Feiyu Peng | Searcharxiv

arXiv2026

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

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.

hep-th↗

arXiv2024

Truncating Dyson-Schwinger Equations Based on Lefschetz Thimble Decomposition and Borel Resummation

We study the zero-dimensional prototype of the path integrals in quantum mechanics and quantum field theory, with the action $S(\phi)=\frac{\sigma }{2}\phi^{2}+\frac{\lambda}{4}\phi^{4}$. Using the Lefschetz thimble decomposition and the saddle point expansion, we derive multiple asymptotic formal series of the correlation function associated with the perturbative and non-perturbative saddle points. Furthermore, we reconstruct the exact correlation function employing the Borel resummation. We then consider how to truncate the Dyson-Schwinger (DS) equations beginning with the perturbation expansion of the correlation functions, analogous to the one obtained from the Feynmann diagram in higher dimensions. For the case $\sigma<0$, we find that although the asymptotic series around the perturbative saddle point is Borel summable, it does not capture the full information. Consequently, contributions from non-perturbative saddle points must be included to ensure a complete truncation procedure.

hep-th↗