A First Course in Topological Field Theory

Interlacing algebra, topology, and quantum physics, topological quantum field theories are framed as symmetric monoidal functors mapping cobordism categories to vector spaces. Using Dijkgraaf–Witten examples, the text introduces category theory, Hopf algebras, and tensor categories to link core concepts with research.

Topological quantum field theories lie at the intersection of algebra, topology, and quantum field theory. The area is highly active, driven by recent developments in representation theory and topology, as well as developments in theoretical physics, including the study of topological phases of matter and topological symmetries. This book provides an accessible introduction to topological quantum field theories, along with basic notions of category theory - the natural mathematical setting for topological field theories - by describing these theories as a symmetric monoidal functor from a cobordism category to the category of vector spaces.

Using Dijkgraaf–Witten theories as a recurring example, the book introduces key tools such as category theory, Hopf algebras, and tensor categories, providing a solid foundation for students aiming to pursue a more advanced study in quantum topology and related areas.

The book concludes with an outlook connecting the foundational theory to recent developments and current research directions.

The text is intended for advanced undergraduates and beginning graduate students in mathematics and theoretical physics. It is largely self-contained and assumes no specific background beyond standard undergraduate algebra and topology. While some familiarity with basic ideas from quantum theory is helpful, it is not a prerequisite.