On the construction of supersymmetric symmetries

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

We study supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions.We first study the berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries, which results in a natural construction of the equivariant elliptic cohomologyvariety of the higgs branch.We then investigate supersymmetric boundary conditions and show from an analysis of boundary't hooft anomalies that their boundary amplitudes represent equivariant elliptic cohomology classes.We analyze two distinguished classes of boundary conditions known as exceptional dirichlet and enriched neumann, which are exchanged under mirror symmetry.We show that the boundary amplitudes of the latter reproduce elliptic stable envelopes introduced by aganagic-okounkov, and relate boundary amplitudes of the mirror symmetry interface to the mother function in equivariant ellipticcohomology.Finally, we consider correlation functions of janus interfaces for varying mass parameters, recovering the chamber r-matrices of elliptic integrable systems.