Nonperturbative Superpotentials and Compactification to Three Dimensions

TL;DR

This paper explores the nonperturbative superpotentials in 4d N=2 supersymmetric gauge theories compactified to three dimensions, revealing a deep connection with integrable systems, matrix models, and a novel mirror symmetry.

Contribution

It introduces a method to derive the quantum superpotential using the Lax matrix of integrable systems, linking 4d vacua to 3d reductions and matrix models.

Findings

Vacuum structures match between 4d and 3d frameworks

Simplification of vacuum lifting from U(N) to U(tN)

Identification of a new mirror symmetry involving matrix models

Abstract

We consider four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, in the presence of a classical superpotential. The low-energy quantum superpotential is obtained by simply replacing the adjoint scalar superfield in the classical superpotential by the Lax matrix of the integrable system that underlies the 4d field theory. We verify in a number of examples that the vacuum structure obtained in this way matches precisely that in 4d, although the degrees of freedom that appear are quite distinct. Several features of 4d field theories, such as the possibility of lifting vacua from U(N) to U(tN), become particularly simple in this framework. It turns out that supersymmetric vacua give rise to a reduction of the integrable system which contains information about the field theory but also about the Dijkgraaf-Vafa matrix model. The relation between the matrix model and the quantum superpotential on R^3 x S^1 appears to involve a novel kind of mirror symmetry.