Topological Quantum Field Theories in Dimension Four

This dissertation explores Topological Quantum Field Theories (TQFTs) in four dimensions. It examines the Crane-Yetter theory through three distinct applications. First, it demonstrates the equivalence between Crane-Yetter theory and the Turaev shadow state sum as a 4-manifold invariant. Second, it investigates the values of Crane-Yetter theory for 2-manifolds, identifying these values as linear categories and describing their structures for oriented surfaces with at least one puncture. Finally, an application of Crane-Yetter theory to a problem within tensor categories is presented. The work builds upon previous research by the author and offers insights into representing mathematical objects and their relationships using advanced algebraic structures.

Curriculum

The dissertation is structured into several chapters, covering foundational concepts, the core theory, and specific applications. It progresses from the introduction of invariants and TQFTs to detailed discussions of algebraic and topological preliminaries, the Crane-Yetter state sum, its relation to Turaev shadows, and its application in categorical settings.

1. Chapter 1
   • Introduction to Invariants as Data Representations
   • TQFTs as Higher Invariants
2. Chapter 2
   • Preliminaries: Algebra (Premodular categories, Graphical calculus, Coordinates, Symbols)
   • Preliminaries: Topology (4-manifolds, Triangulations, Handle decompositions)
   • Crane-Yetter State Sum
3. Chapter 3
   • Crane-Yetter Theory and Turaev Shadow
   • Main Result: Equivalence of State Sums
4. Chapter 4
   • Categorical Center of Higher Genera
   • Topological Theory (String nets, Crane-Yetter theory in dimension two, Surface presentations)
   • Algebraic Theory (Drinfeld categorical center, Categorical center of higher genera)
   • Proof of the Main Theorem
5. Chapter 5
   • Explicit Factorization of Categorical Center
6. Chapter 6
   • Appendix (Abelian categories, Semisimple categories, Tensor categories, Adjunctions)