I will discuss 3d N=4 supersymmetric gauge theories compactified on an elliptic curve, and how this set-up physically realises recent mathematical results on the equivariant elliptic cohomology of symplectic resolutions. In particular, I will describe the Berry connection for supersymmetric ground states, and in doing so connect the elliptic cohomology of the Higgs branch with spectral data of doubly periodic monopoles. I will show that boundary conditions, via a consideration of boundary ’t Hooft anomalies, naturally represent elliptic cohomology classes. Finally, if I have time, I will discuss mirror symmetry/symplectic duality in our framework, and physically recover concepts in elliptic cohomology such as the mother function, and the elliptic stable envelopes of Aganagic-Okounkov.
This talk will be based on https://arxiv.org/abs/2109.10907 with Mathew Bullimore.