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The Coulomb branch of the Leigh-Strassler deformation and matrix models

Francesco Benini

TL;DR

This work uses the Dijkgraaf-Vafa matrix-model framework to analyze the Coulomb branch of the Leigh-Strassler deformation of N=4 SYM with gauge group U(N). By introducing an $N+1$-th order tree-level potential, the moduli space is lifted to discrete vacua, and the resulting generalized Seiberg-Witten curve becomes an $N$-sheeted covering of a base torus, described by multi-valued functions on the torus. The on-shell curve, resolvents, and period integrals are derived, revealing that the effective superpotential is determined by matrix-model data and that the curve encodes the moduli and chiral operator vevs; the construction extends to SU(N) with appropriate residue conditions. The semiclassical limit shows eigenvalues condensing at the critical points of the potential, with vanishing condensates and a consistent reduction to the expected weak-coupling regime. Overall, the paper provides explicit, geometry-driven expressions for the generalized Seiberg-Witten curve and the associated resolvents, enabling computation of chiral-ring data in this N=1, LS-deformed setup.

Abstract

The Dijkgraaf-Vafa approach is used in order to study the Coulomb branch of the Leigh-Strassler massive deformation of N=4 SYM with gauge group U(N). The theory has N=1 SUSY and an N-dimensional Coulomb branch of vacua, which can be described by a family of ``generalized'' Seiberg-Witten curves. The matrix model analysis is performed by adding a tree level potential that selects particular vacua. The family of curves is found: it consists of order N branched coverings of a base torus, and it is described by multi-valued functions on the latter. The relation between the potential and the vacuum is made explicit. The gauge group SU(N) is also considered. Finally the resolvents from which expectation values of chiral operators can be extracted are presented.

The Coulomb branch of the Leigh-Strassler deformation and matrix models

TL;DR

This work uses the Dijkgraaf-Vafa matrix-model framework to analyze the Coulomb branch of the Leigh-Strassler deformation of N=4 SYM with gauge group U(N). By introducing an -th order tree-level potential, the moduli space is lifted to discrete vacua, and the resulting generalized Seiberg-Witten curve becomes an -sheeted covering of a base torus, described by multi-valued functions on the torus. The on-shell curve, resolvents, and period integrals are derived, revealing that the effective superpotential is determined by matrix-model data and that the curve encodes the moduli and chiral operator vevs; the construction extends to SU(N) with appropriate residue conditions. The semiclassical limit shows eigenvalues condensing at the critical points of the potential, with vanishing condensates and a consistent reduction to the expected weak-coupling regime. Overall, the paper provides explicit, geometry-driven expressions for the generalized Seiberg-Witten curve and the associated resolvents, enabling computation of chiral-ring data in this N=1, LS-deformed setup.

Abstract

The Dijkgraaf-Vafa approach is used in order to study the Coulomb branch of the Leigh-Strassler massive deformation of N=4 SYM with gauge group U(N). The theory has N=1 SUSY and an N-dimensional Coulomb branch of vacua, which can be described by a family of ``generalized'' Seiberg-Witten curves. The matrix model analysis is performed by adding a tree level potential that selects particular vacua. The family of curves is found: it consists of order N branched coverings of a base torus, and it is described by multi-valued functions on the latter. The relation between the potential and the vacuum is made explicit. The gauge group SU(N) is also considered. Finally the resolvents from which expectation values of chiral operators can be extracted are presented.

Paper Structure

This paper contains 12 sections, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The matrix model and the Riemann surface
  3. The on-shell theory
  4. The curve
  5. Order of the covering
  6. Curve and factorization
  7. Resolvents
  8. The effective potential
  9. The curve
  10. Gauge group $SU(N)$
  11. The function $G(x)$
  12. The semiclassical limit

Figures (2)

  • Figure 1: Degeneration of the surface. When two branching points collide, we can take the cut wound around a cycle, obtaining a torus of modified modular parameter.
  • Figure 2: Simply connected surface. If we take into account the points at infinity, the system of cycles cuts off the surface and makes it simply connected.