Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases

Understand

We study the nonlinear $\sigma$-model in ${(d+1)}$-dimensional spacetime with connected target space $K$ and show that, at energy scales below singular field configurations (such as vortices), it has an emergent non-invertible 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 $d$-th Postnikov stage of $K$.