Complex vector bundles and Jacobi forms

TL;DR

This paper introduces a modified Witten genus as an automorphic correction to the elliptic genus, extending its automorphic properties to arbitrary vector bundles over compact complex manifolds, with implications in geometry and physics.

Contribution

It defines a new automorphic function called the modified Witten genus applicable to any holomorphic vector bundle, broadening the scope beyond Calabi-Yau manifolds.

Findings

Defines the modified Witten genus as an automorphic correction.

Extends automorphic properties of elliptic genus to arbitrary bundles.

Connects the concept to applications in physics and geometry.

Abstract

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).