Evaluation of state integrals at rational points
type: Item
instance of: scholarly article
title: Evaluation of state integrals at rational points
zbMATH Open document ID: 1364.81216
author: Stavros Garoufalidis
author: R. M. Kashaev
published in: Communications in Number Theory and Physics
publication date: 2015-11-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.6062
review text: 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)].
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 81T45
Mathematics Subject Classification ID: 11G55
zbMATH DE Number: 6505258
zbMATH Keywords: state-integrals
zbMATH Keywords: \(q\)-series
zbMATH Keywords: quantum dilogarithm
zbMATH Keywords: cyclic dilogarithm
zbMATH Keywords: Rogers dilogarithm
zbMATH Keywords: quasi-periodic functions
zbMATH Keywords: Nahm equation
zbMATH Keywords: gluing equations
zbMATH Keywords: quantum hyperbolic geometry
MaRDI profile type: Publication
arXiv ID: 1411.6062
DOI: 10.4310/CNTP.2015.V9.N3.A3
description: scientific article