The threefold way to quantum periods: WKB, TBA equations and q-Painlevé
Authors: Fabrizio Del Monte, Pietro Longhi
Preprint number: UUITP-19/23
We show that TBA equations defined by the BPS spectrum of 5d N=1 SU(2) Yang-Mills on S1×R4 encode the q-Painlevé 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-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 P1×P1.