# Permutations of Massive Vacua

## TL;DR

This paper explores the permutation group of massive vacua in supersymmetric gauge theories, linking it to Galois groups of characteristic polynomials, and examines how duality symmetries break depending on the theory's specifics.

## Contribution

It introduces methods to compute characteristic polynomials of vacua and reveals how symmetry breaking patterns depend on gauge algebra and matter content.

## Key findings

- Permutation group G is the Galois group of characteristic polynomials.
- Symmetry breaking patterns vary with gauge algebra and matter content.
- Examples lead to field extensions of modular form spaces.

## Abstract

We discuss the permutation group G of massive vacua of four-dimensional gauge theories with N=1 supersymmetry that arises upon tracing loops in the space of couplings. We concentrate on superconformal N=4 and N=2 theories with N=1 supersymmetry preserving mass deformations. The permutation group G of massive vacua is the Galois group of characteristic polynomials for the vacuum expectation values of chiral observables. We provide various techniques to effectively compute characteristic polynomials in given theories, and we deduce the existence of varying symmetry breaking patterns of the duality group depending on the gauge algebra and matter content of the theory. Our examples give rise to interesting field extensions of spaces of modular forms.