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

Abstract

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.

101 pages, 21 figures. v2: version published in SciPost. References updated, typos corrected

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• Remarks on Berry Connection in QFT, Anomalies, and Applications
• di-Langlands correspondence and extended observables
• Towards the theory of Yangians
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• ${\text{AdS}}_4$ Holography and the Hilbert Scheme
• Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence
• Difference Equations: from Berry Connections to the Coulomb Branch