turkmath.org

I will discuss classification of topological quantum field theories ($TQFT$s) with non-invertible generalized/categorical symmetries. From a condensed matter point of view, this is related to the classification of gapped phases of systems with non-invertible symmetries. Although the general formalism will be applicable to any spacetime dimension, I will provide concrete details in spacetime dimension $d=2$. As main examples, I will describe the only ($1+1$)d gapped phase with Ising symmetry which carries $3$ vacua along with relative Euler terms, and four possible ($1+1$)d gapped phases with $Rep(S_3)$ symmetry. Along the way, I will also discuss the order parameters for such gapped phases, which carry generalized charges under non-invertible symmetries.