Dilogarithm identities, fusion rules and structure constants of CFTs

TL;DR

This paper explores the connection between dilogarithm identities, fusion rules, and structure constants in conformal field theories, revealing how quantum dimensions relate to dilogarithm arguments and establishing conditions for modular covariance.

Contribution

It generalizes previous proofs by linking quantum dimensions to dilogarithm arguments and derives infinite consistency conditions for Rogers-Ramanujan type partitions.

Findings

Quantum dimensions relate to dilogarithm function arguments.

Derived infinite consistency conditions for modular covariance.

Connected dilogarithm identities to structure constants in CFTs.

Abstract

Recently dilogarithm identities have made their appearance in the physics literature. These identities seem to allow to calculate structure constants like, in particular, the effective central charge of certain conformal field theories from their fusion rules. In Nahm, Recknagel, Terhoeven (1992) a proof of identities of this type was given by considering the asymptotics of character functions in the so-called Rogers-Ramanujan sum form and comparing with the asymptotics predicted by modular covariance. Refining the argument, we obtain {\it the general connection of quantum dimensions of certain conformal field theories to the arguments of the dilogarithm function} in the identities in question and {\it an infinite set of consistency conditions on the parameters of Rogers-Ramanujan type partitions for them to be modular covariant}.