Resurgence of Faddeev's quantum dilogarithm

TL;DR

This paper proves that Faddeev's quantum dilogarithm can be obtained via Borel summation of formal solutions to certain difference equations, revealing its resurgence properties and detailed analytic structure.

Contribution

It establishes a link between the quantum dilogarithm and resurgence theory, providing explicit formulas, pole analysis, and Stokes phenomena insights.

Findings

Borel summation yields Faddeev's quantum dilogarithm.

Explicit formula for the Borel plane meromorphic function.

Identification of poles, residues, and Stokes phenomena.

Abstract

The quantum dilogarithm function of Faddeev is a special function that plays a key role as the building block of quantum invariants of knots and 3-manifolds, of quantum Teichm\"uller theory and of complex Chern-Simons theory. Motivated by conjectures on resurgence and recent interest in wall-crossing phenomena, we prove that the Borel summation of a formal power series solution of a linear difference equation produces Faddeev's quantum dilogarithm. Along the way, we give an explicit formula for the meromorphic function in Borel plane, locate its poles and residues, and describe the Stokes phenomenon of its Laplace transforms along the Stokes rays.