SO(N) Superpotential, Seiberg-Witten Curves and Loop Equations

TL;DR

This paper derives the exact superpotential for SO(N) N=1 super Yang-Mills theory using Seiberg-Witten curves and confirms the matrix model predictions, including non-orientable surface contributions.

Contribution

It provides a field-theoretic derivation of the SO(N) superpotential and explicitly solves loop equations to verify matrix model conjectures.

Findings

Superpotential matches matrix model predictions.

Explicit solution of loop equations relates free energy to non-orientable surfaces.

Verification of Dijkgraaf-Vafa conjecture for SO(N) gauge theories.

Abstract

We consider the exact superpotential of N=1 super Yang-Mills theory with gauge group SO(N) and arbitrary tree-level polynomial superpotential of one adjoint Higgs field. A field-theoretic derivation of the glueball superpotential is given, based on factorization of the N=2 Seiberg-Witten curve. Following the conjecture of Dijkgraaf and Vafa, the result is matched with the corresponding SO(N) matrix model prediction. The verification involves an explicit solution of the first non-trivial loop equation, relating the spherical free energy to that of the non-orientable surfaces with topology $RP^2$.