The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

Abstract

We show that TBA equations defined by the BPS spectrum of 5d5d\mathcal{N}=1𝒩=1SU(2)SU(2) Yang-Mills on S^1\times \mathbb{R}^4S1×ℝ4 encode the q-Painlevé III_33 equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to q-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify two one-parameter deformations of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition functions continues to provide explicit TBA solutions. Motivated by these observations, we propose a further extensions of the range of validity of this correspondence, under a suitable identification of moduli. As further checks of our proposal, we study the behavior of exact WKB quantum periods for the quantum curve of local \mathbb{P}^1\times\mathbb{P}^1ℙ1×ℙ1.

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

“…Together the NS free energies and the NS Wilson loop amplitudes completely determine the quantum periods [55], and the Stokes automorphisms of quantum periods can also be derived from those of NS free energies and Wilson loop amplitudes. A pattern of the latter was recognised, from which one concludes that the Stokes automorphisms of quantum periods in 5d gauge theories follow the same DDP formulas as 4d gauge theories [55] (see also [56]), further confirming the horizontal arrows in the second arrow in figure 1. In addition, Stokes constants of NS free energies are themselves identified with Stokes constants of quantum periods, providing the vertical arrows on the right hand side in figure 1.…”

“…The forms of the alien derivatives (2.32) and the properties of the Stokes constants S NS A have profound consequences. As pointed out in [55] (see also [56]) and will be reviewed in appendix A, they imply, together with the Stokes transformation properties of Wilson loop amplitudes, that the quantum periods satisfy the DDP type of formulas for Stokes automorphism [44][45][46], so that the Stokes constants of quantum periods can be identified with BPS invariants. More importantly, the Stokes constants S NS A are identified with those of quantum periods [55], so that S NS A themselves are given by BPS invariants.…”

Resurgent Structure of the Topological String and the First Painlevé Equation

“…Although topological strings are closely related to quantum periods (also known as Voros symbols), their resurgent structure is quite different. For example, it is expected that the Stokes automorphisms of quantum periods are given by the Delabaere-Dillinger-Pham (DDP) formula [23,24,25] (see [22,43] for recent arguments in that direction), but the Stokes automorphisms act on the topological string partition function with a more complicated structure, and no closed formula is known for them.…”