SL(2,R) Chern-Simons Theories with Rational Charges and Two-dimensional Conformal Field Theories

TL;DR

This paper develops a Hamiltonian quantization of SL(2,R) Chern-Simons theory with fractional charges, revealing connections to conformal field theories and 2+1 dimensional gravity with negative cosmological constant.

Contribution

It introduces a fractional charge framework for SL(2,R) Chern-Simons theory, linking it to admissible representations of current algebra and conformal minimal models.

Findings

Wave functions correspond to Kac-Wakimoto characters.

Quantum Hilbert space satisfies Verlinde and Vafa constraints.

Chern-Simons theory exhibits modular properties of Virasoro discrete series.

Abstract

This paper (completed March 1992) is an extensively revised and expanded version of work which appeared July 1991 on the initial incarnation of the hepth bulletin board, and which was published in the Proceedings of the Workshop on String Theory, Trieste, March 1991. Abstract We present a hamiltonian quantization of the $SL(2,R)$ 3-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 ${\hat A}_1$ 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 2-dimensional counterparts of Chern-Simons $SU(2)$ theories with half-integer charge $k=p/2$