Wall-crossing formulas via spectral networks

TL;DR

This paper provides a self-contained proof of the Kontsevich-Soibelman wall-crossing formula using spectral networks and quadratic differentials, avoiding DT theory and establishing convergence and path-lifting rules.

Contribution

It introduces a new framework for proving the wall-crossing formula purely through spectral networks and quadratic differentials, extending path-lifting rules and demonstrating convergence.

Findings

Established convergence of path liftings in spectral networks.

Defined path lifting rules for quadratic differentials.

Generated the hat-homology lattice via extended path liftings.

Abstract

We give a self-contained proof of the Kontsevich-Soibelman wall-crossing formula entirely in the scope of quadratic differentials without relying on input from DT theory. Our approach is based on path-lifting rules for spectral networks introduced by Gaiotto, Moore and Neitzke. We provide a framework to justify the convergence of the path liftings, including the cases with spiral domains. In particular, we define path lifting rules for spectral networks associated to holomorphic quadratic differentials. As an intermediate step in the proof of the wall-crossing formula, we show that upon extending the path lifting rules to $\mathcal{A}_0$-laminations we generate the hat-homology lattice.