We study the resurgent structure of the refined topological string partition function on a non-compact Calabi-Yau threefold, at large orders in the string coupling constant g_s and fixed refinement parameter b. For b\neq 1, the Borel transform admits two families of simple poles, corresponding to integral periods rescaled by b and 1/b. We show that the corresponding Stokes automorphism is expressed in terms of a generalization of the non-compact quantum dilogarithm, and we conjecture that the Stokes constants are determined by the refined DT invariants counting spin-xxxxxxxx BPS states. This jump in the refined topological string partition function is a special case (unit five-brane charge) of a more general transformation property of wave functions on quantum twisted tori introduced in earlier work by two of the authors. We show that this property follows from the transformation of a suitable refined dual partition function across BPS rays, defined by extending the Moyal star product to the realm of contact geometry.
