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

Abstract

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.

3d $$ \mathcal{N} $$ = 2 brane webs and quiver matrices

“…These different brane webs are different phases (chambers) of 3d theories. Notice that some mass parameters between two chambers have opposite signs, which is similar to the interface discussed in [55,62,63].…”

di-Langlands correspondence and extended observables

“…Meanwhile, the R-matrices on highest-weight modules admit a geometric construction from stable envelopes [99,100]. The stable envelope was realized in the context of the Bethe/gauge correspondence by the Janus interface interpolating different values of real masses [6,[101][102][103]. It would be interesting to extend this gauge theoretical construction of the stable envelopes so that the R-matrices on bi-infinite modules would be incorporated, perhaps by regarding the C ε 2 -plane as a cigar and compactifying along its circle fiber.…”

Web of Seiberg-like dualities for 3D N=2 quivers

“…Half-BPS boundary conditions can preserve either N ¼ ð1; 1Þ which have been described in [35,36], or N ¼ ð0; 2Þ which we focus here. In particular, N ¼ ð0; 2Þ half-BPS chiral UV boundary conditions preserving the Uð1Þ R symmetry lead to various applications [35,[37][38][39][40][41][42][43][44][45][46][47]. For pairs of UV boundary conditions, there exist dual boundary conditions which flow to the same IR boundary conditions.…”