Elliptic Genus of Calabi–yau Manifolds and Jacobi and Siegel Modular Forms

In the paper we study two types of relations: a one is between the elliptic genus of Calabi–Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac–Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applications to the theory of elliptic genus and of the second quantized elliptic genus. Introduction For a compact complex manifold one can define its elliptic genus as a function in two complex variables. If the first Chern class c1(M) of the complex manifold is equal to zero in H(M,R), then the elliptic genus is an automorphic form in variables τ ∈ H1 (H1 is the upper-half plane) and z ∈ C. More exactly, it is a Jacobi modular form with integral Fourier coefficients of weight 0 and index d/2, where d =dimC(M). The q -term of the Fourier expansion (q = e ) of the elliptic genus is essentially equal to the Hirzebruch χy-genus. Thus we can analyze the arithmetic properties of the χy-genus of the Calabi– Yau manifolds and its special values such as signature (y = 1) and Euler number (y = −1) in terms of Jacobi modular forms. The famous Rokhlin theorem about divisibility by 16 of the signature of a compact, oriented, differentiable spin manifold of dimension 4 was one of the starting points of the theory of elliptic genera. Ochanine generalized this Rokhlin result to the manifolds of dimRM ≡ 4 mod 8. One can find an elegant proof the Ochanine’s theorem using modular forms in one variable with respect to Γ0(2) in the lectures of Hirzebruch [HBJ, Chapter 8]. In this paper we study the Z-structure of the graded ring J 0,∗ = ⊕m≥1J Z 0,m of all weak Jacobi forms with integral Fourier coefficients (Theorem 1.9). We prove that this ring has four generators J 0,∗ = Z[φ0,1, φ0,2, φ0,3, φ0,4] where φ0,1, . . . , φ0,4 are some fundamental Jacobi forms related to Calabi–Yau manifolds of dimension d = 2, 3, 4, 8. We consider some applications of this result to Calabi–Yau manifolds. Properties of the signature (i.e. the value of χy-genus at y = 1) modulo some powers of 2 are well known (see (2.8)). We analyze properties of the value of the elliptic Supported by Max-Planck-Institut für Mathematik in Bonn