Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

Abstract

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 P GL(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 modulidependent 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.

Generalized Mathieu Moonshine

“…For some conjugacy classes [g] ∈ M 24 the associated second-quantized twining genera Φ g also give rise to denominator formulas of GKMs [56], which should presumably be identified with the wall-crossing algebras found in CHL-models [51,[57][58][59][60]. Although these observations are very suggestive, the precise role of the GKMs for Mathieu Moonshine remains to be understood.…”

From Special Geometry to Black Hole Partition Functions

“…The issue of choosing integration countours is actually more subtle than apparent from our review, see [64,65] for a detailed account. For a comprehensive account of Siegel modular forms, see [86].…”

BPS-saturated string amplitudes: K3 elliptic genus and Igusa cusp form χ10

“…This suggests that degeneracies of dyons also become related to the root multiplicities of the associated BKM-algebra. The physical role of this BKMalgebra has been further clarified in [32] (see also [33][34][35]), where it has been shown, that the wall-crossing behaviour of the dyon spectrum is controlled by the hyperbolic Weyl group of this BKM-algebra.…”