Generalized symmetries in singularity-free nonlinear <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> models and their disordered phases

We study the nonlinear $\ensuremath{\sigma}$-model in $(d+1)$-dimensional space-time with connected target space $K$ and show that, at energy scales below singular field configurations (such as vortices), it has an emergent noninvertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $d$-representations of a discrete $d$-group ${\mathbb{G}}^{(d)}$ (i.e., the emergent symmetry is the dual of the invertible $d$-group ${\mathbb{G}}^{(d)}$ symmetry). The $d$-group ${\mathbb{G}}^{(d)}$ is determined such that its classifying space $B{\mathbb{G}}^{(d)}$ is given by the $d\mathrm{th}$ Postnikov stage of $K$. In $(2+1)$ dimensions and for finite ${\mathbb{G}}^{(2)}$, this symmetry is always holo-equivalent to an invertible 0-form---ordinary---symmetry with a potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $\ensuremath{\sigma}$ model spontaneously breaks this symmetry, and when ${\mathbb{G}}^{(d)}$ is finite, it is described by the deconfined phase of ${\mathbb{G}}^{(d)}$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free ${S}^{2}$ nonlinear $\ensuremath{\sigma}$ model in $(3+1)$ dimensions and show that it has an emergent noninvertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this nonperturbative result is different from the results obtained using the ${S}^{N}$ and $\mathbb{C}{P}^{N\ensuremath{-}1}$ nonlinear $\ensuremath{\sigma}$ models in the large-$N$ limit.