### A quantum refinement of equivariant cohomology

#### Hekmati Pedram

The algebra of polynomials on the dual of a Lie algebra carries a natural Poisson structure given by the Kostant-Kirillov-Souriau bracket. The celebrated Duflo homomorphism quantises this Poisson algebra to the enveloping algebra, in such a way that its restriction to the centers is an isomorphism of commutative algebras. In this talk I will first review the Duflo map and its relation to equivariant cohomology. I will then introduce a refinement of equivariant cohomology based on the affine Kac-Moody vertex algebra. Interestingly this construction only works at the so called critical level. I will show that the extended equivariant cohomology of a point coincides with the Feigin-Frenkel center generated by Segal-Sugawara vectors and discuss a jet analogue of the Duflo homomorphism. This is joint work with Andrew Linshaw.