Contour Integrals and the Modular S-Matrix

We investigate a conjecture to describe the characters of large families of RCFT's in terms of contour integrals of Feigin-Fuchs type. We provide a simple algorithm to determine the modular S-matrix for arbitrary numbers of characters as a sum over paths. Thereafter we focus on the case of 2, 3 and 4 characters, where agreement between the critical exponents of the integrals and the characters implies that the conjecture is true. In these cases, we compute the modular S-matrix explicitly, verify that it agrees with expectations for known theories, and use it to compute degeneracies and multiplicities of primaries. We also compute S in an 8-character example to provide additional evidence for the original conjecture. On the way we note that the Verlinde formula provides interesting constraints on the critical exponents of RCFT in this context.