Quantum Dilogarithm

TL;DR

This paper introduces a quantum version of Rogers' pentagon dilogarithm identity, explores its classical limit, and constructs a finite-dimensional realization linked to Fermat curves and star-triangle relations.

Contribution

It presents a novel quantum generalization of Rogers' dilogarithm identity and connects it to finite-dimensional representations and integrable models.

Findings

Quantum dilogarithm generalizes classical Rogers' identity.

Classical limit recovers the original Rogers' identity.

Finite-dimensional realization relates to Fermat curves and star-triangle relations.

Abstract

A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter.