The Quantum GIT conjecture

Let X be a Fano variety with G action. The quantum GIT conjecture predicts a formula for the quantum cohomology of “anti-canonical” GIT quotients X//G in terms of the equivariant quantum cohomology of X. The formula is motivated by ideas from 3- dimensional gauge theory (“Coulomb branches”) and provides a vast generalization of Batyrev’s formula for the quantum cohomology of a toric Fano variety. I will explain describe our ongoing work with C. Teleman proving this conjecture. Along the way, I will also discuss integral versions of certain classical facts in the theory of Hamiltonian G-manifolds which are of independent interest.