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

We show that TBA equations defined by the BPS spectrum of 5d N = 1 SU(2) Yang-Mills on S1 x R4 encode the q-Painleve III3 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-Painleve. 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-Painleve 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 P1 x P1.