Hyperbolization of Affine Lie Algebras

Abstract

In 1983, Feingold and Frenkel discovered a relation between Siegel modular forms of genus two and a rank-three hyperbolic Kac--Moody algebra extending the affine Lie algebra of type $A_1$. It inspires a problem to explore more general relations between affine Lie algebras, hyperbolic Kac--Moody algebras and modular forms. In this paper, we give an automorphic answer to this problem. We classify hyperbolic Borcherds--Kac--Moody superalgebras whose super-denominators define reflective automorphic products of singular weight on lattices of type $2U\oplus L$. As a consequence, we prove that there are exactly $81$ affine Lie algebras $\widehat{\mathfrak{g}}$ which have extensions to hyperbolic BKM superalgebras for which the leading Fourier--Jacobi coefficients of super-denominators coincide with the denominators of $\widehat{\mathfrak{g}}$. We find that $69$ of them appear in Schellekens' list of semi-simple $V_1$ structures of holomorphic CFT of central charge $24$, while $8$ of them correspond to the $N=1$ structures of holomorphic SCFT of central charge $12$ composed of $24$ chiral fermions. The last $4$ cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. This clarifies the relationship of affine Lie algebras, vertex algebras and hyperbolic BKM superalgebras at the level of modular forms.