Evaluation of state integrals at rational points

TL;DR

This paper evaluates multi-dimensional state-integrals involving quantum dilogarithms at rational points, expressing results through classical and quantum dilogarithms and finite sums, with applications to knot invariants.

Contribution

It introduces a method to evaluate state-integrals at rational points using quasi-periodicity, connecting quantum and classical dilogarithms, and applies it to knot theory.

Findings

Explicit evaluations of state-integrals for specific knots.

Connection between quantum dilogarithms and classical dilogarithm at rational points.

New formulas involving finite sums at roots of unity.

Abstract

Multi-dimensional state-integrals of products of Faddeev's quantum dilogarithms arise frequently in Quantum Topology, quantum Teichm\"uller theory and complex Chern--Simons theory. Using the quasi-periodicity property of the quantum dilogarithm, we evaluate 1-dimensional state-integrals at rational points and express the answer in terms of the Rogers dilogarithm, the cyclic (quantum) dilogarithm and finite state-sums at roots of unity. We illustrate our results with the evaluation of the state-integrals of the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots at rational points.