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Deformation Theory Seminar

Wednesday, September 27, 2017 - 2:00pm

Corbett Redden

LIU Post


University of Pennsylvania


This is now our canonical room

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.