Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 068, 33 pages      arXiv:0806.2337      http://dx.doi.org/10.3842/SIGMA.2008.068
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

Miranda C.N. Cheng a and Erik P. Verlinde b
a) Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02128, USA
b) Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE, Amsterdam, the Netherlands

Received July 01, 2008, in final form September 23, 2008; Published online October 07, 2008

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.

Key words: generalized Kac-Moody algebra; black hole; dyons.

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