TBA equations for $SU(r+1)$ quantum Seiberg-Witten curve: higher-order Mathieu equation
TLDR
This paper develops the ODE/IM correspondence for the higher-order Mathieu equation, deriving TBA equations and an effective central charge for $SU(r+1)$ quantum Seiberg-Witten curves.
Key contributions
- Develops ODE/IM correspondence for higher-order Mathieu equation in $SU(r+1)$ quantum Seiberg-Witten theory.
- Constructs Q-/Y-systems and derives TBA equations from subdominant solutions.
- Derives an analytic expression for the effective central charge from Y-function boundary conditions.
- Confirms agreement between TBA-derived Q-function expansion and WKB method at subleading and higher orders.
Why it matters
This work advances our understanding of the ODE/IM correspondence in quantum field theories, specifically for $SU(r+1)$ N=2 SYM. The derivation of TBA equations and the effective central charge provides new tools for analyzing these complex systems. The analytical agreement with WKB methods validates the approach.
Original 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 $θ\to -\infty$. From these boundary data, we derive an analytic expression for the effective central charge, which also governs the subleading contribution in the large-$θ$ expansion of the TBA equations. Finally, we compare the large-$θ$ 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.
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