TBA equations for $SU(r+1)$ quantum Seiberg-Witten curve: higher-order Mathieu equation

TL;DR

This paper extends the ODE/IM correspondence to higher-order Mathieu equations from quantum Seiberg-Witten curves, deriving TBA equations and analyzing their boundary conditions, effective central charge, and agreement with WKB methods.

Contribution

It introduces a novel application of the ODE/IM correspondence to higher-order Mathieu equations in supersymmetric Yang-Mills theory, deriving TBA equations and boundary conditions.

Findings

Derived TBA equations from subdominant solutions.

Obtained an analytic expression for the effective central charge.

Achieved agreement between TBA and WKB methods at subleading order.

Abstract

We develop the ODE/IM correspondence for the higher-order Mathieu equation arising from the quantum Seiberg-Witten curve of the pure $SU(r+1)$ ${\cal N}=2$ supersymmetric Yang-Mills theory. From the subdominant solutions, we construct the Q-/Y-systems and derive the corresponding TBA equations. The dependence of the moduli parameters is found to be encoded in the boundary conditions of the Y-functions at $\theta \to -\infty$. From these boundary data, we derive an analytic expression for the effective central charge, which also governs the subleading contribution in the large-$\theta$ expansion of the TBA equations. Finally, we compare the large-$\theta$ expansion of the Q-function derived from the TBA equations with that obtained from the WKB method, which yields analytic agreement at subleading order and precise numerical agreement at the higher-order corrections.