Dilogarithm identities for constant solutions of level-restricted Y-systems

Prove that the sums of Rogers dilogarithms evaluated at the character-theoretic constant solutions of each level-restricted Y-system, after adding the corresponding logarithmic correction terms, are expressed in terms of the central charge and scaling dimensions of the associated parafermion conformal field theory.

Background

The paper recalls that constant solutions of level-restricted Y-systems can be constructed from character values of finite-dimensional representations evaluated at torus elements determined by dominant integral weights of the relevant level. Earlier results establish dilogarithm identities for the positive-real constant solution, relating the resulting sum to the central charge of the associated parafermion conformal field theory.

The unresolved conjecture concerns the more general character-theoretic constant solutions. It asserts that their Rogers-dilogarithm sums, supplemented by logarithmic correction terms, encode not only the central charge but also the scaling dimensions of the corresponding parafermion conformal field theory. The present paper realizes fusion rings geometrically through rotation fixed points, but does not prove this conjectural dilogarithm statement.