differential equivariant cohomology

Differential equivariant cohomology is the interaction of the notion of

with that of

Difference to naive equivariant gerbes with connection

With the definition of differential cohomology as described at differential cohomology the equivariant case behaves a little differently than one might expect naively. For the case of connections on abelian bundle gerbe this correct non-naive version of the equivariant case is known: the necessary loosening of the notion of connection is known as pseudoconnections cite or alternatively the loosening of the notion of morphisms of bundle gerbes with connection is known under the name of gerbe bimodule cite.

To see the issue, consider some \infty-groupoid X\mathbf{X} whose differential cohomology with coefficients in some \infty-groupoid AA we are interested in. For simplicitly and definiteness we should assume for the purpose of this discussion that X=X//G\mathbf{X} = X//G is the global action groupoid of a finite group GG acting on a space XX. The general case is conceptually not really different.

In the naive approach one would consider the 2-stack U(1)Grb ()U(1)Grb_\nabla(-) of U(1)U(1)-gerbes with connection and connection-preserving morphisms between them. Then one would say that a GG-equivariant gerbe with connection on XX is a cocycle in

H(X//G,U(1)Grb ) \mathbf{H}(X//G, U(1)Grb_\nabla)

such a cocycle is

  • a gerbe with connection aU(1)Grp (X)a \in U(1)Grp_\nabla(X) on XX;

  • for each gGg \in G a morphism (ag *a)U(1)Grb (X)(a \to g^* a) \in U(1)Grb_\nabla(X) of gerbes with connection

  • such that the group structure is respected in the suitable sense.

Write P(a)Ω 3(X)P(a) \in \Omega^3(X) for the curvature 3-form of the gerbe with connection aa. Notice that for every morphism aba \to b of gerbes with connection, and in particular for our morphism ag *aa \to g^* a, the curvature forms of domain and codomain are equal: P(a)=P(b)P(a) = P(b). From the fact alone that these are curvature forms on equivalent gerbes follows already that these closed curvature forms must represent the same element in deRham cohomology. But the existence of a connection-preserving morphism between the gerbes then forces these forms to be not just cohomologous, but equal.

And this is not the right condition in the equivariant case.

This was noticed originally in examples where it was clear from the context that examples of equivariant gerbes with connection ought to exist, but where the naive definition yielded an empty set of those.

We point out that with the definition of gerbes with connection as following from our notion of differential cohomology, the problem is resolved automatically.

For by its very definition, a differential cocycle is not required to have a curvature 3-form in H(X,B 3U(1) flat)\mathbf{H}(X,\mathbf{B}^3U(1)_{flat}), but in H(X//G,B 3U(1) flat)\mathbf{H}(X//G,\mathbf{B}^3U(1)_{flat}). This means in words that we allow a curvature 3-form which is itself GG-equivariant in a possibly non-trivial way: there may be non-trivial coboundaries between H 3Ω 3(X)H_3 \in \Omega^3(X) and g *H 3Ω 3(X)g^* H_3 \in \Omega^3(X). So these two forms are required to be just cohomologous, not required to be equals. (And the coboundaries inducing the cohomology must respect the group structure, of course.).

Revised on September 17, 2009 12:13:27 by Urs Schreiber (