
Manifolds and Modular Forms
Friedrich Hirzebruch(Author)
Vieweg+Teubner Verlag
Published on 1. January 1992
Book
Paperback/Softback
XI, 212 pages
978-3-528-06414-3 (ISBN)
Description
During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". Iwanted to develop the theory of "Elliptic Genera" and to leam 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 Thom 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 Chem 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 o 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.
More details
Series
Edition
1992
Language
German
Place of publication
Wiesbaden
Germany
Publishing group
Vieweg & Teubner
Target group
Professional and scholarly
Research
Illustrations
XI, 212 S.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 13 mm
Weight
353 gr
ISBN-13
978-3-528-06414-3 (9783528064143)
DOI
10.1007/978-3-663-14045-0
Schweitzer Classification
Other editions
Additional editions

Friedrich Hirzebruch
Manifolds and Modular Forms
E-Book
09/2013
Vieweg+Teubner Verlag
€59.99
Available for download
Content
Background.- Elliptic genera.- A universal addition theorem for genera.- Multiplicativity in fibre bundles.- The Atiyah-Singer index theorem.- Twisted operators and genera.- Riemann-Roch and elliptic genera in the complex case.- A divisibility theorem for elliptic genera.