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



see also



The goal here is to define 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 Bord dBord_d has

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 YY denotes dd-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

RBord dR Bord_d is defined using bicollars from the beginning

an object in RBord dR Bord_d is a quintuple

consisting of

  • a dd-dimensional Riemannian manifold YY;

  • a core (d1)(d-1)-manifold Y cY^c sitting Y cYY^c \hookrightarrow Y in a thickening Y dY^d – being a dd-manifold –

  • Y +,Y YY^+, Y^- \hookrightarrow Y two disjointly embedded open dd-dimensional manifolds such that

    • Y cY^c is in the closure of both Y +Y^+ and Y Y^-

    • that Y +Y =Y\Y CY^+ \coprod Y^- = Y \backslash Y^C

so the picture of an object, which is missing in this writeup here for the moment, is a d1d-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 ±)(Y_0,Y_0^c, Y_0^{\pm}) to (Y 1,Y 1 c,Y 1 ±)(Y_1,Y_1^c, Y_1^{\pm}) is a triple (Σ,i 0,i 1)(\Sigma, i_0, i_1) where

  • Σ\Sigma is a dd-dimensional Riemannian manifold without boundary

  • for i=0,1i=0,1 an open neighbourhood of the core Y i cW iopenY iY_i^c \hookrightarrow W_i \stackrel{open}{\hookrightarrow} Y_i

    this defines the intersections W k ±:=W kY k ±W^\pm_k := W_k \cap Y^\pm_k with the two collars for each k=0,1k = 0,1.

  • a smooth map i k:W kΣi_k : W_k \to \Sigma

    such that

    • i k:W k +Y k cZi_k : W^+_k \cup Y_k^c \to Z is a proper map;

    • (+) for i k +:=i k/W k +i^+_k := i_k/W^+_k are isometric embeddings into Σ\i 1(W 1 Y 1 c)\Sigma \backslash i_1(W^-_1 \cup Y^c_1)

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

    • the core Σ c:=Σ\(i 0(W 0 +)i 1(W 1 ))\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 TVTV of topological vector spaces will have a similar asymmetry (from the fact that for \infty-dimensional vector spaces there is an evaluation map but not necessarily a coevaluation/unit for VV *V \otimes V^*), similarly, with the above asymmetric definition we have a cobordims YY *Y \coprod Y^* \to \emptyset (where Y *Y^* is obtained from YY 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

ϕ:W 0W 1 \phi : W_0 \to W_1

such that ϕ\phi preserves the decomposition W k ±,Y k cW_k^\pm, Y_k^c we get a Riemannian cobordism using

Σ:=W 1 \Sigma := W_1


i 1=Id W 1,i 0=ϕ i_1 = Id_{W_1}\,,\;\;\;\;\; i_0 = \phi

definition (morphisms in RBord dR Bord_d) morphisms from Y 0Y_0 to YY in RBord dR Bord_d (or dRBd-RB or whatever the notation is) are isometry classes rel. boundary (see below) of Riemannian cobordisms from Y 0Y_0 to Y 1Y_1.

We require the commutativity of the following diagram

V 1 i 1 X i 0 V 0 f 1 f 0 V 1 i 1 X i 0 V 0 \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)(F,f_0, f_1) is “rel. boundary” if f 0=Idf_0 = Id and f 1=Idf_1 = Id

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

V 1 i 1 X i 0 V 0 Id Id V 1 i 1 X i 0 V 0 \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=1d=1

we decribe RBord 1R Bord_1 explicitly

it has at least the object

pt=(pt pt c pt + )=(,{0}, ±) pt = \left( \array{ pt^- & pt^c & pt^+ \\ -- & \bullet & -- } \right) = (\mathbb{R}, \{0\}, \mathbb{R}_\pm)

which is a point with collar all of \mathbb{R}.

Lemma every object in RBord 1R Bord_1 which is connected and not the empty set is isomorphic to this ptpt

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

I tRBord 1(pt,pt) I_t \in R Bord_1(pt,pt)

defined as the triple (,i 0,i 1)(\mathbb{R}, i_0, i_1) where i 0:i_0 : \mathbb{R} \to \mathbb{R} is the identity map, and where i 1:i_1 : \mathbb{R} \to \mathbb{R} is translation by tt.

This means that i 0i_0 takes the core of in the incoming point to 00 \in \mathbb{R} while i 1i_1 takes the core of the outgoing point to tt \in \mathbb{R}. Everything in \mathbb{R} outside of [0,1][0,1] is hence “collar” and this describes what naively one would think of as just the interval [0,1][0,1] regarded as a Riemannian cobordism.

Lemma The composition of these cobordisms is given by

I tI t=I t+t I_t \circ I_{t'} = I_{t+t'}

There are also morphisms

L +:ptpt L_+ : pt \coprod pt \to \emptyset


R +:ptpt R_+ : \emptyset \to pt \coprod pt

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

Here L tL_t is formall given exactly as I tI_t only that the map i 0:i_0 : \mathbb{R} \to \mathbb{R} is not the identity, but reflection at the origin. This encodes the orientation reversal at that end.

This is defined for t>0t \gt 0. For t=0t= 0 the morphism L 0L_0 is still defined, but R 0R_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 0R_0 does not satisfy the axioms above.

So this means that we have a cobordism of length 0 going ptpt\emptyset \to pt \coprod pt, but all cobordisms going the other way round ptptpt \coprod pt \to \emptyset will have to have non-vanishing length.

Another morphism in RBord 1R Bord_1 is the morphism

σ:ptptptpt \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 tσ=L tL_t \circ \sigma = L_t

  • R t=σR tR_t = \sigma \circ R_t

  • R t L 0R t=R t+tR_t \circ_{L_0} R_{t'} = R_{t+t'}

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

(Id ptL 0Id pt)(R tR t) (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 RBord 1R Bord_1 is generated as a symmetric monoidal category by

  • the object ptpt

  • the morphisms L 0L_0, {R t} t>0\{R_t\}_{t \gt 0}

subject to the relations

L 0σ=L 0 L_0 \circ \sigma = L_0
σR t=R t \sigma \circ R_t = R_t
t,t>0:R t L 0R t=R t+t \forall t,t' \gt 0 : R_t \circ_{L_0} R_{t'} = R_{t + t'}

corollary symmetric monoidal functors

EFun (RBord 1,TV) E \in Fun^\otimes(R Bord_1, TV)

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

  • E:ptVE : pt \mapsto V

  • E:L 0(λ:VV)E : L_0 \mapsto (\lambda : V\otimes V \to \mathbb{R})

  • E:R tρ tVVE : R_t \mapsto \rho_t \in V \otimes V

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

This isomorphism is used to get an embedding

VVtoVV *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 ρ\rho is to be understood. For varying tt the ρ t\rho_t form a semigroup (for instance a typical example would be V=Γ(E)V = \Gamma(E) a space of sections of a vector bundle and ρ t=e tΔ\rho_t = e^{-t \Delta} for Δ\Delta a Laplace operator on EE).

note for λ:VV\lambda : V \otimes V \to \mathbb{R} to be continuous, one cannot use the Hilbert tensor product H\otimes_H

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

λV algebraicVfiniterankEnd(V) \mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes_{algebraic} V \stackrel{finite rank}{\hookrightarrow} End(V)
λVVtraceclassEnd(V) \mathbb{R} \stackrel{\lambda}{\leftarrow} V \otimes V \stackrel{trace class}{\hookrightarrow} End(V)
λV HVHilbertSchmitdtEnd(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 RBord dR Bord_d and TVTV 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, RBord d famDiffR Bord_d^{fam} \to Diff and TV famDiffTV^{fam} \to Diff are the Grothendieck construction of these stacks.

definition of TV famTV^{fam}

recall that TVTV denotes the category of locally convex Hausdorff topological vector space

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

let V,WTVV, W \in TV

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

V f W U \array{ V &\stackrel{f}{\to}& W \\ \uparrow^\subset & \nearrow \\ U }

– called C 1C^1 at uUu \in U in the direction vVv \in V if

lim t0F(u+tv)F(u)t \lim_{t \to 0} \frac{F(u+t v) - F(u)}{t}

exists in WW and

U×VW U \times V \to W
(u,v)dF u(v) (u,v) \mapsto d F_u(v)

is continuous.

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

V f˜ V S f S \array{ V' &\stackrel{\tilde f}{\to}& V \\ \downarrow &&\downarrow \\ S' &\stackrel{f}{\to}& S }

definition of RBord d famR Bord_d^{fam}

Similarly RBord d famR Bord_d^{fam} has as objects submersions YSY \to S and Y cSY^c \to S (not necessarily surjective) with a smooth rank-2 tensor on YY that fiberwise induces the structure of a Riemannian manifold (so these are SS-families of Riemannian manifolds) such that

Y Y c submersion propersubm. S \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 YSY \to S, then what varies as we vary the fibers is the Riemannian metric on the fibers, so here each SS 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 XX here will play as a kind of “moduli space of field theories”.

a morphism in RBord d famR Bord_d^{fam} in

RBord d fam(Y 0 S 0,Y 1 S 1) 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 ΣS 0\Sigma \to S_0 such that

Σ i 1 f *Y 1 Y 1 i 0 Y 0 S 0 f S 1 \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 Σ\Sigma is an S 0S_0-family of cobordisms.

Riemannian field theories


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

EFun Diff (RBord d fam,TV fam) E \in Fun^\otimes_{Diff}(R Bord_d^{fam}, TV^{fam})

such that

RBord d fam TV fam Diff \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)

Invertible Field Theories

In the study of invertible field theories, let Bord n G\mid \mid Bord^G_n \mid \mid be the fundamental groupoid of the classifying space of Bord n GBord^G_n. An invertible field theory factors through this and the Picard subgroupoid 𝒞 *\mathcal{C}^* of the target.

Kreck, S. Stolz, and P. Teichner.Invertible topological field theories are SKK invariants.

Last revised on December 19, 2019 at 09:52:53. See the history of this page for a list of all contributions to it.