TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.15.3.112
TI  - The threefold way to quantum periods: WKB, TBA equations and q-Painlevé
PY  - 2023/09/25
UR  - https://www.scipost.org/SciPostPhys.15.3.112
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 15
IS  - 3
SP  - 112
A1  - Del Monte, Fabrizio
AU  - Longhi, Pietro
AB  - We show that TBA equations defined by the BPS spectrum of $5d$ $\mathcal{N}=1$ $SU(2)$ Yang-Mills on $S^1\times \mathbb{R}^4$ encode the q-Painlevé III$_3$ 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 $\mathbb{P}^1\times\mathbb{P}^1$.
ER  -

@Article{10.21468/SciPostPhys.15.3.112,
	title={{The threefold way to quantum periods: WKB, TBA equations and q-Painlevé}},
	author={Fabrizio  Del Monte and Pietro Longhi},
	journal={SciPost Phys.},
	volume={15},
	pages={112},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.15.3.112},
	url={https://scipost.org/10.21468/SciPostPhys.15.3.112},
}