SciPost Phys. 13, 005 (2022) ·
published 22 July 2022
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This paper studies $3d$ $\mathcal{N}=4$ 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 cohomology
variety 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
analyse 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 elliptic
cohomology. Finally, we consider correlation functions of Janus interfaces for
varying mass parameters, recovering the chamber R-matrices of elliptic
integrable systems.
