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 ${\mathrm{RCob}}_d\to \mathrm{TopVect}$ 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 $BG=*//G$ 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

$${\mathrm{RCob}}_d(X)$$RCob_d(X)

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

Notice that $d{\mathrm{RRCob}}_d(X)$ does depend covariantly on $X$. This means that ${\mathrm{Fun}}^{\otimes }({\mathrm{RCob}}_d(X),\mathrm{TopVect})$ 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

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

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

acts on field theories $E_X$ over $X$

$$E_X\mapsto E$$E_X \mapsto E

by the assignment

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

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

category: reference