Wall-crossing from supersymmetric galaxies

TL;DR

The paper provides an elementary physical derivation of the Kontsevich-Soibelman wall-crossing formula within 4d $N=2$ supergravity by modeling BPS bound states as halos around a fixed, supermassive core (BPS galaxies). It introduces framed BPS indices and a halo wall-crossing operator that acts on a generating function, showing that for contractible loops the sequence of wall crossings composes to the identity, recovering the KS formula, and extending naturally to noncontractible loops to include monodromies. A concrete monodromy relation $M\cdot \Gamma = \Gamma + \sum_k k^2 \Omega(k\gamma)\langle \gamma,\Gamma\rangle\,\gamma$ is derived, connecting monodromy to the BPS spectrum and recovering Picard-Lefschetz in the conifold case. The approach clarifies when bound-state mixing is suppressed in the large-core limit and provides a framework for relating wall-crossing to geometric transitions in moduli space, with implications for refined (motivic) variants under suitable symmetry assumptions.

Abstract

We give an elementary physical derivation of the Kontsevich-Soibelman wall crossing formula, valid for any theory with a 4d N=2 supergravity description. Our argument leads to a slight generalization of the formula, which relates monodromy to the BPS spectrum.

Figures (2)

• Figure 1: This shows the neighborhood ${\cal U}$ in the normal bundle to $W_{\gamma_1}\cap W_{\gamma_2}$. The wall of marginal stability is given by ${\rm Im}[Z(\gamma_1;t)\overline{Z(\gamma_2;t)}] =0$ since ${\rm Re}[Z(\gamma_1;t)\overline{Z(\gamma_2;t)}]$ is nonzero throughout ${\cal U}$. We choose the ordering of $\gamma_1, \gamma_2$ so that $W_{\gamma_1}$ is counterclockwise from $W_{\gamma_2}$ with opening angle smaller than $\pi$. Then the BPS walls $W_{r_1\gamma_1 + r_2 \gamma_2}$ are ordered so that increasing $r_1/r_2$ gives walls in the counterclockwise direction. We consider a path ${\cal P }$ in ${\cal U}$ circling the origin in the counterclockwise direction. The central charges of vectors $r_1\gamma_1 + r_2 \gamma_2$ with $r_1, r_2 \geq 0$ at representative points $t_0, \dots, t_7$ along ${\cal P }$ are illustrated in the next figure.
• Figure 2: As $t$ moves along the path ${\cal P }$ the central charges evolve as in this figure. Note that ${\rm Im}(Z_1 \overline{Z_2}) >0$ means that $Z_1$ is counterclockwise to $Z_2$ and rotated by a phase less than $\pi$. In that case the rays parallel to $r_1 Z_2 + r_2 Z_2$ for $r_1, r_2\geq 0$ are contained in the cone bounded by $Z_1 {\mathbb{R}}_+$ and $Z_2{\mathbb{R}}_+$, and ordered so that increasing $r_1/r_2$ corresponds to moving counterclockwise. When $t$ crosses the marginal stability wall the cone collapses and the rays reverse order. As $t$ moves in the region $t_2$ the quantity ${\rm arg} [Z_\gamma e^{-i \alpha_0}] > 0$ is increasing for all $\gamma_{r_1,r_2}$ with $r_1, r_2\geq 0$ while at the point $t_6$ the argument is decreasing.