of a) moduli space of Riemannian metrics on a given topological space. Don’t confuse this with the role the space always called <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> here will play as a kind of “moduli space of field theories”.</p> <p>a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup></mrow><annotation encoding="application/x-tex">R Bord_d^{fam}</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msub><mi>Y</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> R Bord_d^{fam}\left( \array{ Y_0 \\ \downarrow \\ S_0 }, \;\; \array{ Y_1 \\ \downarrow \\ S_1 } \right) </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/isometry">isometric</a> rel boundary classes of submersions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><msub><mi>S</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma \to S_0</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>Y</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>S</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msub><mi>S</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma &\stackrel{i_1}{\leftarrow}&f^* Y_1 &\to& Y_1 \\ \uparrow^{i_0} &&\downarrow && \downarrow \\ Y_0& \to&S_0 &\stackrel{f}{\to}& S_1 } </annotation></semantics></math></div> <p>so here <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 an <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>-family of cobordisms.</p> <h2 id="fieldtheories">Riemannian field theories</h2> <p><strong>definition</strong></p> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional Riemannian quantum field theory</strong> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msubsup><mi>Fun</mi> <mi>Diff</mi> <mo>⊗</mo></msubsup><mo stretchy="false">(</mo><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup><mo>,</mo><msup><mi>TV</mi> <mi>fam</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E \in Fun^\otimes_{Diff}(R Bord_d^{fam}, TV^{fam}) </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>R</mi><msubsup><mi>Bord</mi> <mi>d</mi> <mi>fam</mi></msubsup></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><msup><mi>TV</mi> <mi>fam</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Diff</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ R Bord_d^{fam} &&\stackrel{}{\to}&& TV^{fam} \\ & \searrow && \swarrow \\ && Diff } </annotation></semantics></math></div> <p>and such that it preserves <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> <p>(so its a <a class="existingWikiWord" href="/nlab/show/cartesian+functor">cartesian functor</a> between these <a class="existingWikiWord" href="/nlab/show/fibered+category">fibered categories</a> that is also symmetric monoidal)</p> <h2 id="invertible_field_theories">Invertible Field Theories</h2> <p>In the study of invertible field theories, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><msubsup><mi>Bord</mi> <mi>n</mi> <mi>G</mi></msubsup><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo></mrow><annotation encoding="application/x-tex">\mid \mid Bord^G_n \mid \mid</annotation></semantics></math> be the fundamental groupoid of the classifying space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord^G_n</annotation></semantics></math>. An <a class="existingWikiWord" href="/nlab/show/invertible+field+theory">invertible field theory</a> factors through this and the Picard subgroupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^*</annotation></semantics></math> of the target.</p> <p>Kreck, S. Stolz, and P. Teichner.Invertible topological field theories are SKK invariants.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on December 19, 2019 at 09:52:53. 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