Manifolds and Modular Forms

Friedrich Hirzebruch

ISBN 10: 352816414X / ISBN 13: 9783528164140

Editore: Vieweg+Teubner, Vieweg+Teubner Verlag Jan 1994, 1994

Riguardo questo articolo

This item is printed on demand - it takes 3-4 days longer - Neuware -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. 212 pp. Englisch.

Codice articolo 9783528164140

Dati bibliografici

• Titolo: Manifolds and Modular Forms
• Editore: Vieweg+Teubner, Vieweg+Teubner Verlag Jan 1994
• Data di pubblicazione: 1994
• Lingua: Inglese
• ISBN 10: 352816414X
• ISBN 13: 9783528164140
• Rilegatura: Buch
• Edizione: seconda edizione
• Condizione: Neu
• Dimensioni: 233 L x 166 A x 13 P (millimetri) 233 millimetri di larghezza per 166 millimetri di altezza per 13 millimetri di profondità
• Peso: 361 grammi

Descrizione articolo

Riassunto

During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes.

Contenuti

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.