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

by Mathew Bullimore, Daniel Zhang

Submission summary

As Contributors: Daniel Zhang
Arxiv Link: https://arxiv.org/abs/2109.10907v1 (pdf)
Date submitted: 2021-10-04 18:01
Submitted by: Zhang, Daniel
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approach: Theoretical

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.

Current status:
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Submission 2109.10907v1 on 4 October 2021

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