Communications in Number Theory and Physics
Volume 9 (2015)
Evaluation of state integrals at rational points
Pages: 549 – 582
Multi-dimensional state-integrals of products of Faddeev’s quantum dilogarithms arise frequently in Quantum Topology, quantum Teichmüller 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.
state-integrals, $q$-series, quantum dilogarithm, cyclic dilogarithm, Rogers dilogarithm, quasi-periodic functions, Nahm equation, gluing equations, $4_1$, $5_2$ and $(-2, 3, 7)$ pretzel knot