Evaluation of state integrals at rational points
Multi-dimensional state-integrals of products of Faddeev's quantum dilogarithms arise frequently in Quantum Topology, quantum Teichmüller theory and complex Chern-Simons theory, but have proved difficult to evaluate explicitly. The present paper calculates \(1\)-dimensional state integrals of the form: \[ \mathcal{I}_{A,B}(\mathbf{b})= \int_{\mathbb{R}+i\varepsilon} \Phi_\mathbf{b}(x)^{B} e^{-A\pi i x^2}\, dx \] where \(\mathbf{b}^2\) is a rational number. The answer is expressed in terms of the classical dilogarithm and elementary functions. Such state integrals appear in the context of state-sum invariants in quantum topology. The technique to evaluate them has since been used to evaluate other similar integrals, for example by the second author and \textit{M. Mariño} [Commun. Math. Phys. 346, No. 3, 967--994 (2016; Zbl 1348.81436)].
scientific article