Voros Coefficients for the Hypergeometric Differential Equations and Eynard–Orantin’s Topological Recursion: Part I—For the Weber Equation

Quantum curves from refined topological recursion: The genus 0 case

“…We conjecture these hold for all curves of hypergeometric type, though due to technical difficulties we were only able to confirm this for low values of g in the case with second order poles. These statements generalize the results of both [6,7] and [1] to the refined setting, and require several new ingredients. We propose values for Ω in Definition 3.5, some of which do not seem to have appeared before in the literature.…”

“…[22,23,24]. Most relevant for us, for spectral curves of hypergeometric type, Iwaki-Koike-Takei [6,7] were able to show existence of and obtain explicit expressions for quantum curves (in the unrefined case). Furthermore, they used the form of the quantum curves to obtain closed formulae for certain topological recursion invariants called free energies.…”

“…In general, it is difficult to find a closed formula for such coefficients. For unrefined quantum curves of hypergeometric type, an explicit computation of Voros coefficients was given in [7,6], following earlier progress [25,26]. One of the goals of the present work is to obtain an explicit formula for the Voros coefficients for the refined quantum curves of hypergeometric type, in the case without second order poles.…”

“…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.…”

On the integrable hierarchy for the resolved conifold

“…The main ingredients of our proof are two difference equations. One of these is obtained from a certain functional representation of the AL hierarchy due to Vekslerchik [32, 33], while the other was proved by one of us in [3] by adapting ideas which appeared in the study of the exact Wentzel–Kramers–Brillouin (WKB) method in [25]. In particular, by combining this with the results of Bridgeland in [8], we are able to obtain a closed‐form expression for the tau function, thus providing a new, but perhaps expected, link between integrable hierarchies of GW theory and Bridgeland's DT Riemann–Hilbert problem.…”