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Wall crossing, discrete attractor flow and Borcherds algebra. (English) Zbl 1164.81009

Summary: 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 \(PGL(2,\mathbb{Z})\) 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.

MSC:

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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