Path Integral Formulation of the Conformal Wess-Zumino-Witten to Liouville Reduction

The quantum Wess-Zumino-Witten $\to$ Liouville reduction is formulated using the phase space path integral method of Batalin, Fradkin, and Vilkovisky, adapted to theories on compact two dimensional manifolds. The importance of the zero modes of the Lagrange multipliers in producing the Liouville potential and the WZW anomaly, and in proving gauge invariance, is emphasised. A previous problem concerning the gauge dependence of the Virasoro centre is solved.