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

Abstract

We show that TBA equations defined by the BPS spectrum of 5d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>5</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> \mathcal{N}=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>𝒩</mml:mi> </mml:mstyle> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> SU(2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> Yang-Mills on S^1\times \mathbb{R}^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℝ</mml:mi> </mml:mstyle> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> encode the q-Painlevé III _3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>3</mml:mn> </mml:msub> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℙ</mml:mi> </mml:mstyle> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℙ</mml:mi> </mml:mstyle> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> .

Topics & Concepts