Boundary Effects on Anyon Dynamics in Chern-Simons Theory

This work investigates the boundary and defect effects on the modular data in SU$(N)_k$ Chern-Simons theories, focusing on how different boundary conditions and symmetry defects modify the fusion rules and braiding statistics of anyons. Using the framework of modular tensor categories (MTCs) and Frobenius algebra objects, explicit expressions for the modified $S$-matrix, $S'$, are derived in the presence of heterogeneous boundary conditions, and the connection between the bulk modular data and edge CFTs is analyzed. The approach includes the computation of modular matrix deformations in the presence of junctions between different boundary conditions, as well as the influence of global symmetry defect lines, which introduce twisted sectors into the MTC framework. The ideas are applied to SU$(2)_k$, SU$(3)_2$, and SU$(4)_1$ Chern-Simons theories, providing examples of boundary algebras and fusion rules at junctions. Additionally, the implications of categorical anomaly inflow and central charge matching across boundary and defect sectors are explored. This work lays the groundwork for further studies of boundary and defect effects in topological quantum field theories and their connections to topological quantum computation and holography.