Four ways across the wall

TL;DR

The paper surveys four complementary approaches to computing jumps in the BPS spectrum across walls of marginal stability in $\mathcal{N}=2$ theories: Kontsevich-Soibelman and Joyce-Song formalisms based on generalized Donaldson-Thomas invariants, and two physical pictures from multi-centered black hole solutions on the Coulomb and Higgs branches. It presents the motivic KS formula with a Lie-algebraic encoding of wall-crossing, the explicit JS combinatorics for DT invariants, and a Coulomb-branch localization calculation of the configurational index $g_{ref}(\\{\\alpha_i\\},y)$ from multi-centered dynamics, plus an Abelian-quiver (Higgs branch) derivation via Reineke’s formula. Explicit checks show agreement among the four approaches for $n\le 5$, though a complete combinatorial proof of equivalence remains open. The work highlights deep connections between DT theory, quiver moduli, and supergravity/multi-centered dynamics, and points to localization as a powerful tool for understanding BPS spectrum transitions in broader contexts.

Abstract

An important question in the study of N=2 supersymmetric string or field theories is to compute the jump of the BPS spectrum across walls of marginal stability in the space of parameters or vacua. I survey four apparently different answers for this problem, two of which are based on the mathematics of generalized Donaldson-Thomas invariants (the Kontsevich-Soibelman and the Joyce-Song formulae), while the other two are based on the physics of multi-centered black hole solutions (the Coulomb branch and the Higgs branch formulae, discovered in joint work with Jan Manschot and Ashoke Sen). Explicit computations indicate that these formulae are equivalent, though a combinatorial proof is currently lacking.

Figures (2)

• Figure 1: Chamber structure of the $u$-plane and BPS spectrum in $\mathcal{N}=2, D=4$ SYM theory with $SU(2)$ gauge group and no flavor. The line $\,{\rm Im}\, (a/a_D)=0$ separates the strong and weak coupling chambers. The only stable BPS states in the strong coupling chamber are the monopole and dyons with charges $(q,p)=\pm(0,1)$, $\pm(2,-1)$, in the conventions of Seiberg:1994aj. The weak coupling spectrum consists of these same states and their images around the monodromy at infinity, plus the $W$-boson with charge $(2,0)$.
• Figure 2: Phase structure of the moduli space $\mathcal{M}_n$ of 3-centered solutions as a function of $c_i$, for fixed charges such that $\alpha_{12}>0,\alpha_{23}>0,\alpha_{13}>0$, $\alpha_{12}<\alpha_{23}$. The shaded area represent the values of $c_i$ in the two-dimensional section $c_1+c_2+c_3=0$ which are spanned as the location of the 3rd center is varied, keeping the centers 1 and 2 fixed. Conversely, if the values of $c_i$ is fixed, the range of distances between the centers 1 and 2 can be read off by intersecting the shaded area with a radial line which joins $c_i$ to the origin. Thus, for this choice of charges, 3-centered solutions only exist in the region $c_1>0, c_3<0$. Inside this region, the range of $r_{12}$ is bounded from below and from above, except on the wall of marginal stability $c_2=0$. The boundaries of the shaded region correspond to collinear solutions whose order is indicated. As the wall is crossed, the topology of the collinear solutions changes from $(321),(132)$ to $(321),(312)$.