Galois symmetries in Super Yang-Mills Theories

TL;DR

This paper explores the use of Galois theory to classify phases in supersymmetric gauge theories, providing a new mathematical framework that captures quantum gauge symmetry breaking beyond traditional methods.

Contribution

It introduces a novel approach using Galois symmetries to analyze gauge phases, extending the concept of gauge symmetry breaking into the quantum regime.

Findings

Galois symmetry is a phase invariant in supersymmetric gauge theories.

The algebraic structure of vacua supports Galois-theoretic classification.

This framework offers a rigorous mathematical tool for understanding gauge phases.

Abstract

Classifying the phases of gauge theories is hindered by the lack of local order parameters. In particular, the standard Wilson's and 't Hooft's non-local order parameters are known to be insufficient to explain the existence of the plethora of phases that are found in supersymmetric gauge theories. Motivated by these observations, we reanalyze the concept of gauge symmetry breaking using Galois theory. Unlike the ordinary classical notion of unbroken gauge group, the Galois symmetry makes sense in the full quantum theory and must be a phase invariant. The algebraic structure underlying the space of vacua of supersymmetric gauge theories, that we have developed recently, is precisely designed to allow a rigorous mathematical implementation of these ideas.