Dilogarithm Identities in Conformal Field Theory and Group Homology

TL;DR

This paper links Rogers' dilogarithm identities in conformal field theory to group homology and algebraic K-theory, clarifying their mathematical origins and correcting previous conjectures.

Contribution

It interprets key dilogarithm identities as lifts of generators in third homology of certain groups, connecting conformal field theory with algebraic topology and K-theory.

Findings

Identifies the Richmond-Szekeres identity as a lift in third homology.

Clarifies the role of algebraic K-theory in dilogarithm identities.

Resolves conjectures related to hyperbolic 3-manifolds and group homology.

Abstract

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all $2 \times 2$ real matrices viewed as a {\it discrete} group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic $K$-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all $2 \times 2$ real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with the summary of a number of open conjectures on the mathematical side.