Quantization of Chern-Simons gauge theory with complex gauge group

Abstract

The canonical quantization of Chern-Simons gauge theory in 2+1 dimensions is generalized from the case in which the gauge group is a compact Lie groupG to the case in which the gauge group is a complex Lie groupG ℂ. Though the physical Hilbert spaces become infinite dimensional in the latter case, the quantization can be described as precisely as for compact gauge groups and using similar methods. The special case in which the gauge group isSL(2,ℂ) gives a description of 2+1 dimensional quantum gravity with Lorentz signature and positive cosmological constant or with Euclidean signature and negative cosmological constant. While it is not clear whether there is a 1+1 dimensional conformal field theory related to these 2+1 dimensional models, there are natural, computable candidates for the central charge and the conformal blocks of such a hypothetical theory.