Axiomatic field theories and their motivation from topology

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 indicates, generally, how FQFTs may be related to cohomology theories.

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


All of the following assumes that the reader is well familiar with the basic ideas indicated at FQFT.

In the following dRBd-RB or RBord dR Bord_d and the like denotes a category of dd-dimensiomnal Riemannian cobordisms that are equipped with Riemannian structure, i.e. with Riemannian metric. Similarly dEBd-E B or EBord dE Bord_d or the like denotes a category of cobordisms with Eudlidean structure, by which is meant a flat Riemannian metric.

The definition of this is discussed in detail in


dd-dimensional Riemannian field theories are symmetric monoidal functors dRBTVd-RB \to TV from dd-dimensional Riemannian bordisms to topological vector spaces.

A field theory is very similar to a representation of a group. Only where a representation of a group GG is a functor from the delooping BG=*//G\mathbf{B}G = {*}//G of GG to Vect, an FQFT is a representation of a more complicated domain category.

how does topology enter?

for XX some topological space there is also a symmetric monoidal category

dRB(X) d-RB(X)

of Riemannian bordisms equipped with a continuous map to XX.

Notice that dRB(X)d RB(X) does depend covariantly on XX. This means that Fun (dRB(X),TV)Fun^\otimes(d RB(X), TV) is contravariant in XX.

When special structure is around, however, we also have a push-forward of such functors along morphisms.

Example: push-forward to the point: for XX as above and X*X \to {*} the unique map to the point heuristically we want a map

dRFT(X)p *dRFT(pt) d RFT(X) \stackrel{p_*}{\to} d RFT(pt)

notice that this push-forward is not an adjoint functor. Instead, it is a map that comes from integration over fibers. In particular it will change the degree of cohomology theories.

heuristically the pushforward

dRFT(X)p *dRFT(pt) d RFT(X) \stackrel{p_*}{\to} d RFT(pt)

acts on field theories E XE_X over XX

E XE E_X \mapsto E

by the assignment

E(Y d1)Γ(E X(Y) Maps(Y,X)regardedasavectorbundle) E(Y^{d-1}) \mapsto \Gamma\left( \array{ E_X(Y) \\ \downarrow \\ Maps(Y,X) } regarded as a vector bundle \right)

for instance when E X(Y)=E_X(Y) = \mathbb{C} then E(Y)=Γ(Maps(Y,X))E(Y) = \Gamma(Maps(Y,X)).

For instance take Σ\Sigma to be the pair of pants with input Y 0Y_0 and output Y 1Y_1 and let F:ΣXF : \Sigma \to X be a map.

Then E(Σ)E(\Sigma) will take some section ΨE(Y 0)\Psi \in E(Y_0) to

E(Σ)(Ψ):(f 1:Y 1X) {F:ΣX}F/Y 1=f 1E X(F)(Ψ(f 0))1Zexp(S(F)dvol). E(\Sigma)(\Psi) : (f_1 : Y_1 \to X) \mapsto \int_{\{F : \Sigma \to X\} | F/Y_1 = f_1} E_X(F)(\Psi(f_0)) \frac{1}{Z}\exp(-S(F) dvol) \,.

Here the expression 1Zexp(S(F)dvol)\frac{1}{Z}\exp(-S(F) dvol) denotes a would-be measure which is still to be defined.

Here f 0=F/Y 0f_0 = F/Y_0 is the restriction of FF to Y 0Y_0.

Contravariant functors with push-forward also arise as part of a cohomology theory

h n:Diff opAb h^n : Diff^{op} \to Ab

from the category Diff of smooth manifolds to the category Ab of abelian groups that satisfies the axioms of generalized (Eilenberg-Steenrod) cohomology theory.

These Eilenberg-Steenrod axioms are

  1. homotopy axiom for h nh^n

  2. Mayer-Vietoris axiom for h nh^n

  3. suspension isomorphism h n(X)h cvs n+1(X×)h^n(X) \simeq h^{n+1}_{cvs}(X \times \mathbb{R})

here in the last axiom for topological spaces we’d simply have the suspension h n+1(ΣX)h^{n+1}(\Sigma X). Here in order to stay within manifolds we instead do as indicated, where cvs{}_{cvs} means “compact vertical support”.

Concerning the homotopy axiom: given any functor that sends manifolds to a category of FQFTs over XX

dFTs:=h:Diff opC d-FTs := h : Diff^{op} \to C

in dRFTdRFTs we can make it a homotopy functor by defining

ω 0,ω 1h(X)\omega_0, \omega_1 \in h(X) to be concordant if ωh(X×)\exists \omega \in h(X \times \mathbb{R}) such that ω/(X×{i})ω i\omega/(X\times \{i\}) \simeq \omega_i in h(X)h(X), i=0,1i = 0,1

then under this relation

XdRFT(X)/ X \mapsto d-RFT(X)/\simeq

is a homotopy invariant functor

Homework: for h(X)=Ω closed n(X)h(X) = \Omega^n_{closed}(X)

we have that ω 0\omega_0 concordant to ω 1\omega_1 exactly when ω 0=ω 1+dα\omega_0 = \omega_1 + d \alpha for some αΩ n1(X)\alpha \in \Omega^{n-1}(X)

Concerning the Mayer-Vietoris axiom: if X=UVX = U \cup V then the functor should respect this gluing.

suppose E2RFT(X)E \in 2-RFT(X) is a 2d Riemmanian field theory. let γ:S 1X\gamma : S^1 \to X be a loop in XX. then E(γ)E(\gamma) is a vector space.

If γ\gamma sits neither entirely in UU or in VV, then there is no way that the vector space E(γ)E(\gamma) can be reconstructed by knowing just the restriction of RR to UU and VV.

So this is a problem for the definition of field theories so far.

The proposed solution (from What is an elliptic object?) is to use extended FQFTs instead. This introduces locality into FQFTs, at the expense of working with n-categories.

This will however not be studied here for the moment.

Concerning the suspension isomorphism: for that first we need for nn \in \mathbb{Z} the notion of a field theory over XX of degree nn, i.e.

XdRFTs n(X) X \mapsto d-RFTs^n(X)

such that for n=0n=0 this is an ordinary Riemannian field theory, in dRFT(X)d-RFT(X).

This requires to replace manifolds by supermanifolds.

Example let d=0d = 0 and consider 0-dimensional TFTs over XX.

so consider

Fun (0Bord(X),TV) Fun^\otimes(0 Bord(X), TV)

in 0Bord(X)0 Bord(X) there is only a single object: the empty set \emptyset which has a unique map X\emptyset \to X

the collecton of morphisms is {finitesetX}\{finite set \to X\} in there is Hom Diff(pt,X)Hom_{Diff}(pt,X).

here composition of morphisms is the same as tensor product of objects: both comes from the disjoint union of these finite set domains.

E:0Bord(X)TV E : 0 Bord(X) \to TV_{\mathbb{R}}
\emptyset \mapsto \mathbb{R}
xXE(X)mathbR x \in X \mapsto E(X) \in \mathb{R}

so a priori we have

Fun (0Bord(X),TV)Maps(X,), Fun^\otimes(0 Bord(X), TV) \simeq Maps(X, \mathbb{R}) \,,

where on the right we have all maps.

This is not quite what is intended. We want to see smooth maps on the right. To get that, we need to talk about smooth functors on the left. This is the topic of later discussion, which will yield

SmoothFun (0Bord(X),TV)C (X,). SmoothFun^\otimes(0 Bord(X), TV) \simeq C^\infty(X, \mathbb{R}) \,.

But one other ting goes wrong the: the corresponding homotopy functor is C (X)/={0}C^\infty(X)/\simeq = \{0\}. So turning this into an Eilenberg-Steenrod theory yields the trivial theory.

The way out to that will be to go to supermanifolds.

Punchline of this session here:

(01)TFTs(X)SmoothFun ((01)Bord(X),TV) (0|1)-TFTs(X) \simeq SmoothFun^\otimes((0|1)Bord(X), TV)

and then it is a lemma Glemma^G that

C (SuperDiff( 01,X)) GΩ (X) G=( n=0 Ω n(X)) G C^\infty(SuperDiff(\mathbb{R}^{0|1}, X))^G \simeq \Omega^\bullet(X)^G = (\oplus_{n = 0}^{\infty} \Omega^n(X))^G

where GG is the supergroup G=Diff( 01)wim42 01 ×G = Diff(\mathbb{R}^{0|1}) \wim42 \mathbb{R}^{0|1} \rtimes \mathbb{R}^\times.

The degree decomposition on forms then turns out to be the eigenvalue decompositon of the ×\mathbb{R}^\times-part, while the deRham differential is the 01\mathbb{R}^{0|1}-action (cite Kontsevich, Severa here …)

so then from that one finds that the GG-invariant bit is the closed 0-forms

Ω (X) GΩ closed 0(X) \Omega^\bullet(X)^G \simeq \Omega^0_{closed}(X)

So we get nn-Lemma :

(01)TFT n(X)Ω cl n(X) (0|1)-TFT^n(X) \simeq \Omega^n_{cl}(X)


(01)TFT n(X) concordH dR n(X) (0|1)-TFT^n(X)_{concord} \simeq H^n_{dR}(X)


Now with the super-directions included, there is a notion of push-forward of the TFTs that does shift the degree, and we get the following

Theorem (Stolz-Teichner-Kreck-Hohnhold)

when XX is oriented

(01)TFT n(X)push(01)TFT 0 (0|1)TFT^n(X) \stackrel{push}{\to} (0|1)TFT^0 \simeq \mathbb{R}
Ω cl n(X) X nΩ cl 0(pt) \Omega^n_{cl}(X) \stackrel{\int_{X_n}}{\to} \Omega^0_{cl}(pt) \simeq \mathbb{R}

and similarly after dividing out concordance, the push-forward becomes the push-forward in deRham cohomology.


  • Stefan Stolz (notes by Arlo Caine), Supersymmetric Euclidean field theories and generalized cohomology Lecture notes (2009) (pdf)
Revised on June 7, 2011 11:36:05 by Urs Schreiber (