Contour integrals and the modular \( \mathcal{S} \) -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 $$ \mathcal{S} $$ -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 verify that our algorithm reproduces the correct $$ \mathcal{S} $$ -matrix for SU (2)k for all k ≤ 18 which provides 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.