Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

The appearance of a generalized (or Borcherds–) Kac–Moody algebra in the spectrum of BPS dyons in $\mathcal N=4$, $d=4$ 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 $PGL(2,\mathbb Z)$ 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.