Date of Award

Fall 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

First Advisor

Cheng, Meng

Abstract

The thesis 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 Z_2 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 i.e. whenever any of the fusion multiplicities involving three fixed points is odd. This obstruction characterizes topological phases with a Z_2 symmetry and fixed points that do not admit a fermion in their Drinfeld center. We also derive a fermionized version of the hexagon identities using our mapping, and find that there are additional contributions from the braiding of the anyons with the fermion present in these identities. 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.

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