# Higher-form symmetries and spontaneous symmetry breaking

## TL;DR

This paper explores spontaneous symmetry breaking in theories with higher-form symmetries, extending classical theorems like Goldstone's and Coleman-Mermin-Wagner to higher-form contexts, and discusses boundary conditions and gauge issues.

## Contribution

It provides a higher-form version of Goldstone's theorem, generalizes the Coleman-Mermin-Wagner theorem for higher-form symmetries, and analyzes boundary and gauge-fixing issues in such theories.

## Key findings

- Higher-form Goldstone's theorem established.
- Continuous p-form symmetries cannot be broken if p ≥ D-2.
- Relations between higher symmetries and asymptotic symmetries discussed.

## Abstract

We study various aspects of spontaneous symmetry breaking in theories that possess higher-form symmetries, which are symmetries whose charged objects have a dimension $p>0$. We first sketch a proof of a higher version of Goldstone's theorem, and then discuss how boundary conditions and gauge-fixing issues are dealt with in theories with spontaneously broken higher symmetries, focusing in particular on $p$-form $U(1)$ gauge theories. We then elaborate on a generalization of the Coleman-Mermin-Wagner theorem for higher-form symmetries, namely that in spacetime dimension $D$, continuous $p$-form symmetries can never be spontaneously broken if $p\geq D-2$. We also make a few comments on relations between higher symmetries and asymptotic symmetries in Abelian gauge theory.