On Three Dimensional Quiver Gauge Theories and Integrability

TL;DR

This paper explores the various descriptions of the moduli space of vacua in 3D N=4 superconformal theories, revealing deep connections with integrable systems and establishing dualities between different models.

Contribution

It establishes a precise dictionary linking different gauge theory descriptions of the vacua space and relates these to integrable systems, uncovering new dualities.

Findings

Different descriptions of vacua are shown to be equivalent via a detailed dictionary.

The vacua in linear quiver theories relate to quantum integrable spin chains.

The vacua in 4D theories connect to classical integrable many-body problems.

Abstract

In this work we compare different descriptions of the space of vacua of certain three dimensional N=4 superconformal field theories, compactified on a circle and mass-deformed to N=2 in a canonical way. The original N=4 theories are known to admit two distinct mirror descriptions as linear quiver gauge theories, and many more descriptions which involve the compactification on a segment of four-dimensional N=4 super Yang-Mills theory. Each description gives a distinct presentation of the moduli space of vacua. Our main result is to establish the precise dictionary between these presentations. We also study the relationship between this gauge theory problem and integrable systems. The space of vacua in the linear quiver gauge theory description is related by Nekrasov-Shatashvili duality to the eigenvalues of quantum integrable spin chain Hamiltonians. The space of vacua in the four-dimensional gauge theory description is related to the solution of certain integrable classical many-body problems. Thus we obtain numerous dualities between these integrable models.