Abstract
A new construction of BPS monodromies for 4d {mathcal {N}}=2 theories of class {mathcal {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.
Highlights
Advances on wall crossing have been seminal for a wealth of developments that reshaped our understanding of BPS spectra of supersymmetric gauge theories [1,2,3,4,5]
A universal feature of wall crossing across all these contexts is the existence of an invariant quantity, known as the BPS monodromy, which controls how BPS spectra change across moduli spaces
As we show with some examples, a consequence of this topological nature is that the monodromy is manifestly symmetric under permutations of identical punctures, reflecting symmetry of the index under the exchange of the corresponding fugacities
Summary
Advances on wall crossing have been seminal for a wealth of developments that reshaped our understanding of BPS spectra of supersymmetric gauge theories [1,2,3,4,5]. The combinatorial data attached to a spectral network encode the spectrum of 2d–4d BPS states for a particular type of surface defect, termed canonical defect [14,21,22] These data are determined entirely by the topology of the network W(θ, u) and exhibit wall crossing behavior simultaneously with the topological degeneration of the network, at the critical phase θc in our setup. A refined version of S was constructed in [29], which captures the spectrum of 2d–4d BPS states for canonical surface defects of A1 theories In this case it is less clear if there is a connection to the present work, but it is natural to ask whether the construction of [29] admits a generalization to higher rank theories which relies on critical graphs. In the appendix we collect some conventions and several technical results, worth mentioning is the formalism developed in “Appendix D,” which may be of separate interest for handling complicated computations in generic spectral networks
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