Borcherds-Kac-Moody Symmetry of N=4 Dyons

TL;DR

The paper uncovers a Borcherds-Kac-Moody superalgebra symmetry underlying the exact dyon spectrum in CHL orbifolds with ${\cal N}=4$, showing that for $N=1,2,3$ the dyon partition function can be realized as the denominator of this algebra via a Weyl denominator identity for a finite root system. The degeneracies are encoded by a genus-two Siegel modular form $\Φ_t(\Omega)$, built from a multiplicative lift of weak Jacobi forms and supplemented by a moduli-dependent contour that tracks wall-crossing through Weyl reflections corresponding to marginal stability walls. The framework provides a microscopic, algebraic interpretation of wall-crossing and attractor flows, identifying the Weyl group with the discrete attractor flow group and the dyon spectrum with a second-quantized representation of the algebra. For $N\ge4$, the finite-wall structure disappears and no such denominator interpretation exists, highlighting a sharp boundary between highly symmetric finite CHL models and the more elusive infinite-wall cases.

Abstract

We consider compactifications of heterotic string theory to four dimensions on CHL orbifolds of the type T^6 /Z_N with 16 supersymmetries. The exact partition functions of the quarter-BPS dyons in these models are given in terms of genus-two Siegel modular forms. Only the N=1,2,3 models satisfy a certain finiteness condition, and in these cases one can identify a Borcherds-Kac-Moody superalgebra underlying the symmetry structure of the dyon spectrum. We identify the real roots, and find that the corresponding Cartan matrices exhaust a known classification. We show that the Siegel modular form satisfies the Weyl denominator identity of the algebra, which enables the determination of all root multiplicities. Furthermore, the Weyl group determines the structure of wall-crossings and the attractor flows of the theory. For N> 4, no such interpretation appears to be possible.

Figures (1)

• Figure 1: The structure of walls when projected onto the Poincar� disk. The fundamental Weyl Chamber is bounded by three walls labeled by $\alpha_1, \alpha_2, \alpha_3$ for $N=1$; by four walls labeled by $\alpha_1, \alpha_2, \alpha^{(2)}_3, \alpha^{(2)}_4$ for $N=2$; and by six walls labeled by $\alpha_1, \alpha_2, \alpha^{(3)}_3, \alpha^{(3)}_4, \alpha^{(3)}_5, \alpha^{(3)}_6$ for $N=3$. Recall that each triangle is really equivalent and any triangle can be mapped to the central one by some conformal transformation.