N=1 G_2 SYM theory and Compactification to Three Dimensions

TL;DR

This paper investigates the exact superpotential of four-dimensional N=2 G_2 supersymmetric gauge theory compactified on R^3×S^1, utilizing integrable models and algebraic geometry to understand its low-energy dynamics.

Contribution

It introduces a novel approach to derive the superpotential using the Lax matrix of the integrable model based on the dual affine G_2 Lie algebra, extending methods beyond U(N) theories.

Findings

Exact superpotential obtained via integrable model techniques

Factorization of Seiberg-Witten curve analyzed for G_2 theories

Evidence for an auxiliary hyperelliptic curve describing low-energy dynamics

Abstract

We study four dimensional N=2 G_2 supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice based on the dual of the affine G_2 Lie algebra. At extrema of the superpotential the Seiberg-Witten curve typically factorizes, and we study the algebraic equations underlying this factorization. For U(N) theories the factorization was closely related to the geometrical engineering of such gauge theories and to matrix model descriptions, but here we will find that the geometrical interpretation is more mysterious. Along the way we give a method to compute the gauge theory resolvent and a suitable set of one-forms on the Seiberg-Witten curve. We will also find evidence that the low-energy dynamics of G_2 gauge theories can effectively be described in terms of an auxiliary hyperelliptic curve.