Manifolds and Modular Forms

Author:   Friedrich Hirzebruch ,  Peter S. Translated by Landweber ,  Thomas Berger ,  Rainer Jung
Publisher:   Springer Fachmedien Wiesbaden
Edition:   2nded. 1994
Volume:   20
ISBN:  

9783528164140


Pages:   212
Publication Date:   October 1994
Format:   Paperback
Availability:   Manufactured on demand   Availability explained
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Manifolds and Modular Forms


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Overview

"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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Author:   Friedrich Hirzebruch ,  Peter S. Translated by Landweber ,  Thomas Berger ,  Rainer Jung
Publisher:   Springer Fachmedien Wiesbaden
Imprint:   Vieweg+Teubner Verlag
Edition:   2nded. 1994
Volume:   20
Weight:   0.503kg
ISBN:  

9783528164140


ISBN 10:   352816414
Pages:   212
Publication Date:   October 1994
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand   Availability explained
We will order this item for you from a manufactured on demand supplier.
Language:   English

Table of Contents

1 Background.- 2 Elliptic genera.- 3 A universal addition theorem for genera.- 4 Multiplicativity in fibre bundles.- 5 The Atiyah-Singer index theorem.- 6 Twisted operators and genera.- 7 Riemann-Roch and elliptic genera in the complex case.- 8 A divisibility theorem for elliptic genera.- Appendix I Modular forms.- 1 Fundamental concepts.- 2 Examples of modular forms.- 3 The Weierstraß ?-function as a Jacobi form.- 4 Some special functions and modular forms.- 5 Theta functions, divisors, and elliptic functions.- Appendix II The Dirac operator.- 1 The solution.- 2 The problem.- 1 Zolotarev polynomials.- 2 Interpretation as an algebraic curve.- 3 The differential equation — revisited.- 4 Modular interpretation of Zolotarev polynomials.- 5 The embedding of the modular curve.- 6 Applications to elliptic genera.- Symbols.

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