Defects and symmetries in three-dimensional topological field theories

Schulz, Jonathan (2022): Defects and symmetries in three-dimensional topological field theories. Dissertation, LMU München: Fakultät für Physik

Abstract

Despite the extensive success of quantum field theories (QFTs) in particle and solid state physics there are still unsolved conceptual problems, in particular regarding the underlying mathematical foundations. In recent years, research has focused on special cases like topological QFTs (TFTs) where mathematically rigorous descriptions in the language of category theory have been found. Two of these descriptions, namely those using bordisms and higher categories, are also capable of describing defects including boundaries, interfaces between different TFTs, and point insertions. Translating examples of defect TFTs from a physics description to a rigorous mathematical model is, however, a challenging problem. A multifaceted example is given by the affine Rozansky-Witten model, which from a physics point of view is a topologically twisted supersymmetric 3D N=4 QFT. On the mathematics side, it features a description in terms of a higher category RW which covers many aspects of this model, in particular regarding its defects. For example, previous fundamental analysis of RW has shown that its two-dimensional defects are closely related to the topological Landau-Ginzburg model which forms a well-studied 2D defect TFT described by the bicategory LG. However, many aspects of the tricategory RW have not yet been studied in detail. This thesis consists of two parts: The first part begins with a summary of the mathematical description of TFTs in general and RW in particular. The latter prominently features matrix factorisations which are introduced in detail, followed by several new results. Afterwards, an introduction to the description of RW as a tricategory is presented, including novel details required for a future proof of the tricategory axioms. With the goal of applying the orbifold procedure, adjunctions and pivotal structures in RW are discussed subsequently, yielding the first major result of this thesis: a generalisation of several established results in LG including the well-known Kapustin-Li formula. The second major result is the construction of a pivotal tricategory with duals T that is a subcategory of RW. Finally, an orbifold datum in T is constructed and significant progress is made towards proving its defining relations. The second part of this thesis discusses models with less supersymmetry, namely 3D N=2, which admit a holomorphic half-twist. While the latter is only capable of making QFTs partially topological, ruling out a mathematical description in the above sense, it nevertheless enables several exact (non-perturbative) constructions like supercurrent multiplets on the level of Lagrangians. The latter are generalised to 3D N=2 QFTs with boundaries and degrees of freedom on the boundary and then applied to three-dimensional Landau-Ginzburg models as an example.

• Dokumententyp: Dissertationen (Dissertation, LMU München)
• Keywords: topological field theory, defects, mathematical physics, Landau-Ginzburg, Rozansky-Witten
• Themengebiete: 500 Naturwissenschaften und Mathematik ; 500 Naturwissenschaften und Mathematik > 530 Physik
• Fakultäten: Fakultät für Physik
• Sprache der Hochschulschrift: Englisch
• Datum der mündlichen Prüfung: 26. Oktober 2022
• 1. Bericht­erstatter:in: Brunner, Ilka