nLab Geometric Models for Elliptic Cohomology

this is a sub-entry of A Survey of Elliptic Cohomology, see there for background and context.

definition

$d$-dimensional Riemanninan field theories are symmetric monoidal functors $RCob_d \to TopVect$ 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

$$RCob_d(X)$$

of Riemannian bordisms equipped with a continuous map to $X$.

Notice that $d RRCob_d(X)$ does depend covariantly on $X$. This means that $Fun^\otimes(RCob_d(X), TopVect)$ 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

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

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

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

acts on field theories $E_X$ over $X$

$$E_X \mapsto E$$

by the assignment

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

for instance when $E_X(Y) = \mathbb{C}$ then $E(Y) = \Gamma(Maps(Y,X);\mathbb{C})$. This is clearly reminiscent of the pushforward of a sheaf along a continuous functions and suggests that $d RFT(X)$ should be looked at as a sheaf on $X$. It is however not so, since if $X=X_1\cup X_2$ then an object $Y$ in $RCob_d(X)$ (i.e., a $(d-1)$-dimensional closed manifold with a map to $X$) cannot in general be reconstructed from $RCob_d(X_1)$ and $RCob_d(X_2)$. On the other hand, such a reconstruction is possible if one allows objects in $RCob_d(X_i)$ to have $(d-2)$-dimensional boundaries. This point of view leads to extended topological quantum field theory.

category: reference