Spectral networks for polynomial cubic differentials

2025/07/10 by Omar Kidwai, Kidwai, Omar, Guillaume Tahar +1
Computer Science · Physics and Astronomy · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Polynomial and algebraic computation #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2507.07971

openalex publication_date 2025/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study cubic differentials and their spectral networks on Riemann surfaces, focusing on the polynomial case on the Riemann sphere. We introduce the notion of spectral core as the primary tool for our study, refining the classical notion of core in the theory of flat surfaces, and show that it controls the birthing process of spectral network trajectories. As an application, we completely characterize the polynomial cubic differentials having saddle connections or critical tripods when the degree d is at most 3; in particular, we obtain the relevant degenerations as the phase is varied and determine explicitly the wall-and-chamber structure. In this case, we obtain the BPS structure according to Gaiotto-Moore-Neitzke's algorithm, and verify that it satisfies the Kontsevich-Soibelman wall-crossing formula. In physics language, this corresponds to computing the BPS spectrum of a certain four-dimensional N=2 quantum field theory, known as the (A₂,A_d-1) generalized Argyres-Douglas theory.

Citations

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• Spectral Networks and Snakes
• Wall-crossing, Hitchin Systems, and the WKB Approximation
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Related

• Riemann-Hilbert problems from rank 3 WKB spectral networks
• Wall-crossing formulas via spectral networks
• Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories
• Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies
• Network, cluster coordinates and N = 2 theory II: Irregular singularity