Dilogarithm identities for the central charges and conformal dimensions exist for at least large classes of rational conformally invariant quantum field theories in two dimensions. In many cases, proofs are not yet known but the numerical and structural evidence is convincing. In particular, close relations exist to fusion rules and partition identities. We describe some examples and ideas, and present conjectures useful for the classification of conformal theories. The mathematical structures seem to be dual to Thurston's program for the classification of 3-manifolds.
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 the general connection of quantum dimensions of certain conformal field theories to the arguments of the dilogarithm function in the identities in question and an infinite set of consistency conditions on the parameters of RogersRamanujan type partitions for them to be modular covariant.
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