A 5d/3d duality from relativistic integrable system

TL;DR

This paper establishes a precise duality between 5d and 3d supersymmetric gauge theories with adjoint matter, linking their vacua to the eigenstates of a relativistic elliptic Calogero-Moser integrable system.

Contribution

It introduces and proves a new duality connecting 5d and 3d gauge theories via their effective superpotentials and integrable systems, extending to elliptic quiver theories.

Findings

Matching of effective twisted superpotentials with the Yang-Yang functional.

Derivation of Bethe ansatz equations from the 5d partition function.

Extension of duality to elliptic quiver gauge theories.

Abstract

We propose and prove a new exact duality between the F-terms of supersymmetric gauge theories in five and three dimensions with adjoint matter fields. The theories are compactified on a circle and are subject to the Omega deformation. In the limit proposed by Nekrasov and Shatashvili, the supersymmetric vacua become isolated and are identified with the eigenstates of a quantum integrable system. The effective twisted superpotentials are the Yang-Yang functional of the relativistic elliptic Calogero-Moser model. We show that they match on-shell by deriving the Bethe ansatz equation from the saddle point of the five-dimensional partition function. We also show that the Chern-Simons terms match and extend our proposal to the elliptic quiver generalizations.