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We study the resurgent structure of the refined topological string partitionfunction on a non-compact Calabi-Yau threefold, at large orders in the stringcoupling constant $g_s$ and fixed refinement parameter $b$. For $b\neq 1$, theBorel transform admits two families of simple poles, corresponding to integralperiods rescaled by $b$ and $1/b$. We show that the corresponding Stokesautomorphism is expressed in terms of a generalization of the non-compactquantum dilogarithm, and we conjecture that the Stokes constants are determinedby the refined DT invariants counting spin-$j$ BPS states. This jump in therefined topological string partition function is a special case (unitfive-brane charge) of a more general transformation property of wave functionson quantum twisted tori introduced in earlier work by two of the authors. Weshow that this property follows from the transformation of a suitable refineddual partition function across BPS rays, defined by extending the Moyal starproduct to the realm of contact geometry.

Resurgence of refined topological strings and dual partition functions