Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

TL;DR

The paper demonstrates that the spectrum of 1/4-BPS dyons in ${\cal N}=4$, $d=4$ string theory is governed by a Borcherds--Kac--Moody algebra whose root lattice matches the $\,T$-duality invariants and whose Weyl group, extended by parity, coincides with the duality symmetry ${\rm PGL}(2,\mathbb{Z})$. Marginal-stability walls align with Weyl-chamber walls, making the Weyl group the discrete attractor-flow group that organizes moduli dependence of the BPS spectrum; the authors propose a microscopic interpretation of dyon degeneracies as a second-quantized highest-weight multiplicity, and show that the resulting wall-crossing formula matches the supergravity analysis. They connect the moduli dependence to a moduli-driven shift of highest weights in Verma modules and derive the wall-crossing formula purely from algebraic data. The work also reveals an arithmetic facet of attractor flows, linking wall-crossing to a Stern–Brocot-like labeling of decay channels. Overall, the study provides a concrete algebraic and geometric framework for understanding BPS spectra, wall-crossing, and attractor dynamics in a highly symmetric string theory context.

Abstract

The appearance of a generalized (or Borcherds-) Kac-Moody algebra in the spectrum of BPS dyons in N=4, d=4 string theory is elucidated. From the low-energy supergravity analysis, we identify its root lattice as the lattice of the T-duality invariants of the dyonic charges, the symmetry group of the root system as the extended S-duality group PGL(2,Z) of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant two-centered solutions. This leads to an interpretation for the Weyl group as the group of wall-crossing, or the group of discrete attractor flows. Furthermore we propose an equivalence between a "second-quantized multiplicity" of a charge- and moduli-dependent highest weight vector and the dyon degeneracy, and show that the wall-crossing formula following from our proposal agrees with the wall-crossing formula obtained from the supergravity analysis. This can be thought of as providing a microscopic derivation of the wall-crossing formula of this theory.

Figures (7)

• Figure 1: The Coxeter graph of the hyperbolic reflection group generated by (\ref{['generator_weyl']}). See (\ref{['def_coxeter']}) for the definition of the Coxeter graph.
• Figure 2: The dihedral group $D_3$, which is the symmetry group of an equilateral triangle, or the outer automorphism group of the real roots of the Borcherds--Kac--Moody algebra (the group of symmetry mod the Weyl group), is generated by an order two element corresponding to a reflection and an order three element corresponding to the $120^\circ$ rotation.
• Figure 3: A plane $(X,\alpha)=0$ of orthogonality to a positive real root $\alpha$ always intersects the hyperboloid $|X|^2=1$, or equivalently the upper-half plane or the Poincaré disk. And the root $\alpha$ defines two lightcone directions $\{\alpha^+$, $\alpha^-\}$ perpendicular to it, given by the intersection of the plane with the future light-cone.
• Figure 4: (i) The wall of marginal stability for the two-centered solution with charges $(P,0)$ and $(0,Q)$, projected onto the two-dimensional slice of moduli space equipped with a natural hyperbolic metric, and mapped to the Poincaré disk and the upper-half plane. (ii) The three simple walls $(X,\alpha_i)=0$.
• Figure 5: Tessellation of the Poincaré disk using the group $W$ generated by the reflections with respect to the three "mirrors", namely the three sides of the regular triangle in the middle. Walls of the same color are mirror images of each other. Notice that each triangle has the same volume using the hyperbolic metric. The slight inhomogeneity of colours on the edge is an artifact of the computer algorithm we use.

Theorems & Definitions (8)

• Definition 1: Coxeter system
• Definition 2: Length function
• Definition 3: Roots of the Coxeter group
• Theorem 1
• Corollary 1
• Theorem 2
• Definition 4: Bruhat order
• Definition 5: Weak Bruhat order