Boundary Effects on Anyon Dynamics in Chern-Simons Theory

TL;DR

This paper explores how boundaries and defects influence the modular data and anyon behavior in SU(N)_k Chern-Simons theories, revealing modifications to fusion rules and braiding statistics through categorical and algebraic methods.

Contribution

It introduces explicit formulas for the modified modular S-matrix in the presence of boundaries and defects, connecting bulk data with edge conformal field theories and analyzing junctions and symmetry defects.

Findings

Derived explicit boundary-modified S-matrices for SU(N)_k theories

Analyzed the impact of defects on fusion and braiding statistics

Explored anomaly inflow and central charge consistency across sectors

Abstract

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.