Supermanifolds: Theory And Applications

Supermanifolds Theory and Applications This page intentionally left blank Supermanifolds Theory and Applications Alice Rogers King's College London World Scientific NEW JERSEY . LONDON . SINGAPORE . BEIJING . SHANGHAI . HONG KONG . TAIPEI . CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. SUPERMANIFOLDS: THEORY AND APPLICATIONS Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-02-1228-5 ISBN-10 981-02-1228-3 Printed in Singapore. ZhangJi - Supermanifolds.pmd 1 4/25/2007, 3:09 PM April 25, 2007 9:45 WSPC/Book Trim Size for 9in x 6in to Richard v RogersSupermanifolds April 25, 2007 9:45 WSPC/Book Trim Size for 9in x 6in RogersSupermanifolds Preface The subject of supermanifolds has to some extent ‘just growed’ like Topsy, as the idea of adjoining anticommuting variables to conventional commuting variables proved useful in a variety of contexts. This has led to a bewildering variety of approaches which has often obscured the underlying unity of the ideas. Supermanifolds have been with me through much of this period of growth, and now in writing this book I intend not a formal mathematical treatise, but rather a working man or woman’s guide to the geometry and analysis of supermanifolds, together with applications of the theory. Supermanifolds continue to find new uses, the underlying ideas have proved robust, powerful and adaptable. My aim is to provide a unified picture, distilling the key ideas from a welter of sources. I have tried to give references to relevant original work, and can only apologise for any failures, which will have been unintentional. I am very grateful for many conversations about supermanifolds that I have enjoyed and benefited from over the years – these have been with almost everyone whose work is referred to in this book, a sad exception being Berezin. I would particularly like to thank Bryce DeWitt and Marjorie Batchelor, both of whom have patiently explained many things to me. I am also grateful to my colleagues in the mathematics department of King’s College London for a stimulating and agreeable working environment. F. A. Ro