Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

• Main Authors: Miranda C.N. Cheng, Erik P. Verlinde
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2008-10-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: generalized Kac-Moody algebra; black hole; dyons
• Online Access: http://dx.doi.org/10.3842/SIGMA.2008.068

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.