عنوان

Manifolds and Modular Forms

(Number (ISBN

3663107264

(Number (ISBN

9783663107262

Number

b584621

Title Proper

Manifolds and Modular Forms

General Material Designation

[Book]

First Statement of Responsibility

by Friedrich Hirzebruch, Thomas Berger, Rainer Jung.

Edition Statement

Second Edition

Place of Publication, Distribution, etc.

Wiesbaden

Name of Publisher, Distributor, etc.

Vieweg+Teubner Verlag : Imprint: Vieweg+Teubner Verlag

Date of Publication, Distribution, etc.

1994

Specific Material Designation and Extent of Item

(XI, 212 p.).

Series Title

Aspects of Mathematics, 20.

Text of Note

A publication of the Max-Planck-Institut für Mathematik, Bonn.

Text of Note

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.

Text of Note

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.

Engineering, general.

Engineering.

by Friedrich Hirzebruch, Thomas Berger, Rainer Jung.

Friedrich Hirzebruch

Rainer Jung

Thomas Berger

Electronic name

[Book]

Y