Classical Spin Chains and Exact Three-dimensional Superpotentials

TL;DR

This paper explores the exact superpotentials of 4D N=2 supersymmetric gauge theories with fundamental matter, using integrable spin chains to parametrize the moduli space and derive various vacua and superpotentials.

Contribution

It establishes a novel connection between gauge theory moduli space and classical integrable spin chains, providing explicit superpotentials and vacua classifications.

Findings

Maximally confining Seiberg-Witten curves are obtained.

Coordinate patches from spin chains parametrize gauge theory moduli space.

Explicit superpotentials for N_f ≤ N_c are derived.

Abstract

We study exact effective superpotentials of four-dimensional {\cal N} = 2 supersymmetric gauge theories with gauge group U(N) and various amounts of fundamental matter on R^3 x S^1, broken to {\cal N} = 1 by turning on a classical superpotential for the adjoint scalar. On general grounds these superpotentials can easily be constructed once we identify a suitable set of coordinates on the moduli space of the gauge theory. These coordinates have been conjectured to be the phase space variables of the classical integrable system which underlies the {\cal N} = 2 gauge theory. For the gauge theory under study these integrable systems are degenerations of the classical, inhomogeneous, periodic SL(2,C) spin chain. Ambiguities in the degeneration provide multiple coordinate patches on the gauge theory moduli space. By studying the vacua of the superpotentials in several examples we find that the spin chain provides coordinate patches that parametrize holomorphically the part of the gauge theory moduli space which is connected to the electric (as opposed to magnetic or baryonic) Higgs and Coulomb branch vacua. The baryonic branch root is on the edge of some coordinate patches. As a product of our analysis all maximally confining (non-baryonic) Seiberg-Witten curve factorizations for N_f \leq N_c are obtained, explicit up to one constraint for equal mass flavors and up to two constraints for unequal mass flavors. Gauge theory addition and multiplication maps are shown to have a natural counterpart in this construction. Furthermore it is shown how to integrate in the meson fields in this formulation in order to obtain three and four dimensional Affleck-Dine-Seiberg-like superpotentials.