An Introduction to Supersymmetric Gauge Theories and Matrix Models

TL;DR

The paper develops and systematizes a bridge between four-dimensional $\\mathcal{N}=1$ supersymmetric gauge theories and matrix models. It introduces the superspace formalism, holomorphy, and the chiral ring, then presents two complementary routes—the diagrammatic approach and the Konishi anomaly method—to derive the glueball superpotential and the DV conjecture that matrix-model planar free energy encodes non-perturbative F-terms. Key contributions include a clear treatment of the Wilsonian holomorphic superpotential, the role of the glueball superfield $S$, and the use of the Konishi anomaly to compute exact effective superpotentials in several settings, with planarity and disk amplitudes providing the central computational link to matrix models. The results offer a calculational framework for non-perturbative vacua and low-energy dynamics in SUSY gauge theories, linking field theory to matrix-model tools and laying groundwork for broader applications in moduli spaces and dualities.

Abstract

We give an introduction to the recently established connection between supersymmetric gauge theories and matrix models. We begin by reviewing previous material that is required in order to follow the latest developments. This includes the superfield formulation of gauge theories, holomorphy, the chiral ring, the Konishi anomaly and the large N limit. We then present both the diagrammatic proof of the connection and the one based on the anomaly. Our discussion is entirely field theoretical and self contained.

Figures (7)

• Figure 1: The two-loop contribution to the one-particle effective superpotential.
• Figure 2: The different energy scales
• Figure 3: The parts $1^{a b}_{c d}$ and $G^{ab}_{cd}$ of the propagator and the cubic vertex.
• Figure 4: The stop sign diagram.
• Figure 5: The stop sign with two $W_\alpha$ insertions at two of the propagators (left) and the stop sign with one $W_\alpha$ insertion at two of the propagators and one insertion of two $W_\alpha$ at the third propagator (right).