Elliptic genera
by
Valery A. Gritsenko
Steklov Institute of Mathematics in St. Petersburg

The elliptic genus (EG) of a compact complex manifold (or a Spin^c-manifold) was introduced as a holomorphic Euler characteristic of a formal power series with vector bundle coefficients. In physics such a function appears as the partition function of N=2 superconformal field theories. If the first Chern class of the complex manifold is equal to zero in H²(M, R), then the elliptic genus is a weak Jacobi form with integral Fourier coefficients. In this talk we consider new relations between the elliptic genus and Jacobi modular forms. As applications we get new results about the Euler number of Calabi-Yau manifolds and the integral cobordism ring of SU-manifolds. We discuss also special values of the Hirzebruch y-genus at the point of finite order (the signature, [^A]-genus, etc.)

Besides the elliptic genus we consider the second quantized elliptic genus and its relations with Lorentzian Kac-Moody algebras of Borcherds type.

Date received: August 11, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-57.