Quantization of Diff S 1/S 1, Liouville theory, and the Kac relations

Abstract

Using a convenient parametrization of the manifold of diffeomorphisms of the circle modulo global rotations Diff S 1/S 1, we derive its two-parameter set of invariant symplectic structures and explicit form of the corresponding Poisson brackets. The quantization is performed explicitly, and the resulting quantum systems are verified to correspond to highest weight representations of the Virasoro group (in fact, to Verma modules), as expected from Kirillov-Kostant theory. A close link with quantum Liouville theory is pointed out, which is used to construct a model space which yields representations with null-vectors upon quantization, thereby giving some geometrical insight into the form of the Kac relations.