Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

TL;DR

This paper connects Borcherds-Kac-Moody algebras to BPS dyon spectra in N=4 string theory, interpreting wall-crossing phenomena via algebraic and geometric structures, and offers a microscopic derivation of the wall-crossing formula.

Contribution

It identifies the root lattice with T-duality invariants, relates walls of Weyl chambers to stability walls, and proposes an algebraic interpretation of dyon degeneracies and wall-crossing.

Findings

Root lattice matches T-duality invariants

Walls of Weyl chambers correspond to stability walls

Wall-crossing formula derived from algebraic perspective

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.