Geometric Models for Elliptic Cohomology

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


dd-dimensional Riemanninan field theories are symmetric monoidal functors RCob dTopVectRCob_d \to TopVect 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

RCob d(X) RCob_d(X)

of Riemannian bordisms equipped with a continuous map to XX.

Notice that dRRCob d(X)d RRCob_d(X) does depend covariantly on XX. This means that Fun (RCob d(X),TopVect)Fun^\otimes(RCob_d(X), TopVect) 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);\mathbb{C}). This is clearly reminiscent of the pushforward of a sheaf along a continuous functions and suggests that dRFT(X)d RFT(X) should be looked at as a sheaf on XX. It is however not so, since if X=X 1X 2X=X_1\cup X_2 then an object YY in RCob d(X)RCob_d(X) (i.e., a (d1)(d-1)-dimensional closed manifold with a map to XX) cannot in general be reconstructed from RCob d(X 1)RCob_d(X_1) and RCob d(X 2)RCob_d(X_2). On the other hand, such a reconstruction is possible if one allows objects in RCob d(X i)RCob_d(X_i) to have (d2)(d-2)-dimensional boundaries. This point of view leads to extended topological quantum field theory.

category: reference

Last revised on December 16, 2009 at 19:30:37. See the history of this page for a list of all contributions to it.