SL(2, [formula omitted]) Chern-Simons theories with rational charges and two-dimensional conformal field theories

Abstract

We present a hamiltonian quantization of the SL(2, R ) three-dimensional Chern-Simons theory with fractional coupling constant k = s/ r on a space manifold with torus topology in the “constrain-first” framework. By generalizing the “Weyl-odd” projection to the fractional charge case, we obtain multi-components holomorphic wave functions whose components are the Kac-Wakimoto characters of the modular-invariant admissible representations of  1 current algebra with fractional level. The modular representations carried by the quantum Hilbert space satisfy both Verlinde's and Vafa's constraints coming from conformal field theory. They are the “square roots” of the representations associated to the conformal ( r, s) minimal models. Our results imply that Chern-Simons theory with SO(2, 2) as gauge group, which describes (2 + 1)-dimensional gravity with negative cosmological constant, has the modular properties of the Virasoro discrete series. On the way, we show that the two-dimensional counterparts of Chern-Simons SU(2) theories with half-integer charge k = 1 2 p are the modular invariant D p+1 series of  1 current algebra of level 2 p − 2.