On the Coulomb Branch of a Marginal Deformation of N=4 SUSY Yang-Mills

TL;DR

This work determines the exact vacuum structure of a marginal ${\cal N}=1$ deformation of ${\cal N}=4$ ${\rm U}(N)$ SYM, showing that the quantum Coulomb branch splits into sub-branches described by a union of curves $\Sigma_{n_1}\cup\Sigma_{n_2}\cup\Sigma_{n_3}$ with genus $N=n_1+n_2+n_3$, permuted Higgs/confining branches under ${\rm SL}(2,\mathbb{Z})$. The defining curve is derived both from the Dijkgraaf–Vafa matrix integral and from instanton localization, and each $\Sigma_n$ is identified with the spectral curve of the $n$-body Ruijsenaars–Schneider integrable system. The results imply a holomorphic equivalence between the 4d beta-deformed theory and a 5d ${\cal N}=2^*$ gauge theory, predicting novel confining branches in five dimensions and unifying disparate descriptions via brane constructions and modular properties. This work thus links exact holomorphic data on the Coulomb branch to higher-dimensional dynamics through a shared RS spectral curve and explicit matrix-model/instanton analyses.

Abstract

We determine the exact vacuum structure of a marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N). The Coulomb branch of the theory consists of several sub-branches which are governed by complex curves of the form Sigma_{n_{1}} U Sigma_{n_{2}} U Sigma_{n_{3}} of genus N=n_{1}+n_{2}+n_{3}. Each sub-branch intersects with a family of Higgs and Confining branches permuted by SL(2,Z) transformations. We determine the curve by solving a related matrix model in the planar limit according to the prescription of Dijkgraaf and Vafa, and also by explicit instanton calculations using a form of localization on the instanton moduli space. We find that Sigma_{n} coincides with the spectral curve of the n-body Ruijsenaars-Schneider system. Our results imply that the theory on each sub-branch is holomorphically equivalent to certain five-dimensional gauge theory with eight supercharges. This equivalence also implies the existence of novel confining branches in five dimensions.