  Higher-group symmetries are combinations of higher-form symmetries which
appear in various field theories. In this paper, we explain how higher-group
symmetries arise in 10d and 11d supergravities when the latter are coupled to
brane sources. Motivated by this observation, we study field theories at zero
and finite temperature invariant under a class of continuous Abelian
higher-group symmetries. We restrict the analysis to the low-energy regime
where the dynamical field content exclusively consists of Goldstone fields
arising from the spontaneous breaking of higher-group and spacetime symmetries.
Invariant quantities are constructed and the phases of matter are classified
according to the pattern of spontaneous symmetry breaking. With respect to
supergravity, we highlight how such Goldstone effective theories provide a
symmetry-based interpretation for the theories living on D/M-branes. As an
explicit example we construct a 6-group invariant action for the bosonic M5
brane, consistent with the self-duality of the 3-form field strength on the
brane. While the self-duality condition in the bosonic case needs to be imposed
externally as a constraint at zero temperature, we find an equilibrium
effective action for the bosonic M5 brane at finite temperature that inherently
implements self-duality.
