defined as elements of Grassmann algebra (an algebra with anticom muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, during the next twenty years, the algebraic apparatus de veloped by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G 1. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticom muting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanaly sis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" == maps of sets and not as elements of Grassmann algebras. In 1974, Salam and Strathdee proposed a very apt name for a set of super points. They called this set a superspace.

TO SUPERANAL YSIS Edited by A.A. KIRILLOV Translated from the Russian by J. Niederle and R. Kotecky English translation edited and revised by Dimitri Leites SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Berezin, F.A. (Feliks Aleksandrovich) Introduction to superanalysis. (Mathematical physics and applied mathematics; v. 9) Part I is translation of: Vvedenie v algebru i analiz s antikommutirurushchimi peremennymi. Bibliography: p. Includes index. 1. Mathetical analysis. I. Title. II. Title: Superanalysis. III. Series. QA300. B459 1987 530. 15'5 87-16293 ISBN 978-90-481-8392-0 ISBN 978-94-017-1963-6 (eBook) DOI 10. 1007/978-94-017-1963-6 All Rights Reserved © 1987 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1987 No part of the material protected by this copyright notice may be reproduced in whole or in part or utilized in any form or by any means electronic or mechanical including photocopying recording or storing in any electronic information system without first obtaining the written permission of the copyright owner. CONTENTS EDITOR'S FOREWORD ix INTRODUCTION 1 1. The Sources 1 2. Supermanifolds 3 3. Additional Structures on Supermanifolds 11 4. Representations of Lie Superalgebras and Supergroups 21 5. Conclusion 23 References 24 PART I CHAPTER 1. GRASSMANN ALGEBRA 29 1. Basic Facts on Associative Algebras 29 2. Grassmann Algebras 35 3. Algebras A(U) 55 CHAPTER 2. SUPERANAL YSIS 74 1. Derivatives 74 2. Integral 76 CHAPTER 3. LINEAR ALGEBRA IN Zz-GRADED SPACES 90 1.

defined as elements of Grassmann algebra (an algebra with anticom muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, during the next twenty years, the algebraic apparatus de veloped by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G 1. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticom muting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanaly sis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" == maps of sets and not as elements of Grassmann algebras. In 1974, Salam and Strathdee proposed a very apt name for a set of super points. They called this set a superspace.

This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1|1. The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed. The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gravitino field. A natural geometric action functional on super Riemann surfaces is shown to reproduce the action functional of the non-linear supersymmetric sigma model using a component field formalism. The conserved currents of this action can be identified as infinitesimal deformations of the super Riemann surface. This is in surprising analogy to the theory of Riemann surfaces and the harmonic action functional on them. This volume is aimed at both theoretical physicists interested in a careful treatment of the subject and mathematicians who want to become acquainted with the potential applications of this beautiful theory.

Numbers ... , natural, rational, real, complex, p-adic .... What do you know about p-adic numbers? Probably, you have never used any p-adic (nonrational) number before now. I was in the same situation few years ago. p-adic numbers were considered as an exotic part of pure mathematics without any application. I have also used only real and complex numbers in my investigations in functional analysis and its applications to the quantum field theory and I was sure that these number fields can be a basis of every physical model generated by nature. But recently new models of the quantum physics were proposed on the basis of p-adic numbers field Qp. What are p-adic numbers, p-adic analysis, p-adic physics, p-adic probability? p-adic numbers were introduced by K. Hensel (1904) in connection with problems of the pure theory of numbers. The construction of Qp is very similar to the construction of (p is a fixed prime number, p = 2,3,5, ... ,127, ... ). Both these number fields are completions of the field of rational numbers Q. But another valuation 1 . Ip is introduced on Q instead of the usual real valuation 1 . I· We get an infinite sequence of non isomorphic completions of Q : Q2, Q3, ... , Q127, ... , IR = Qoo· These fields are the only possibilities to com plete Q according to the famous theorem of Ostrowsky.

Contains articles based on lectures given at the International Conference on Pseudo-differential Operators and Related Topics at Vaxjo University in Sweden from June 22 to June 25, 2005. Sixteen refereed articles cover a spectrum of topics such as partial differential equations, Wigner transforms, mathematical physics, and more.

Based on a conference held in Trento, Italy, and sponsored by the Centro Internazionale per la Ricera Matematica, this work presents advances in several complex variables and related topics such as transcendental algebraic geometry, infinite dimensional supermanifolds, and foliations. It covers the unfoldings of singularities, Levi foliations, Cauchy-Reimann manifolds, infinite dimensional supermanifolds, conformal structures, algebraic groups, instantons and more.