Gauging Rep(D₈) symmetry on the lattice

Gauging serves as a powerful probe on symmetries in a physical system, it connects different theories and reveals hidden properties of a given theory. With a broadened understanding of symmetries, we investigate gauging non-invertible symmetries in a lattice system and propose a systematic approach to lattice gauging using algebra object. In particular, we gauge one of the non-maximal algebra objects A = 1 \oplus η \oplus D of Rep(D₈) symmetry on the lattice, examples of which include cluster SPT state and two copies of the Ising model. We show that cluster state is self-dual under such a gauging. We compare our result with the analysis in the continuum, in which cluster state is the deformation of Ising² CFT. We identify the gauging map with another non-invertible operator and show that two methods produce consistent results.