Elliptic Genera and Vertex Operator Super-Algebras
This monograph deals with two aspects of the theory of elliptic genus: its topological aspect involving elliptic functions, and its representation theoretic aspect involving vertex operator super-algebras. For the second aspect, elliptic genera are shown
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1704
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1704
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Hirotaka Tamanoi
Elliptic Genera and Vertex Operator Super-Algebras
Springer
Author Hirotaka Tamanoi Department of Mathematics University of California Santa Cruz, CA 95064, USA E-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Tamanoi, Hirotaka:
Elliptic genera and vertex operator super algebras / Hirotaka Tamanoi. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo: Springer, 1999 (Lecture notes in mathematics; 1704) ISBN 3-540-66006-2
Mathematics Subject Classification (1991): Primary: 17B65, 55N22, 81RIO Secondary: 17A70, 17B68, 53C55, 81T40 ISSN 0075-8434 ISBN 3-540-66006-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10650205 41/3143-543210 - Printed on acid-free paper
CONTENTS
Introduction and summary of results. 0.1 Towards the signature of loop spaces: Elliptic genus 0.2 Elliptic genera 0.3 Vertex operator super-algebras 0.4 G-invariant vertex operator super-subalgebras and their vertex operators 0.5 Classifying spaces of geometric structures on manifolds 0.6 Elliptic genera as modules over vertex operator superalgebras 0.7 Infinite dimensional symmetries of elliptic genera for Kahler manifolds
1 1 4 7 9 13 16 17
Chapter I. Elliptic genera 1.1 Elliptic genera 1.2 Theta functions and elliptic functions 1.3 Proof of Theorem 4 of §1.1 1.4 The modular properties of elliptic genera 1.5 On the divisibility of coefficients of the elliptic genus 1.6 Spinors on loop spaces: the Witten genus
22 22 29 32 35 42 46
Chapter II. Vertex operator super-algebras 2.1 Vertex operator super-algebras: A brief introduction 2.2 The infinite dimensional Clifford modules as vertex operator super-algebras and their modules 2.3 Clifford algebras, Virasoro algebras and affine Lie algebras as subalgebras of the vertex oper
