Topological Quantum Field Theories from Compact Lie Groups
Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, Constantin Teleman
TL;DR
The paper constructs fully extended 0-1-2-3 TQFTs for compact Lie groups by using classes in $H^{n+1}(BG;{\mathbb Z})$ to encode actions and anomalies, with a focus on torus and finite groups. It shows that an invertible 4D anomaly theory $\mathscr{A}$ explains the Chern-Simons anomaly and that Chern-Simons for torus groups arises as a truncated extended theory, while the 3D theory is captured by the center of twisted group algebras and related modular tensor categories. A finite-path-integral framework and Morita theory relate toral theories to finite theories, via lattice approximations and the Sum construction, connecting to $L^2$ Chern-Simons, Verlinde data, and boundary/bulk compatibility. Overall, the work provides a higher-categorical, anomaly-aware construction that extends Chern-Simons theory to torus and finite gauge groups, with deep ties to centers, dualizability, and higher algebra.
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.