Topological Recursion and Uncoupled BPS Structures II: Voros Symbols and the $$\tau $$-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 $$\Omega $$ Ω 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 $$\tau $$ τ -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 $$Z_\textrm{TR}$$ Z TR , up to a simple factor.

Resurgent Structure of the Topological String and the First Painlevé Equation

“…Here, the cycle γ is dual to the Borel singularity A underlying the Stokes discontinuity, and as we explain below A is an integral period of the CY geometry (up to an overall normalization). This identification between Stokes constants appearing in the resurgent structure of the topological string and BPS invariants was already suggested in [41,42], and it is also consistent with the results of [46,47]. Further evidence for this identification appears in the recent paper [40].…”

Quantum curves from refined topological recursion: The genus 0 case

“…• The motivation for this work came from attempting to generalize the results of [6,7,1,8] to the refined setting. In particular, having established the formulas of this paper, we expect it should play a role in solving Barbieri-Bridgeland-Stoppa's so-called quantum Riemann-Hilbert problem [16] for corresponding refined BPS structures.…”

“…Let us first fix some notation. For simplicity, we only state the formula for the Weber and Whittaker curve, though analogous results hold for other hypergeometric curves 8 . We denote by ∞ ± ∈ P the (unique two) poles of λ, which satisfy π(∞ ± ) = ∞, and the sign convention in (2.25) We denote by α any oriented path which begins at ∞ − and ends at ∞ + , avoiding ramification points.…”

“…Requiring Ω n to be integer-valued obviously limits this, but an infinite amibiguity remains. Further investigation is needed to justify fully the definition above (for example, to see how the approach of [8] generalizes to the quantum Riemann-Hilbert problem [16]). In a slightly different context, some of this jumping behaviour is foreshadowed in [45,27], where a similar expression appears in the description of the connection formula for WKB solutions across a type II saddle.…”

“…The results of [6,7,1,8] provided explicit formulas for invariants of the quantum curve, the so-called Voros coefficients, as well as an explicit formula and relation to the free energy of the topological recursion. We generalize these results for the mentioned curves, and conjecture them for the remaining ones.…”