Karan Bhatia

This talk investigates the processes of bosonization and fermionization within the framework of fusion category symmetries, which serve as the mathematical language for understanding topological phases of matter. We show that all fermionic fusion categories can be bosonized via a minimal braiding construction, providing a systematic method to relate fermionic and bosonic topological orders. Conversely, certain bosonic fusion categories that contain a Z2 symmetry object can be fermionized using the same construction. However, a concrete obstruction arises when attempting to fermionize the remaining bosonic fusion categories, specifically those whose fusion rules involve two fixed points fusing into an odd number of fixed points. This obstruction characterizes topological phases with a Z2 symmetry that do not admit a fermion in their Drinfeld center. Applications to bosonization and fermionization in conformal field theories on the torus as well as implications for the fusion algebra in the non-fermionizable cases are briefly discussed, highlighting the broader physical significance of these categorical constructions.