String theory and the vacuum structure of confining gauge theories

We discuss recent progress in the understanding of the vacuum structure (effective superpotentials) of confining gauge theories with N = 1 supersymmetry, in particular theories with softly broken N = 2 supersymmetry. We show how the new techniques improve upon older calculations in non-supersymmetric quantum field theories. A common feature of both approaches is that appropriate perturbative field theory calculations (e.g. using the background field method) give non-perturbative information about the vacuum structure of the theory. However, in supersymmetric theories, these results are often exact.; The geometric engineering of supersymmetric gauge theories in string theory provides powerful tools for studying gauge theories. Central to the analysis is a particular class of hyperelliptic curve, which emerges from the Calabi-Yau geometry of the string theory background and encodes the gauge theory effective superpotential. These curves may be rederived using other techniques based on zero-dimensional matrix integrals, the dynamics of integrable systems and the factorization of Seiberg-Witten curves, and we describe in detail how each technique highlights complementary aspects of the gauge theory.; We find that the use of the spectral curve requires the introduction of additional fundamental matter fields, which act as regulators for the UV divergences of the calculation by embedding the gauge theory in a UV finite theory. Theories with 0 ≤ Nf < 2 Nc fundamental multiplets may thus be treated uniformly. We focus in detail on maximally-confining vacua of N = 1 gauge theories with fundamental matter, and of gauge 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.