Critical Points of Glueball Superpotentials and Equilibria of Integrable Systems

TL;DR

This paper establishes a precise correspondence between critical points of the glueball superpotential in N=1 theories and equilibrium states of integrable systems like the Toda chain and Calogero-Moser system, linking vacuum structure to integrable models.

Contribution

It demonstrates a one-to-one correspondence between superpotential critical points and integrable system equilibria, connecting gauge theory vacua with classical integrable models.

Findings

Critical points correspond to equilibrium configurations of integrable systems.

Glueball superpotential at critical points equals the Hamiltonian of the associated integrable system.

Analysis includes vacuum structure of N=1* theory with arbitrary superpotential.

Abstract

We compare the matrix model and integrable system approaches to calculating the exact vacuum structure of general N=1 deformations of either the basic N=2 theory or its generalization with a massive adjoint hypermultiplet, the N=2* theory. We show that there is a one-to-one correspondence between arbitrary critical points of the Dijkgraaf-Vafa glueball superpotential and equilibrium configurations of the associated integrable system. The latter being either the periodic Toda chain, for N=2, or the elliptic Calogero-Moser system, for N=2*. We show in both cases that the glueball superpotential at the crtical point equals the associated Hamiltonian. Our discussion includes an analysis of the vacuum structure of the N=1* theory with an arbitrary tree-level superpotential for one of the adjoint chiral fields.