Caterpillar Degenerations of Spectral Curves and Double Gelfand-Zeitlin Geometry

RESUMO

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. As a result, the variations produce both the perturbative coefficients of isomonodromy tau and the perturbative partition function, and thus give an identification of them without using conformal blocks. Finally, using the gluing technique for spectral curves and the DGZ system, we predict the perturbative coefficients of the conjectural higher-rank $P_{\mathrm{III}}$ isomonodromic tau function from the viewpoint of spectral geometry.