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Manifolds and Modular Forms

Friedrich Hirzebruch, Thomas Berger, Rainer Jung (Autoren)

Buch | Softcover

XI, 212 Seiten

1994 | 2nded. 1994
Vieweg & Teubner (Verlag)
978-3-528-16414-0 (ISBN)

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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.

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.

Erscheint lt. Verlag	1.1.1994
Reihe/Serie	Aspects of Mathematics
Übersetzer	Peter S. Translated by Landweber
Zusatzinfo	XI, 212 p.
Verlagsort	Wiesbaden
Sprache	englisch
Maße	162 x 229 mm
Gewicht	500 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Themenwelt	Technik
Schlagworte	Algebra • Algebraische Topologie • Elliptische Funktion • manifold • Mannigfaltigkeit (Mathematik) • Material • Modulform • SIGNATUR
ISBN-10	3-528-16414-X / 352816414X
ISBN-13	978-3-528-16414-0 / 9783528164140
Zustand	Neuware