On October 17, Karan Bhatia successfully defended the thesis “Bosonization and Fermionization of Categorical Symmetries” (advisor: Meng Cheng).
Bhatia explained, “Categorical symmetries play a central role in describing physical phenomena, from their emergence at critical points of physical models to their application in constructing anyon chains. Anyon chains involving fermions are built from corresponding fermionic fusion category symmetries. These fermionic anyon chains can be bosonized via (generalizations of) Jordan-Wigner transformations into anyon chains described by bosonic fusion categories. However, this bosonization process can only be applied on a case-by-case basis. Instead, we focus on mapping between the underlying fermionic and bosonic fusion categories directly to infer the bosonized anyon chain from the fermionic anyon chain, without needing to resort to the case-dependent bosonization process. My thesis focuses on constructing a general process of bosonizing / fermionizing between categorical symmetries which we refer to as the minimal braiding process involving Z2 symmetry lines. We find that any fermionic fusion category can be bosonized into a corresponding minimally-braided bosonic fusion category with Z2 symmetry, and vice versa.
Bhatia continued, “Going forward, I am keen to build a professional career in quantitative finance / investment management where I can apply my research, mathematical and logical reasoning skills gained during the Ph.D. journey. I will also continue to hone and maintain my lifelong passion for physics.”
Thesis abstract
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.