nLab
equivariant ordinary differential cohomology

Contents

Context

Differential cohomology

Representation theory

Contents

Idea

Equivariant ordinary differential cohomology or equivariant Deligne cohomology is the equivariant cohomology-enhancement of ordinary differential cohomology (Deligne cohomology), equivalently the differential cohomology-enhancement of ordinary equivariant cohomology.

There is one evident general abstract definition of what this means concretely: Given a GG-action on some XX, the (Borel-)GG-equivariant ordinary differential cohomology of XX in degree nn is the (∞,1)-categorical hom-space in the cohesive (∞,1)-topos of Smooth∞Groupoids from the quotient stack XGX \sslash G to the Deligne complex B nU(1) conn\mathbf{B}^n U(1)_{conn}.

That various definitions in the literature coincide with this one (in particular Kübel-Thom 15, and for finite groups also Lupercio-Uribe 01, Gomi 05) is discussed in (Park-Redden 19).

References

The main definitions can be found in

An earlier definition thst works (only) for finite groups can be found in:

Other references:

Relation to action functionals for topological field theories (such as Chern-Simons theory):

Last revised on March 8, 2021 at 09:27:33. See the history of this page for a list of all contributions to it.