Jones–Witten Invariants for Nonsimply Connected Lie Groups and the Geometry of the Weyl Alcove

Abstract

The quotient process of Müger and Bruguières is used to construct modular categories and TQFTs out of closed subsets of the Weyl alcove of a simple Lie algebra. In particular it is determined at which levels closed subsets associated to nonsimply connected groups lead to TQFTs. Many of these TQFTs are shown to decompose into a tensor product of TQFTs coming from smaller subsets. The “prime” subsets among these are classified, and apart from some giving TQFTs depending on homology as described by Murakami, Ohtsuki and Okada, they are shown to be in one-to-one correspondence with the TQFTs predicted by Dijkgraaf and Witten to be associated to Chern–Simons theory with a nonsimply connected Lie group. Thus in particular a rigorous construction of the Dijkgraaf–Witten TQFTs is given. As a byproduct, a purely quantum groups proof of the modularity of the full Weyl alcove for arbitrary quantum groups at arbitrary levels is given.