Spectral networks for polynomial cubic differentials

TL;DR

This paper investigates polynomial cubic differentials on Riemann surfaces, introducing the spectral core concept to analyze spectral networks, characterizing degenerations, and computing BPS spectra relevant to certain quantum field theories.

Contribution

It introduces the spectral core as a new tool for studying spectral networks of polynomial cubic differentials and characterizes their degenerations and BPS spectra.

Findings

Complete characterization of polynomial cubic differentials with saddle connections for degree ≤ 3

Explicit description of wall-and-chamber structures during phase variation

Verification of the Kontsevich-Soibelman wall-crossing formula in this context

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 $\mathcal{N}=2$ quantum field theory, known as the $(A_{2},A_{d-1})$ generalized Argyres-Douglas theory.