Modularity Constraint on Topological Defect Lines in Two-Dimensional Conformal Field Theories

Topological defect lines (TDL) provide a framework for describing symmetries in two dimensional field theories, including a description of generalized symmetries, and also topological interfaces between theories, in addition to being the model for the simplest examples of symmetry categories, generalizing the notion of symmetry groups. In this thesis we aim to give an introduction to TDLs in compact, local, unitary two dimensional conformal field theories (CFT). Starting from the foundations of 2d-CFT, definitions and properties of TDLs will be introduced, followed by a discussion on their mathematical structure and relation to fusion categories. Furthermore, using modular invariance of minimal 2d CFTs, the modular bootstrap equation is derived, which constraints all possible TDLs on a given minimal model. The application of this framework is demonstrated in practice with the Ising model, where we recover the Kramers–Wannier duality.