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portada Manifolds and Modular Forms
Type
Physical Book
Author
Translated by
Publisher
Language
English
Pages
212
Format
Paperback
ISBN13
9783528164140
Edition No.
0002
Categories

Manifolds and Modular Forms

Thomas Berger (Author) · Friedrich Hirzebruch (Author) · Peter S. Translated by Landweber (Translated by) · Vieweg+teubner Verlag · Paperback

Manifolds and Modular Forms - Hirzebruch, Friedrich ; Translated by Landweber, Peter S. ; Berger, Thomas

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Synopsis "Manifolds and Modular Forms"

During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". I wanted to develop the theory of "Elliptic Genera" and to learn it myself on this occasion. This theory due to Ochanine, Landweber, Stong and others was relatively new at the time. The word "genus" is meant in the sense of my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in 1956: A genus is a homomorphism of the Thorn cobordism ring of oriented compact manifolds into the complex numbers. Fundamental examples are the signature and the A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure. The elliptic genera depend on a parameter. For special values of the parameter one obtains the signature and the A-genus. Indeed, the universal elliptic genus can be regarded as a modular form with respect to the subgroup r (2) of the modular group; the two cusps 0 giving the signature and the A-genus. Witten and other physicists have given motivations for the elliptic genus by theoretical physics using the free loop space of a manifold.

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