Automorphic Forms and Lorentzian Kac–moody Algebras Part Ii

Abstract

Abstract. We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). The data for these liftings are Jacobi forms of integral and half-integral indices. In particular, we get modular forms which are generalizations of the Dedekind eta-function. Some of these forms define automorphic corrections of Lorentzian Kac-Moody algebras with hyperbolic generalized Cartan matrices of rank three classified in Part I of this paper. We also construct many automorphic forms which give discriminants of moduli of K3 surfaces with conditions on Picard lattice. These results are important for Mirror Symmetry and theory of Lorentzian Kac-Moody algebras. §0. IntroductionIn Part I we developed the general theory of reflective automorphic forms and their particular case of Lie reflective automorphic forms on Hermitian symmetric domains of type IV. These automorphic forms are very important in Mirror Symmetry (for K3's and Calabi-Yau's) and for Lorentzian Kac-Moody algebras. In Part I, in particular, we showed that this theory is similar to the theory of hyperbolic root systems (it is its mirror symmetric variant). We explained that reflective automorphic forms are very exceptional. Conjecturally their number is finite similarly to finiteness results for corresponding hyperbolic root systems with some condition of finiteness of volume for fundamental polyhedron (i.e., of elliptic or parabolic type). We believe and hope to show in further publications that classification of reflective automorphic forms and corresponding hyperbolic root systems is the key step in classification of some important class of Calabi-Yau's (see [GN6]). For example, finiteness results for hyperbolic root systems of elliptic and parabolic type and for reflective automorphic forms are related with finiteness of families of these Calabi-Yau's.We demonstrated in Part I a general method of classification of the hyperbolic root systems on the example of symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type of rank 3 and with a lattice Weyl vector. Let us denote the set of these matrices by A. It contains 60 matrices. In this Part II we consider methods of construction of reflective automorphic forms. In particular, for many generalized Cartan matrices A ∈ A we find their mirror symmetric objects -automorphic forms F defining automorphic corrections of Kac-Moody algebras g(A) corresponding to A. These forms F define so-called automorphic Lorentzian Kac-Moody algebras g F containing g(A) and having good automorphic properties. In this paper we consider mainly 3-dimensional automorphic forms F with respect to the full paramodular groups, i.e. automorphic forms on the Siegel upper half-space H 2 of complex dimension 3 with respect to the paramodular groups Γ t ⊂ Sp 4 (Q) (the threefold A t = Γ t \ H 2 is the moduli space of Abelian surfaces with ...

Fields, Strings, Matrices and Symmetric Products

“…In the special case of K3 the infinite product Φ(Ω) is a well-known automorphic form [41], see also [42,43]. First of all, the elliptic genus of K3 is the unique (up to a scalar) weak Jacobi form of weight 0 and index 1.…”

Theta lifts and local Maass forms

“…These extended theta lifts have more recently appeared in a variety of applications including generalized Kac-Moody algebras [18] and the arithmetic of Shimura varieties [15]. Following Bruinier's [12] application of Borcherds lifts to harmonic weak Maass forms, Bruinier and Funke [13] extended theta lifts to harmonic weak Maass forms.…”

Siegel modular forms and black hole entropy

“…Our focus will be mostly on the later; the additive lift will play a minor role around (3.14). The exponential lift is described in Theorem 2.1 of [32], which first portion states:…”