Higher-form symmetries and spontaneous symmetry breaking

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.