The classifying algebra for defects

TL;DR

The paper extends the algebraic framework used for boundary conditions in rational CFTs to topological defects, introducing a finite-dimensional semisimple unital commutative algebra—the classifying algebra for defects—whose irreducible representations yield defect transmission coefficients. It shows the algebra’s structure constants are realized as traces on spaces of conformal blocks and that defect transmission data determine defect partition functions. The construction leverages the TFT description and folding, clarifying how defects correspond to A-B bimodules in the chiral category and how fusion, duals, and invertible defects organize the defect spectrum. These results provide a concrete, computable tool for analyzing defects and dualities in rational CFTs and suggest avenues for extending the framework to non-rational theories.

Abstract

We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects - a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations give the defect transmission coefficients. We show in particular that the structure constants of the classifying algebra are traces of operators on spaces of conformal blocks and that the defect transmission coefficients determine the defect partition functions.