Equivariant Cohomology Models for Differentiable Stacks

Abstract: We introduce the concept of equivariant cohomology in the smooth manifold case and the notion of differentiable stacks. Then we consider an action of a Lie group on a differentiable stack in the sense of Romagny and consider the stacky quotient associated to this action. Consequently, we construct an atlas that makes these stacky quotient a differentiable stack. Using that the nerve of the associated Lie groupoid of that stack gives its the homotopy type, we provide a Borel model for equivariant cohomology in this context. In order to follow the classical approach for equivariant cohomology, we build a Cartan model for differentiable stacks and we prove that both models compute the same cohomology as the proposed by the Borel model.

mathematical physicsalgebraic geometrydifferential geometrydynamical systemssymplectic geometry

Audience: researchers in the topic

Junior Global Poisson Workshop II