Suppose G is a Lie group acting on a smooth manifold M. There is a well-known correspondence between {G-equivariant vector bundles on M} and {vector bundles on the quotient stack M//G}, but the situation is more complicated for G-equivariant vector bundles with connection. In this talk, we explain how to naturally resolve the problem by using the "differential quotient stack," defined via principal G-bundles with connection. Precisely, {G-equivariant vector bundles with G-invariant connection on M} are equivalent to {vector bundles with connection on the differential quotient stack}.
This perspective immediately generalizes to higher structures, and we can show the analogous: {G-equivariant gerbe connections on M} are equivalent to {gerbe connections on the differential quotient stack}. This is based on joint work with Byungdo Park, and we use the G-equivariant gerbe connections considered by Meinrenken, Stiénon, and Tu-Xu.
Finally, we define differential equivariant cohomology (or equivariant Deligne cohomology) groups. These provide a natural home for equivariant Chern-Weil theory, and in degree 3 they classify equivariant gerbe connections.
This perspective immediately generalizes to higher structures, and we can show the analogous: {G-equivariant gerbe connections on M} are equivalent to {gerbe connections on the differential quotient stack}. This is based on joint work with Byungdo Park, and we use the G-equivariant gerbe connections considered by Meinrenken, Stiénon, and Tu-Xu.
Finally, we define differential equivariant cohomology (or equivariant Deligne cohomology) groups. These provide a natural home for equivariant Chern-Weil theory, and in degree 3 they classify equivariant gerbe connections.