Complex Vector Bundles and Jacobi Forms

The elliptic genus (EG) of a compact complex manifold was introduced as a holomorphic Euler characteristic of some formal power series with vector bundle coefficients. EG is an automorphic form in two variables only if the manifold is a Calabi–Yau manifold. In physics such a function appears as the partition function of N = 2 superconformal field theories. In these notes we define the modified Witten genus or the automorphic correction of elliptic genus. It is an automorphic function in two variables for an arbitrary holomorphic vector bundle over a compact complex manifold. This paper is an exposition of the talks given by the author at Symposium “Automorphic forms and L-functions” at RIMS, Kyoto (January, 27, 1999) and at Arbeitstagung in Bonn (June, 20, 1999). Introduction In these notes we present a link between the theory of automorphic forms and geometry. For an arbitrary compact spin manifold one can define its elliptic genus. It is a modular form in one variable with respect to a congruence subgroup of level 2 (see, for example, [W1], [L], [HBJ]). For a compact complex manifold one can define its elliptic genus φ(M ; τ, z) as a function in two complex variables (see, for example, [W2], [EOTY], [Hö], [FY], [KYY]). In the last case the elliptic genus is the holomorphic Euler characteristic of a formal power series with vector bundle coefficients. If the first Chern class c1(M) of the complex manifold is equal to zero in H (M,R), then the elliptic genus is a weak Jacobi modular form (with integral Fourier coefficients) of weight 0 and index d/2, where d =dimC(M). The same modular form appears in physic as the partition function of N = 2 super-symmetric sigma model whose target space is the given Calabi–Yau manifold. We note that any “good” partition function has appeared in physic is an automorphic form with respect to some group. This reflects the fact that physical models have some additional symmetries. If c1(M) 6= 0, then the elliptic genus φ(M ; τ, z) of M is not an automorphic form. In these notes we define the modified Witten genus (MWG) or the automorphic correction of elliptic genus of an arbitrary holomorphic vector bundle over a compact complex manifold and we briefly study its properties. This new object is always an automorphic form in two variables. We are going to present here automorphic aspects of the theory. In the proof of the theorem that the modified Witten genus is a Jacobi form we use a nice formula which relates the Jacobi theta-series, its logarithmic derivative, the quasi-modular Eisenstein series G2(τ) and all the derivatives of the Weierstrass ℘-function (see Proposition 1.4 To appear in the Proceedings of the Symposium “Automorphic forms and L-functions”, RIMS, Kyoto, Japan, January 25–29, 1999