Minimal models on Riemann surfaces: The partition functions

Abstract

The Coulomb gas representation of the A n series of c = 1 − 6/[ m( m+1)], m ⩾ 3, minimal models is extended to compact Riemann surfaces of genus g > 1. An integral representation of the partitions functions, for any m and g is obtained as the difference of two gaussian correlation functions of a background charge, (background charge on sphere) × (1 − g), and screening charges integrated over the surface. The coupling constant × (compactification radius) 2 of the gaussian expressions are, as on the torus, m( m+1), and m/( m+1). The partition functions obtained are modular invariant, have the correct conformal anomaly and - restricting the propagation of states to a single handle - one can verify explicitly the decoupling of the null states. On the other hand, they are given in terms of coupled surface integrals, and it remains to show how they degenerate consistently to those on lower-genus surfaces. In this work, this is clear only at the lattice level, where no screening charges appear.