Wall-Crossing Invariants from Spectral Networks

TL;DR

This paper introduces a new, invariant construction of BPS monodromies for 4d N=2 theories of class S using spectral networks and graph topology, avoiding direct BPS spectrum data.

Contribution

It presents a novel graph-based method to compute BPS monodromies that is invariant under wall crossing, with an algorithmic approach and applications to specific theories.

Findings

Constructed BPS monodromies from spectral network limits.

Developed an algorithm to solve the monodromy equations.

Connected graph structures to superconformal index symmetries.

Abstract

A new construction of BPS monodromies for 4d ${\mathcal N}=2$ theories of class S is introduced. A novel feature of this construction is its manifest invariance under Kontsevich-Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve $C$ of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations, and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For $A_1$-type theories, the graphs encoding the monodromy are "dessins d'enfants" on $C$, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg-Witten curve.