Higher-form symmetries and spontaneous symmetry breaking

TL;DR

This work extends spontaneous symmetry breaking to theories with higher-form symmetries by formulating a higher Goldstone theorem in which massless $p$-form gauge fields act as Goldstones, and by clarifying boundary conditions, Ward identities, and gauge fixing in these contexts. It establishes a generalized Coleman-Mermin-Wagner result: continuous $p$-form symmetries in $D$ dimensions cannot be spontaneously broken when $p \,\ge\, D-2$, with complementary analyses for continuum, compact, and discrete theories and interpretations via duality. The paper also links higher-form symmetry breaking to confinement diagnostics using Wilson membranes/loops and explores connections to asymptotic symmetries on manifolds with trivial cohomology, outlining how boundary data and nonperturbative effects shape IR behavior. Overall, it provides a coherent framework to understand phase structure, dual descriptions, and boundary phenomena for higher-form symmetries, with implications for topological phases and gauge-theory memory effects. The results suggest rich interplay between symmetry breaking, confinement, duality, and asymptotic symmetries in Abelian gauge theories.

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.

Figures (3)

• Figure 1: An example of the support of two charge operators $Q(M)$ inside a tube $\Sigma$. On the left the 2-manifold $M$ is closed with $\partial M=0$, while on the right it is relatively closed: $\partial M\neq0$, but $\partial M \subset \partial \Sigma$. The left $M$ is a boundary, and thus the associated $Q(M)$ generates a trivial gauge transformation.
• Figure 2: A section of a tubular spatial slice of the form $S^1\times \mathbb{R}$. Integrating $\star F$ along the manifold $M$ (red line) measures the electric flux around the $S^1$. Acting with $Q(M)$ transforms a Wilson loop wrapping around the $S^1$, which is given by the integral of $A$ along $C$ (blue line).
• Figure 3: A two-dimensional spatial slice $\Sigma$, with the dotted circle representing $r=\infty$. The Wilson line $W_C=\exp(i\int_C A)$ (in black) is charged under a 1-form symmetry generated by $\star F$ integrated along the red curve $M$. This shifts the gauge field by $\widehat{M}$, which is the bump function Poincare dual to $M$ and is shown schematically in grey.