Resurgence of Faddeev’s quantum dilogarithm

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 Teichmüller theory and of complex Chern–Simons theory. Motivated by conjectures on resurgence and the 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 Borel transform, a meromorphic function in the Borel plane, locate its poles and residues and describe the Stokes phenomenon of its Laplace transforms along the Stokes rays.