Caterpillar Degenerations of Spectral Curves and Double Gelfand-Zeitlin Geometry
Abstract
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 PIII isomonodromic tau function from the viewpoint of spectral geometry.
Related
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BibTeX
@misc{liu2026caterpillar,
title = {{Caterpillar Degenerations of Spectral Curves and Double Gelfand-Zeitlin Geometry}},
author = {George Y. Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14423},
eprint = {2609.14423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14423},
}
RIS
TY - UNPB
TI - Caterpillar Degenerations of Spectral Curves and Double Gelfand-Zeitlin Geometry
AU - George Y. Liu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.14423
UR - https://arxiv.org/abs/2609.14423
AB - 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_III isomonodromic tau function from the viewpoint of spectral geometry.
ER -