Contour Integrals and the Modular S-Matrix

TL;DR

This paper explores a conjecture linking RCFT characters to contour integrals, providing an algorithm for the modular S-matrix, and verifying it through explicit calculations for multiple character cases, thus advancing understanding of RCFT structures.

Contribution

It introduces a simple algorithm to compute the modular S-matrix from contour integrals for arbitrary characters and verifies the conjecture through explicit examples.

Findings

Agreement between integrals' critical exponents and characters confirms the conjecture for 2-4 characters.

Explicit computation of the S-matrix matches known theories, validating the approach.

Additional evidence from an 8-character example supports the conjecture.

Abstract

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.