谱曲线的毛毛虫退化与双Gelfand-Zeitlin几何
Caterpillar Degenerations of Spectral Curves and Double Gelfand-Zeitlin Geometry
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中文总结 AI 辅助
本文研究(2,2)型谱曲线在弱耦合极限下的分解，将其与双Gelfand-Zeitlin系统联系，通过毛毛虫谱网络得到Stokes矩阵显式表达式，并识别微扰系数与配分函数，预测高秩P_{III} tau函数。
中文摘要 AI 辅助
本文研究了在毛毛虫或弱耦合极限下，具有两个二阶不规则奇点的谱曲线（即(2,2)型谱曲线）分解为两条各具有一个正则奇点（一阶极点）和一个二阶不规则奇点的谱曲线（即(1,2)型或(2,1)型谱曲线）的过程。此类谱曲线可与双Gelfand-Zeitlin（DGZ）系统等同。我们证明了DGZ侧的Duistermaat-Heckman测度可与规范理论侧由胶合两个quiver产生的Weyl测度等同。此外，等单值tau函数的傅里叶展开可解释为DGZ系统实极化的Peter-Weyl展开。另一方面，我们引入了节点毛毛虫谱网络。应用毛毛虫谱网络，我们获得了Stokes矩阵的显式表达式（直至归一化因子），该表达式与现有解析公式一致。此外，我们将Stokes计算中出现的端点归一化变化等同于Coulomb模空间上涨落复形的等变Euler类。因此，这些变化同时产生了等单值tau的微扰系数和微扰配分函数，从而无需共形块即可给出它们的等同性。最后，利用谱曲线胶合技术和DGZ系统，我们从谱几何视角预测了猜想的高秩P_{III}等单值tau函数的微扰系数。
英文摘要
In this paper, we study the decomposition, in the caterpillar or weak-coupling limit, of a spectral curve with two irregular poles of order two, namely a spectral curve of type $(2,2)$, into two spectral curves each having one regular singularity(order-$1$ pole) and one irregular pole of order two, namely spectral curves of type $(1,2)$ or $(2,1)$. Such spectral curves can be identified with a double Gelfand--Zeitlin (DGZ) system. We show that the Duistermaat--Heckman measure on the DGZ side can be identified with the Weyl measure on the gauge-theory side arises from gluing two quivers. Furthermore, the Fourier expansion of the isomonodromic tau function can be interpreted as a Peter--Weyl expansion in the real polarization of the DGZ system. On the other hand, we introduce the nodal caterpillar spectral network. Applying the caterpillar spectral network, we then obtain an explicit expression for the Stokes matrix up to a normalization factor. This expression agrees with the existing analytic formulas. Moreover, we identify the variation of endpoint normalization arising in the Stokes computation with the equivariant Euler class of the fluctuation complex over the Coulomb moduli space. Finally, using the gluing technique for spectral curves and the DGZ system, we conjecturally predict the perturbative coefficients of the higher-rank $P_{\mathrm{III}}$ isomonodromic tau function from the viewpoint of spectral geometry.
