Topological Manipulations of Quantum Field Theories

Abstract

In this thesis we study some topological aspects of Quantum Field Theories (QFTs). In particular, we study the way in which an arbitrary QFT can be separated into “local” and “global” data by means of a “symmetry Topological Field Theory” (symmetry TFT). We also study how various “topological manipulations” of the global data correspond to various well-known operations that previously existed in the literature, and how the symmetry TFT perspective provides a systematic tool for studying these topological manipulations.

We start by reviewing the bijection between G-symmetric d-dimensional QFTs and boundary conditions for G-gauge theories in (d+1)-dimensions, which effectively defines the symmetry TFT. We use this relationship to study the “orbifold groupoids” which control the composition of “topological manipulations,” relating theories with the same local data but different global data. Particular attention is paid to examples in d = 2 dimensions. We also discuss the extension to fermionic symmetry groups and find that the familiar “Jordan-Wigner transformation” (fermionization) and “GSO projection” (bosonization) appear as examples of topological manipulations. We also study applications to fusion categorical symmetries and constraining RG flows in WZW models as well.

After this, we present a short chapter showcasing an application of this symmetry TFT framework to the study of minimal models in 2d CFT. In particular, we complete the classification of 2d fermionic unitary minimal models.

Finally, we discuss how the symmetry TFT intuition can be used to classify duality defects in QFTs. In particular, we focus on Zm duality defects in holomorphic Vertex Operator Algebras (VOAs) (and especially the E8 lattice VOA), where we use symmetry TFT intuition to conjecture, and then rigorously prove, a formula relating (duality-)defected partition functions to Z2 twists of invariant sub-VOAs.