Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

Abstract: We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices $\Omega$ in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's $\tau$-function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function $Z_{\rm TR}$, up to a simple factor.