Topological Quantum Field Theories from Compact Lie Groups

Abstract

It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in case the gauge group is a torus. We also develop from different points of view an associated 4-dimensional invertible topological field theory which encodes the anomaly of Chern-Simons. Finite gauge groups are also revisited, and we describe a theory of "finite path integrals" as a general construction for a certain class of finite topological field theories. Topological pure gauge theories in lower dimension are presented as a warm-up.

39 pages, submission to proceedings of "A Celebration of the Mathematical Legacy of Raoul Bott"; v2 has minor changes

References in corpus (7)

• Spherical Categories
• Classical Chern-Simons theory, Part 1
• Classification of abelian spin Chern-Simons theories
• The Character Theory of a Complex Group
• Gerbes on complex reductive Lie groups
• Abelian Chern-Simons theory
• Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories

Cited by in corpus (7)

• Wall-Crossing in Coupled 2d-4d Systems
• A higher stacky perspective on Chern-Simons theory
• Cohomological Twisting of 2-Linearization and Extended TQFT
• On 2-form gauge models of topological phases
• Quantization via Linear homotopy types
• Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter
• The low-energy TQFT of the generalized double semion model