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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." Full Product DetailsAuthor: Friedrich Hirzebruch , Peter S. Translated by Landweber , Thomas Berger , Rainer JungPublisher: Springer Fachmedien Wiesbaden Imprint: Vieweg+Teubner Verlag Edition: 2nded. 1994 Volume: 20 Weight: 0.503kg ISBN: 9783528164140ISBN 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 We will order this item for you from a manufactured on demand supplier. Language: English Table of Contents1 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.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |