nLab
bordism categories following Stolz-Teichner

This is a sub-entry of geometric models for elliptic cohomology and A Survey of Elliptic Cohomology

See there for background and context.

This entry here is about the definition of cobordism categories for Riemannian cobordisms.

raw material: this are notes taken more or less verbatim in a seminar – needs polishing

Previous:

• Axiomatic field theories and their motivation from topology.

• (1,1)-dimensional Euclidean field theories and K-theory

• (2,1)-dimensional Euclidean field theories and tmf

Next:

• (2,1)-dimensional Euclidean field theories

see also

• cobordism

• Riemannian cobordism

• cobordism category

• (infinity,n)-category of cobordisms

---

Idea

The goal here is to befine a category of cobordisms that carry the structure of Riemannian manifolds. Where a functor on an ordinary cobordism category defines a TQFT, the assignments of a functor on a category of Riemannian cobordisms 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 Euclidean quantum field theory .

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 pseudo-Riemannian. For Stolz-Teichner it means (later on) that the Riemannian metric is flat .

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 smooth category. By the logic of space and quantity, one way to do this is to realize it as a stack on Diff with values in categories. This realization will be described here.

Part 1 (topological) bordism category

definition sketch

the category ${\mathrm{Bord}}_d$ has

• as objects closed $(d-1)$-dimensional smooth manifolds

• and the morphisms are compact $d$-dimensional smooth manifolds with boundary, modulo diffeomorphism “rel boundaries” (i.e. those that restrict to the identy on the boundary)

The composition of morphisms is given by gluing of manifolds along their boundary

Part 2 Riemannian bordism category

in all of the following

• the symbol $Y$ denotes $d$-dimensional a Riemannian manifold

  without boundary.

• note on boundaries 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 cobordisms by injecting collars into them. The manifolds with boundary could be obtained by cutting of at the core of these collars (see the definition below) but, while this is morally the idea, in the construction this is never explicitly considered.

  Also, later when we generalize manifolds to supermanifolds it will be very convenient not to have to talk about boundaries

$R{\mathrm{Bord}}_d$ is defined using bicollars from the beginning

an object in $R{\mathrm{Bord}}_d$ is a quintuple

consisting of

• a $d$-dimensional Riemannian manifold $Y$;

• a core $(d-1)$-manifold $Y^c$ sitting $Y^c\hookrightarrow Y$ in a thickening $Y^d$ – being a $d$-manifold –

• $Y^{+},Y^{-}\hookrightarrow Y$ two disjointly embedded open $d$-dimensional manifolds such that

  • $Y^c$ is in the closure of both $Y^{+}$ and $Y^{-}$

  • that $Y^{+}\coprod Y^{-}=Y\backslash Y^C$

so the picture of an object, which is missing in this writeup here for the moment, is a $d-1$-dimensional Riemannian manifold that is thickened a bit in one further othogonal direction

definition A Riemannian bordism from $(Y_0,Y_0^c,Y_0^{\pm })$ to $(Y_1,Y_1^c,Y_1^{\pm })$ is a triple $(\Sigma ,i_0,i_1)$ where

• $\Sigma$ is a $d$-dimensional Riemannian manifold without boundary

• for $i=0,1$ an open neighbourhood of the core $Y_i^c\hookrightarrow W_i\stackrel{\mathrm{open}}{\hookrightarrow }{Y}_i$

  this defines the intersections $W_k^{\pm }:=W_k\cap Y_k^{\pm }$ with the two collars for each $k=0,1$.

• a smooth map $i_k:W_k\to \Sigma$

  such that

  • $i_k:W_k^{+}\cup Y_k^c\to Z$ is a proper map;

  • (+) for $i_k^{+}:=i_k/W_k^{+}$ are isometric embeddings into $\Sigma \backslash i_1(W_1^{-}\cup Y_1^c)$

    i.e. restricted to the (+)-collar the embedding of the thickened object into the would-be cobordisms is isomertric

  • the core ${\Sigma }^c:=\Sigma \backslash (i_0(W_0^{+})\cup i_1(W_1^{-}))$ is compact

    i.e. cutting of the (+)-collar of the incoming object and the (-)-collar of the outgoing object yields a compact manifold

Remark. Notice that this builds in an asymmetry: the (+)-side is preferred. This is intentionally: also the category $\mathrm{TV}$ of topological vector spaces will have a similar asymmetry (from the fact that for $\mathrm{\infty }$-dimensional vector spaces there is an evaluation map but not necessarily a coevaluation/unit for $V\otimes V^{*}$), similarly, with the above asymmetric definition we have a cobordims $Y\coprod Y^{*}\to \varnothing$ (where $Y^{*}$ is obtained from $Y$ by reversing orientation) but not one going the other way round.

A big difference between TQFTs and the Riemannian QFTs is that for TQFTs 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.

example Given any isometry

$$\varphi :W_0\to W_1$$\phi : W_0 \to W_1

such that $\varphi$ preserves the decomposition $W_k^{\pm },Y_k^c$ we get a Riemannian cobordism using

$$\Sigma :=W_1$$\Sigma := W_1

and

$$i_1={\mathrm{Id}}_{W_1},i_0=\varphi$$i_1 = Id_{W_1}\,,\;\;\;\;\; i_0 = \phi

definition (morphisms in $R{\mathrm{Bord}}_d$) morphisms from $Y_0$ to $Y$ in $R{\mathrm{Bord}}_d$ (or $d-\mathrm{RB}$ or whatever the notation is) are isometry classes rel. boundary (see below) of Riemannian cobordisms from $Y_0$ to $Y_1$.

We require the commutativity of the following diagram

$$\begin{array}{ccccc}{V}_1& \stackrel{i_1}{\to }& X& \stackrel{i_0}{\leftarrow }& V_0\\ {\downarrow }^{f_1}& & \downarrow & & {\downarrow }^{f_0}\\ V{\prime }_1& \stackrel{i{\prime }_1}{\to }& X\prime & \stackrel{i{\prime }_0}{\leftarrow }& V{\prime }_0\end{array}$$\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 }

The isometry $(F,f_0,f_1)$ is “rel. boundary” if $f_0=\mathrm{Id}$ and $f_1=\mathrm{Id}$

so an isomorphism “rel boundary” in the sense here (more “rel collars”, really) is an isometry $F$ sitting in a diagram

$$\begin{array}{ccccc}{V}_1& \stackrel{i_1}{\to }& X& \stackrel{i_0}{\leftarrow }& V_0\\ {\downarrow }^{\mathrm{Id}}& & \downarrow & & {\downarrow }^{\mathrm{Id}}\\ V{\prime }_1& \stackrel{i{\prime }_1}{\to }& X\prime & \stackrel{i{\prime }_0}{\leftarrow }& V{\prime }_0\end{array}$$\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 }

description for $d=1$

we decribe $R{\mathrm{Bord}}_1$ explicitly

it has at least the object

$$\mathrm{pt}=\left(\begin{array}{ccc}{\mathrm{pt}}^{-}& {\mathrm{pt}}^c& {\mathrm{pt}}^{+}\\ --& •& --\end{array}\right)=(ℝ,\{0\},{ℝ}_{\pm })$$pt = \left( \array{ pt^- & pt^c & pt^+ \\ -- & \bullet & -- } \right) = (\mathbb{R}, \{0\}, \mathbb{R}_\pm)

which is a point with collar all of $ℝ$.

Lemma every object in $R{\mathrm{Bord}}_1$ which is connected and not the empty set is isomorphic to this $\mathrm{pt}$

now for $t\in {ℝ}_{+}$ consider the morphism

$$I_t\in R{\mathrm{Bord}}_1(\mathrm{pt},\mathrm{pt})$$I_t \in R Bord_1(pt,pt)

defined as the triple $(ℝ,i_0,i_1)$ where $i_0:ℝ\to ℝ$ is the identity map, and where $i_1:ℝ\to ℝ$ is translation by $t$.

This means that $i_0$ takes the core of in the incoming point to $0\in ℝ$ while $i_1$ takes the core of the outgoing point to $t\in ℝ$. Everything in $ℝ$ outside of $[0,1]$ is hence “collar” and this describes what naively one would think of as just the interval $[0,1]$ regarded as a Riemannian cobordism.

Lemma The composition of these cobordisms is given by

$$I_t\circ I_{t\prime }=I_{t+t\prime }$$I_t \circ I_{t'} = I_{t+t'}

There are also morphisms

$$L_{+}:\mathrm{pt}\coprod \mathrm{pt}\to \varnothing$$L_+ : pt \coprod pt \to \emptyset

and

$$R_{+}:\varnothing \to \mathrm{pt}\coprod \mathrm{pt}$$R_+ : \emptyset \to pt \coprod pt

which describe morally the same cobordisms as $I_t$ does, but where both boundary components are regarded as incoming or noth as outgoing, respectively.

Here $L_t$ is formall given exactly as $I_t$ only that the map $i_0:ℝ\to ℝ$ is not the identity, but reflection at the origin. This encodes the orientation reversal at that end.

This is defined for $t>0$. For $t=0$ the morphism $L_0$ is still defined, but $R_0$ is not!! Exercise: check carefully with the above definition, keeping the asymmetry mentioned there in mind, to show that the obvious definition of $R_0$ does not satisfy the axioms above.

So this means that we have a cobordism of length 0 going $\varnothing \to \mathrm{pt}\coprod \mathrm{pt}$, but all cobordisms going the other way round $\mathrm{pt}\coprod \mathrm{pt}\to \varnothing$ will have to have non-vanishing length.

Another morphism in $R{\mathrm{Bord}}_1$ is the morphism

$$\sigma :\mathrm{pt}\coprod \mathrm{pt}\to \mathrm{pt}\coprod \mathrm{pt}$$\sigma : pt \coprod pt \to pt \coprod pt

which just interchanges the two points, without having any length.

Lemma We have the following composition laws:

• $L_t\circ \sigma =L_t$

• $R_t=\sigma \circ R_t$

• $R_t{\circ }_{L_0}{R}_{t\prime }=R_{t+t\prime }$

where in the last line we have the composition that is obvious once you draw the corresponding picture, which in full beuaty is

$$({\mathrm{Id}}_{\mathrm{pt}}\otimes L_0\otimes {\mathrm{Id}}_{\mathrm{pt}})\circ (R_t\otimes R_{t\prime })$$(Id_{pt} \otimes L_0 \otimes Id_{pt}) \circ (R_t \otimes R_{t'})

where the tensor product $\otimes$ is given by disjoint union.

theorem the symmetric monoidal category $R{\mathrm{Bord}}_1$ is generated as a symmetric monoidal category by

• the object $\mathrm{pt}$

• the morphisms $L_0$, $\{R_t{\}}_{t>0}$

subject to the relations

$$L_0\circ \sigma =L_0$$L_0 \circ \sigma = L_0

$$\sigma \circ R_t=R_t$$\sigma \circ R_t = R_t

$$\forall t,t\prime >0:R_t{\circ }_{L_0}{R}_{t\prime }=R_{t+t\prime }$$\forall t,t' \gt 0 : R_t \circ_{L_0} R_{t'} = R_{t + t'}

corollary symmetric monoidal functors

$$E\in {\mathrm{Fun}}^{\otimes }(R{\mathrm{Bord}}_1,\mathrm{TV})$$E \in Fun^\otimes(R Bord_1, TV)

to the category ${\mathrm{TV}}_{ℝ}$ of topological vector spaces are specified by their imagges of these generators. We have

• $E:\mathrm{pt}\mapsto V$

• $E:L_0\mapsto (\lambda :V\otimes V\to ℝ)$

• $E:R_t\mapsto {\rho }_t\in V\otimes V$

The map $\lambda :V\otimes V\to ℝ$ is necessarily a nondegenerate and symmetric bilinear form and thus may be used to produce and fix an isomorphism $V\simeq V^{*}$.

This isomorphism is used to get an embedding

$$V\otimes V\mathrm{to}V\otimes V^{*}\hookrightarrow \mathrm{End}(V).$$V \otimes V to V \otimes V^* \hookrightarrow End(V) \,.

The image of this embedding is the set of what in this context will be called “trace class” operators.

With respect to this identification the map $\rho$ is to be understood. For varying $t$ the ${\rho }_t$ form a semigroup (for instance a typical example would be $V=\Gamma (E)$ a space of sections of a vector bundle and ${\rho }_t=e^{-t\Delta }$ for $\Delta$ a Laplace operator on $E$).

note for $\lambda :V\otimes V\to ℝ$ to be continuous, one cannot use the Hilbert tensor product ${\otimes }_H$

the reason is that we have the folloing possible mpas out of the following possible tensor products

$$ℝ\stackrel{\lambda }{\leftarrow }V{\otimes }_{\mathrm{algebraic}}V\stackrel{\mathrm{finite}\mathrm{rank}}{\hookrightarrow }\mathrm{End}(V)$$\mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes_{algebraic} V \stackrel{finite rank}{\hookrightarrow} End(V)

$$ℝ\stackrel{\lambda }{\leftarrow }V\otimes V\stackrel{\mathrm{trace}\mathrm{class}}{\hookrightarrow }\mathrm{End}(V)$$\mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes V \stackrel{trace class}{\hookrightarrow} End(V)

$$ℝ\stackrel{\lambda }{\leftarrow }V{\otimes }_{H}V\stackrel{\mathrm{Hilbert}\mathrm{Schmitdt}}{\hookrightarrow }\mathrm{End}(V)$$\mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes_H V \stackrel{Hilbert Schmitdt}{\hookrightarrow} End(V)

(so here the middle is the projective tensor product, the one that we are actually using)

smooth version / families version

We now refine the definition of the categories $R{\mathrm{Bord}}_d$ and $\mathrm{TV}$ such that they remember smooth stucture.

Effectively, what the following implicitly does is to refine these categories to stacks with values in categories over Diff. The fibred categories that appear in the following, $R{\mathrm{Bord}}_d^{\mathrm{fam}}\to \mathrm{Diff}$ and ${\mathrm{TV}}^{\mathrm{fam}}\to \mathrm{Diff}$ are the Grothendieck construction of these stacks.

definition of ${\mathrm{TV}}^{\mathrm{fam}}$

recall that $\mathrm{TV}$ denotes the category of locally convex Hausdorff topological vector space

now let ${\mathrm{TV}}^{\mathrm{fam}}$ be the fibred category over Diff whose fiber over $X\in \mathrm{Diff}$ is the category of topological vector bundles over $X$. This has as objects vector bundles of topological vector spaces, and the morphisms are fiberwise linear $C^{\mathrm{\infty }}$-morphisms of bundles in the following sense:

let $V,W\in \mathrm{TV}$

Then a linear map $F:V\to W$ is – for any inclusion $U\hookrightarrow V$

$$\begin{array}{ccc}V& \stackrel{f}{\to }& W\\ {\uparrow }^{\subset }& \nearrow \\ U\end{array}$$\array{ V &\stackrel{f}{\to}& W \\ \uparrow^\subset & \nearrow \\ U }

– called $C^1$ at $u\in U$ in the direction $v\in V$ if

$$\underset{t\to 0}{\lim }\frac{F(u+tv)-F(u)}{t}$$\lim_{t \to 0} \frac{F(u+t v) - F(u)}{t}

exists in $W$ and

$$U\times V\to W$$U \times V \to W

$$(u,v)\mapsto d{F}_u(v)$$(u,v) \mapsto d F_u(v)

is continuous.

Iteratively one defines $C^n$ and then $C^{\mathrm{\infty }}$. The morphsims of $\mathrm{TV}$-bundles are supposed to be $C^{\mathrm{\infty }}$ maps in this sense (linear in the fibers, of course)

$$\begin{array}{ccc}V\prime & \stackrel{\tilde{f}}{\to }& V\\ \downarrow & & \downarrow \\ S\prime & \stackrel{f}{\to }& S\end{array}$$\array{ V' &\stackrel{\tilde f}{\to}& V \\ \downarrow &&\downarrow \\ S' &\stackrel{f}{\to}& S }

definition of $R{\mathrm{Bord}}_d^{\mathrm{fam}}$

Similarly $R{\mathrm{Bord}}_d^{\mathrm{fam}}$ has as objects submersions $Y\to S$ and $Y^c\to S$ (not necessarily surjective) with a smooth rank-2 tensor on $Y$ that fiberwise induces the structure of a Riemannian manifold (so these are $S$-families of Riemannian manifolds) such that

$$\begin{array}{ccc}Y& {\leftarrow }^{\subset }& Y^c\\ {\downarrow }^{\mathrm{submersion}}& {\swarrow }_{\mathrm{proper}\mathrm{subm}.}\\ S\end{array}$$\array{ Y &\leftarrow^\subset& Y^c \\ \downarrow^{submersion} & \swarrow_{proper subm.} \\ S }

recall that a map is a proper map if inverse images of compact sets are compact.

remark Notice that if we fix the topology of the fibers in $Y\to S$, then what varies as we vary the fibers is the Riemannian metric on the fibers, so here each $S$ 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 $X$ here will play as a kind of “moduli space of field theories”.

a morphism in $R{\mathrm{Bord}}_d^{\mathrm{fam}}$ in

$$R{\mathrm{Bord}}_d^{\mathrm{fam}}(\begin{array}{c}{Y}_0\\ \downarrow \\ S_0\end{array},\begin{array}{c}{Y}_1\\ \downarrow \\ S_1\end{array})$$R Bord_d^{fam}\left( \array{ Y_0 \\ \downarrow \\ S_0 }, \;\; \array{ Y_1 \\ \downarrow \\ S_1 } \right)

are isometric rel boundary classes of submersions $\Sigma \to S_0$ such that

$$\begin{array}{ccccc}\Sigma & \stackrel{i_1}{\leftarrow }& f^{*}{Y}_1& \to & Y_1\\ {\uparrow }^{i_0}& & \downarrow & & \downarrow \\ Y_0& \to & S_0& \stackrel{f}{\to }& S_1\end{array}$$\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 }

so here $\Sigma$ is an $S_0$-family of cobordisms.

Riemannian field theories

definition

A $d$-dimensional Riemannian quantum field theory is a symmetric monoidal functor

$$E\in {\mathrm{Fun}}_{\mathrm{Diff}}^{\otimes }(R{\mathrm{Bord}}_d^{\mathrm{fam}},{\mathrm{TV}}^{\mathrm{fam}})$$E \in Fun^\otimes_{Diff}(R Bord_d^{fam}, TV^{fam})

such that

$$\begin{array}{ccccc}R{\mathrm{Bord}}_d^{\mathrm{fam}}& & \stackrel{}{\to }& & {\mathrm{TV}}^{\mathrm{fam}}\\ & \searrow & & \swarrow \\ & & \mathrm{Diff}\end{array}$$\array{ R Bord_d^{fam} &&\stackrel{}{\to}&& TV^{fam} \\ & \searrow && \swarrow \\ && Diff }

and such that it preserves pullback

(so its a cartesian functor between these fibered categories that is also symmetric monoidal)