Wall-crossing, Hitchin Systems, and the WKB Approximation

TL;DR

The paper develops a concrete, geometry-based algorithm to determine the BPS spectrum and its wall-crossing in a broad class of 4d N=2 theories derived from 6d (2,0) on punctured Riemann surfaces. Central to the approach are Hitchin moduli spaces paired with canonical Darboux coordinates constructed via Fock–Goncharov coordinates on flat connections and refined through WKB (Riemann–Hilbert) data, yielding a spectrum generator built from Kontsevich–Soibelman transformations. Wall-crossing is encoded by Poisson automorphisms associated to BPS states, and the authors provide an explicit computational scheme using WKB triangulations, flips, juggles, and pops to extract the BPS spectrum from triangulation data. Extending beyond regular punctures, the work treats irregular singularities and scaling limits of linear SU(2) quivers, including SU(2) gauge theories with N_f flavors, and demonstrates a rich tapestry of examples including Argyres–Douglas points and a unifying associahedron structure for wall-crossing. The framework offers new geometric insight into hyperkähler metrics on Hitchin moduli spaces, links to geometric Langlands via defects, and a triangulation-based method to compute BPS spectra across a wide landscape of 4d N=2 theories.

Abstract

We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in 0807.4723. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

Figures (88)

• Figure 1: Left: three separated M5-branes, including segments of two M2-branes stretching between them. Right: the corresponding picture in the $A_{K-1}$$(2,0)$ theory with $K=3$. The two M2-brane segments have been projected down to string segments.
• Figure 2: Left: a portion of an M2-brane stretching between two sheets of an M5-brane. The M2-brane is foliated by "vertical" segments, each of which lies in a single fiber of $T^* C$. At the branch point where the two sheets collide, the vertical segments shrink to zero length. Right: the projection of the M2-brane onto $C$ is a string in the $(2,0)$ theory which ends on the branch point.
• Figure 3: Left: An M2-brane wrapped on a disc $D$, stretched between two sheets of the curve $\Sigma$ supporting the M5-brane. Right: Under the projection $T^* C \to C$, the disc projects to a string in the $(2,0)$ theory, which ends on the branch points.
• Figure 4: Left: An M2-brane wrapped on a cylinder, stretched between two sheets of the curve $\Sigma$ supporting the M5-brane. Right: Under the projection $T^* C \to C$, the disc projects to a closed string in the $(2,0)$ theory.
• Figure 5: A configuration of Type IIA NS5-branes (blue), D4-branes (purple), and D6-branes (red circles with crosses). We have chosen $n = 2$ and $(k_0, k_1, k_2, k_3) = (2, 4, 3, 1)$. The D6-branes are at definite values of $x^{4,5,6}$.