Dilogarithm Identities in Conformal Field Theory

TL;DR

This paper explores dilogarithm identities in 2D conformal field theories, highlighting their connections to fusion rules, partition identities, and potential links to 3-manifold classification, with many proofs still conjectural.

Contribution

It presents new conjectures and ideas on dilogarithm identities, contributing to the classification of rational conformal field theories and their mathematical structures.

Findings

Dilogarithm identities relate to central charges and conformal dimensions.

Evidence suggests deep connections to fusion rules and partition identities.

Mathematical structures may be dual to Thurston's 3-manifold classification.

Abstract

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 some 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.