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3d N = 4 Gauge Theories on an Elliptic Curve

Bullimore, Mathew; Zhang, Daniel

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Daniel Zhang


This paper studies 3d 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.


Bullimore, M., & Zhang, D. (2022). 3d N = 4 Gauge Theories on an Elliptic Curve. SciPost Physics, 13(1),

Journal Article Type Article
Acceptance Date Jun 1, 2022
Online Publication Date Jul 22, 2022
Publication Date 2022
Deposit Date Sep 5, 2022
Publicly Available Date Sep 5, 2022
Journal SciPost Physics
Print ISSN 2542-4653
Electronic ISSN 2542-4653
Publisher SciPost
Peer Reviewed Peer Reviewed
Volume 13
Issue 1


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