Resurgent Structure of the Topological String and the First Painlevé Equation

Abstract

We present an explicit formula for the Stokes automorphism acting on the topological string partition function. When written in terms of the dual partition function, our formula implies that flat coordinates in topological string theory transform as quantum periods, and according to the Delabaere-Dillinger-Pham formula. We first show how the formula follows from the non-linear Stokes phenomenon of the Painlevé I equation, together with the connection between its $\tau$-function and topological strings on elliptic curves. Then, we show that this formula is also a consequence of a recent conjecture on the resurgent structure of the topological string, based on the holomorphic anomaly equations, and it is in fact valid for arbitrary Calabi-Yau threefolds.

Relations between Stokes constants of unrefined and Nekrasov-Shatashvili topological strings

“…stable D4-D2-D0 brane bound states in type IIA string. Similary conclusions have been reached in [60], where a closed-form formula for Stokes automorphisms of unrefined topological string free energies has been provided, which resembles the DDP formula of quantum periods.…”

“…4007022316 of the Southeast University. [60]. Although the methods and the concrete results of their paper are different from ours, we both give strong support to the idea that Stokes constants of unrefined free energies are given by BPS invariants.…”

“…Furthermore, argument in this paper is indirect and is only valid for non-compact Calab7-9au threefolds. A direct argument or even proof, potentially also valid for compact Calab7-9au threefold, possibly along the line of [60], will be very desirable.…”

“…We emphysize that our argument only works on topological string on non-compact Calabi-Yau threefolds where NS free energies and quantum periods as intermediary steps can be defined. A direct argument that make the left vertical arrow that works also for compact Calabi-Yau threefolds would be very desirable, possibly along the line of [60].…”