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# Communications in Number Theory and Physics

## Volume 9 (2015)

### Number 3

### Evaluation of state integrals at rational points

Pages: 549 – 582

DOI: http://dx.doi.org/10.4310/CNTP.2015.v9.n3.a3

#### Authors

#### Abstract

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.

#### Keywords

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