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
next:
All of the following assumes that the reader is well familiar with the basic ideas indicated at FQFT.
In the following $d-RB$ or $R Bord_d$ and the like denotes a category of $d$-dimensional Riemannian cobordisms that are equipped with Riemannian structure, i.e. with Riemannian metric. Similarly $d-E B$ or $E Bord_d$ or the like denotes a category of cobordisms with Euclidean structure, by which is meant a flat Riemannian metric.
The definition of this is discussed in detail in
definition
$d$-dimensional Riemannian field theories are symmetric monoidal functors $d-RB \to TV$ from $d$-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 $G$ is a functor from the delooping $\mathbf{B}G = {*}//G$ of $G$ to Vect, an FQFT is a representation of a more complicated domain category.
how does topology enter?
for $X$ some topological space there is also a symmetric monoidal category
of Riemannian bordisms equipped with a continuous map to $X$.
Notice that $d RB(X)$ does depend covariantly on $X$. This means that $Fun^\otimes(d RB(X), TV)$ is contravariant in $X$.
When special structure is around, however, we also have a push-forward of such functors along morphisms.
Example: push-forward to the point: for $X$ as above and $X \to {*}$ the unique map to the point heuristically we want a map
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
acts on field theories $E_X$ over $X$
by the assignment
for instance when $E_X(Y) = \mathbb{C}$ then $E(Y) = \Gamma(Maps(Y,X))$.
For instance take $\Sigma$ to be the pair of pants with input $Y_0$ and output $Y_1$ and let $F : \Sigma \to X$ be a map.
Then $E(\Sigma)$ will take some section $\Psi \in E(Y_0)$ to
Here the expression $\frac{1}{Z}\exp(-S(F) dvol)$ denotes a would-be measure which is still to be defined.
Here $f_0 = F/Y_0$ is the restriction of $F$ to $Y_0$.
Contravariant functors with push-forward also arise as part of a cohomology theory
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
homotopy axiom for $h^n$
Mayer-Vietoris axiom for $h^n$
suspension isomorphism $h^n(X) \simeq h^{n+1}_{cvs}(X \times \mathbb{R})$
here in the last axiom for topological spaces we’d simply have the suspension $h^{n+1}(\Sigma X)$. Here in order to stay within manifolds we instead do as indicated, where ${}_{cvs}$ means “compact vertical support”.
Concerning the homotopy axiom: given any functor that sends manifolds to a category of FQFTs over $X$
in $dRFT$s we can make it a homotopy functor by defining
$\omega_0, \omega_1 \in h(X)$ to be concordant if $\exists \omega \in h(X \times \mathbb{R})$ such that $\omega/(X\times \{i\}) \simeq \omega_i$ in $h(X)$, $i = 0,1$
then under this relation
is a homotopy invariant functor
Homework: for $h(X) = \Omega^n_{closed}(X)$
we have that $\omega_0$ concordant to $\omega_1$ exactly when $\omega_0 = \omega_1 + d \alpha$ for some $\alpha \in \Omega^{n-1}(X)$
Concerning the Mayer-Vietoris axiom: if $X = U \cup V$ then the functor should respect this gluing.
suppose $E \in 2-RFT(X)$ is a 2d Riemmanian field theory. let $\gamma : S^1 \to X$ be a loop in $X$. then $E(\gamma)$ is a vector space.
If $\gamma$ sits neither entirely in $U$ or in $V$, then there is no way that the vector space $E(\gamma)$ can be reconstructed by knowing just the restriction of $R$ to $U$ and $V$.
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 $n \in \mathbb{Z}$ the notion of a field theory over $X$ of degree $n$, i.e.
such that for $n=0$ this is an ordinary Riemannian field theory, in $d-RFT(X)$.
This requires to replace manifolds by supermanifolds.
Example let $d = 0$ and consider 0-dimensional TFTs over $X$.
so consider
in $0 Bord(X)$ there is only a single object: the empty set $\emptyset$ which has a unique map $\emptyset \to X$
the collecton of morphisms is $\{finite set \to X\}$ in there is $Hom_{Diff}(pt,X)$.
here composition of morphisms is the same as tensor product of objects: both comes from the disjoint union of these finite set domains.
so a priori we have
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
But one other ting goes wrong the: the corresponding homotopy functor is $C^\infty(X)/\simeq = \{0\}$. 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:
and then it is a $lemma^G$ that
where $G$ is the supergroup $G = Diff(\mathbb{R}^{0|1}) \wim42 \mathbb{R}^{0|1} \rtimes \mathbb{R}^\times$.
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 $\mathbb{R}^{0|1}$-action (cite Kontsevich, Severa here …)
so then from that one finds that the $G$-invariant bit is the closed 0-forms
So we get $n$-Lemma :
and
push-forward
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 $X$ is oriented
and similarly after dividing out concordance, the push-forward becomes the push-forward in deRham cohomology.