On the Geometry of Super Yang-Mills Theories: Phases and Irreducible Polynomials

TL;DR

This paper explores the algebraic and geometric structures of N=1 super Yang-Mills theories' vacua, revealing polynomial equations governing chiral operators and analyzing phase structures through polynomial factorization, with applications to confining and Higgs vacua.

Contribution

It introduces a novel polynomial framework to study the phase structure of super Yang-Mills theories and provides explicit proofs and phase diagrams using algebraic geometry methods.

Findings

Chiral operators satisfy polynomial equations over specific rings.

Confining and Higgs vacua are in the same phase with fundamental flavors.

Full phase diagram derived for theories with one adjoint and small Nc.

Abstract

We study the algebraic and geometric structures that underly the space of vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(Nc) theories, we provide a simple and completely general proof of the fact that confining and Higgs vacua are in the same phase when fundamental flavors are present, by finding an irreducible polynomial equation satisfied by the glueball operator. We also derive the full phase diagram for the theory with one adjoint when Nc is less than or equal to 7 using computational algebraic geometry programs.