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.