Web of dualities on non-orientable surfaces

A bstract It is known that a two-dimensional bosonic theory with a non-anomalous ℤ 2 symmetry can be fermionized. Recent work shows that if the bosonic theory also has non-anomalous time-reversal symmetry, fermionization extends to non-orientable surfaces and yields a fermionic theory that depends on a Pin − structure. Besides fermionization, one can define various topological manipulations, such as gauging and stacking invertible phases, which together generate a web of dualities. We prove that their group structure is the dihedral group D 8 of order 16. Furthermore, we systematically investigate the web from two perspectives: Symmetry TFT and actions on sectors of the S 1 Hilbert space.