Generalized Kac-Moody Algebras from CHL dyons

TL;DR

This work links CHL dyon counting in heterotic compactifications to a family of generalized Kac-Moody (GKM) algebras by showing that the square roots of CHL Siegel modular forms ${\Delta_{k/2}(Z)}$ act as Weyl-Kac-Borcherds denominator formulas for GKMs ${\mathcal G}_N$. It provides an explicit additive lift construction from half-integral index Jacobi forms to produce ${\Delta_{k/2}(Z)}$, whose squares reproduce the level-$N$ Siegel forms ${\Phi_k(Z)}$ governing 1/4-BPS degeneracies. The algebras share the same real-root data with the base algebra ${A_{1,II}}$ but differ in imaginary-root multiplicities, reflecting the orbifold action on BPS sectors; this suggests an algebraic framework for the CHL BPS-state algebra and illuminates connections to the Gritsenko–Nikulin construction. The results indicate a deep link between automorphic corrections, BPS state counting, and the algebraic structure underlying CHL vacua, with implications for dualities, walls of marginal stability, and higher-derivative corrections in the low-energy theory.

Abstract

We provide evidence for the existence of a family of generalized Kac-Moody(GKM) superalgebras, G_N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Delta_{k/2}(Z), for (N,k)=(1,10), (2,6), (3,4), and possibly (5,2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric Z_N-orbifolds of the heterotic string compactified on T^6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Delta_{k/2}(Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, G_1 leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the `algebra of BPS states' in CHL compactifications.