Wall Crossing Invariants from Spectral Networks

Abstract

Abstract.A new construction of BPS monodromies for 4d 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 A1-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.

An infrared bootstrap of the Schur index with surface defects

“…The spectral network, for ϑ = ϑ c , becomes maximally degenerate, and defines a so-called "BPS graph" [22]. In turn this graph encodes the 4d spectrum generator [21]. 11 Thus, throughout, we shall employ "BPS graphs" to explicitly compute the quantum spectrum generators S and S. 12 2.2 2d-4d wall-crossing and the IR Schur formula with surface defects Supersymmetric quantum field theories can be decorated with general (global) defects, such as Wilson [60] and 't Hooft [61,62] lines as well as their higher dimensional analogues.…”

“…Thus, we conclude that at the Roman locus the 2d-4d spectrum is well-defined, while the 4d spectrum alone is ill-defined. This is not an issue since the 4d spectrum generator still makes sense at the Roman locus [21], and encodes all the information we need about 4d BPS states to compute the 2d-4d spectrum generator. In fact, at the Roman locus the spectral network becomes maximally degenerate, and defines a so-called "BPS graph" [22].…”

“…This is precisely when the network becomes highly degenerate, and turns into the BPS graph. The analysis of soliton data for BPS graphs has been carried our in detail in [21], and we can directly borrow results from this reference. In particular, from [21, eq.…”

“…The Schur index I(q) was introduced in [16,17], as a particular limit of the full 4d N = 2 superconformal index [18,19], and -in analogy with the AGT correspondence -it was shown that I(q) arises as a correlator of a 2d TQFT on the UV curve C. The BPS monodromy M (q) arose in the work of Kontsevich and Soibelman as a wall-crossing invariant of BPS spectra [20]. A series of developments for studying BPS spectra through geometric techniques on C [4,13], led to a construction of M (q) based on topological data of a certain ribbon graph G, embedded in C [21,22]. previous related work [24][25][26][27].…”

“…Another important feature of BPS graphs is that they arise from "spectral networks". 3 One should keep in mind, that BPS graphs arise from spectral networks at special loci in the Coulomb branch -they arise from degenerate Strebel differentials [21,22] (Also see [40] for another relation between Strebel differentials and "Fenchel-Neilsen networks"). However, this is not an issue for computing Mv(q), since the latter is by definition invariant across the moduli space.…”