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

Abstract

This paper studies 3d3d\mathcal{N}=4𝒩=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.

Superspin chains from superstring theory

“…The latter also provides a recipe for the computation of the R-matrix by an A-model localization computation. The systematic gaugetheoretic construction of elliptic stable envelopes was developed in [70,71]. In these works the elliptic stable envelope was identified as certain Janus partition functions of 3d N = 4 theories on I × E τ where I is some interval and E τ is the elliptic curve.…”

“…The gauge-theoretic definition of elliptic stable envelopes as Janus partition functions first appeared in [70,71]. They define the Janus interfaces for 3d N = 2 supersymmetry but provide boundary conditions for 3d N = 4 vacua, which give elliptic stable envelopes of 3d N = 4 Higgs branches.…”

Gauge/Bethe correspondence from quiver BPS algebras

“…If the cylinder is long enough then the theory at each value p(s) can be approximated by the theory at a constant value of the parameter p(s). As a result, the cylinder neck [0, 1] × S 1 can have simultaneously two effective descriptions at p 0 = p(0) and p 1 = p(1), hence the name Janus interface for such a theory in the literature (see [57][58][59] for recent discussions). The Janus interface allows one to parallel transport the theory from theory p 0 to theory p 1 along some path ℘, so that for the partition functions we have:…”

“…In addition to discussing more general algebras, another strategy is to study some special theories in more detail, for example 3d N = 4 theories [58,59] corresponding to quantum toroidal BPS algebras for non-chiral quivers.…”

“…The stable envelopes were defined in [71], see also [73,74,86] and references therein for development and practical applications. In section 4.2 we treated stable envelopes similarly to [58,59,68] as a transformation between bases in the disk boundary conditions: from a naive basis of classical vacua to a basis of stable objects in the corresponding triangulated category of boundary branes.…”