Lectures notes on Chern-Simons-Witten theory (Q2742811)

This book is based on lectures that E. Witten gave at Princeton University in 1989. Certain constructions of knot invariants via representations of the mapping class group following Moore-Seiberg and Kohno and from topological sigma models following Vafa and Witten are further elaborated. The goal of the book is to make physical ideas accessible to mathematicians. Unfortunately, many typographical errors make it hard to read and unsuitable for beginners or as a standard reference.NEWLINENEWLINEThe first two chapters serve as an introduction, explaining the concepts of geometric quantization and gauge equivalent functionals from Chern-Weil and Chern-Simons theory. Chapter three discusses moduli spaces of flat bundles over surfaces and their geometric quantization. Chapter four relates these to representations of the mapping class group and quantum invariants of knots and three-manifolds. Chapter five describes a perturbative Chern-Simons-Witten theory using renormalization techniques. The last chapter applies localisation techniques to topological sigma models.NEWLINENEWLINEAs appendices, the book contains reprints of chapter 7 of \textit{S. S. Chern's} book [Complex manifolds without potential theory (Springer) (1979; Zbl 0444.32004)] and of original articles by \textit{S. Axelrod, S. Della Pietra} and \textit{E. Witten} [J. Differ. Geom. 33, 787-902 (1991; Zbl 0697.53061)] (unfortunately without the list of references) and \textit{E. Witten} [Commun. Math. Phys. 144, 189-212 (1992; Zbl 0766.53068)]. The introduction was written by E. Witten.