String Theory and the Vacuum Structure of Confining Gauge Theories

TL;DR

This paper reviews recent advances in understanding the vacuum structure of confining N=1 supersymmetric gauge theories, emphasizing geometric, matrix model, and integrable system techniques, especially with fundamental matter fields.

Contribution

It provides explicit formulas for effective superpotentials in theories with fundamental matter and extends techniques to SO and Sp gauge groups.

Findings

Effective superpotentials derived for theories with fundamental matter.

Seiberg-Witten curve factorization for N=2 theories with matter.

Refined techniques for SO and Sp gauge groups.

Abstract

We discuss recent progress in the understanding of the vacuum structure (effective superpotentials) of confining gauge theories with N=1 supersymmetry. Even for non-supersymmetric theories, appropriate perturbative calculations (e.g. using the background field method) give non-perturbative information about the vacuum structure. However, in supersymmetric theories, these results are often exact. The gauge theory effective superpotential is encoded by a hyperelliptic curve, which emerges from the geometry of the string theory background, and may be rederived using other techniques based on zero-dimensional matrix integrals, the dynamics of integrable systems and the factorization of Seiberg-Witten curves. We describe in detail how each technique highlights complementary aspects of the gauge theory. The spectral curve requires the introduction of additional fundamental matter fields, which act as UV regulators by embedding the gauge theory in a UV-finite theory. We focus in detail on maximally-confining vacua of N=1 gauge theories with fundamental matter, and of theories with SO and Sp gauge groups. Both cases require refinements to the basic techniques used for SU gauge theory without fundamental matter. We derive explicit general formulae for the effective superpotentials of N=1 theories with fundamental matter and arbitrary tree-level superpotential, which reproduce known results in special cases. The problem of factorizing the Seiberg-Witten curve for N=2 gauge theories with fundamental matter is also solved and used to rederive the corresponding N=1 effective superpotential.