Remarks on Chern-Simons theory

Abstract

In the late 1980s Witten used the Chern-Simons form of a connection to construct new invariants of 3-manifolds and knots, recovering in particular the Jones invariants. Since then the associated topological quantum field theory (TQFT) has served as a key example in understanding the structure of TQFTs in general. We survey some of that structure with a particular focus on the multi-tier aspects. We discuss general axioms, generators-and-relations theorems, a priori constructions, dimensional reduction and K-theory, and Chern-Simons as a 0-1-2-3 theory. An appendix gives a lightening treatment of the Chern-Simons-Weil theory of connections. The paper concludes with general remarks about the Geometry-QFT-Strings interaction.