Grundlehren der

mathematischen Wissenschaften 289

ASeries 0/ Comprehensive Studies in Mathematics

Editors
S. s. ehern B. Eckmann P. de la Harpe H. Hironaka
F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus
A. Kupiainen J. Lannes G. Lebeau M. Ratner
D. Serre Ya G. Sinai N. J. A. Sloane J. Tits
M. Waldschmidt S. Watanabe

Managing Editors
M. Berger J. Coates S. R. S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Yuri I. Manin

Gauge Field Theory

and Complex Geometry

Translated from the Russian by

N. Koblitz and J. R. King

With an Appendix by Sergei Merkulov

Second Edition

Springer
Yuri Ivanovich Manin
Max-Planck-Institut fUr Mathematik
Gottfried-Claren-Stra8e 26
D-53225 Bonn

Tide of the original Russian edition:

Kalibrovocbnye polya i komplesksnaya geometriya
Publisher Nauka, Moscow 1984

Llbrary of Congress Cataloglng-In-Publleatlon Data

Manln, :ID. I.
[Kallbrovochnya polfi I kOlplaksnafi gaolatrlfi. Engllsh]
Gauga flald thaory and cOlplex gaolatry I Yurl 1. Manln ;
translatad frol the Russlan by N. Koblltz and ~.R. Klng. -- 2nd ed.
p. CI. -- (Grundlahren der lathalatlschan Hlssanschaften,
ISSN 0072-7830 ; 289)
Includes blbllographlcal referenees (p. ) and Index.
ISBN 978-3-642-08256-6 ISBN 978-3-662-07386-5 (eBook)
DOI 10.1007/978-3-662-07386-5
1. Geoletry, Dlfferentlal. 2. Gaolatrlc quantlzatlon. 3. Quantui
fleld theory. 1. Ser Ies.
QAS49.M3S13 1997
51S.3·S--dc20 96-35228
CTP

First Reprint 2002

Mathematics Subject Classification (1991): 8IEXX, 14-XX, 18-

XX, 53-XX,35-XX

ISSN 0072-7830

ISBN 978-3-642-08256-6

This work is subject to copyright. Ali rights are reserved, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations, recitation, broadcasting, reproduction on microfilm 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
Berlin Heidelberg GmbH. Violations are liable for prosecution under the German
CopyrightLaw.

© Springer-Verlag Berlin Heidelberg 1988, 1997

Originally published by Springer-Verlag Berlin Heidelberg New York in 1997
Softcover reprint ofthe bardcover 2nd edition 1997

SPIN 10868052 44/3111 - 5 4 3 2 1 - Printed on acid-free paper

PREFACE TO THE SECOND EDITION

In the seven years since this book was published in English, the exciting
interaction between complex geometry and theoretical physics continued to bring
new results and ideas to both domains. For this edition, Prof. S. Merkulov has
written an Addendum and compiled a list of additional references. Thanks to
him, several dozens of misprints in the first edition were corrected as weIl.
I would like to express my thanks to him for this help.

Bonn, January 1997 Yuri 1. Manin

 So here I am, in the middle way, having had twenty years -
Twenty years largely wasted, the years of l'entre deux guerres -
Trying to learn to use words, and every attempt
Is a wholly new start, and a different kind of failure,
Because one has only learnt to get the better of words
For the thing one no longer has to say, or the way in which
One is no longer disposed to say it.
- T. S. Eliot, Four Quartets

FOREWORD

The 19708 were a transitional decade in elementary particle physics. At the

1978 Tokyo conference the standard Weinberg-Salam model, which combined the
weak and the electromagnetic interactions in the framework of spontaneously vio-
lated gauge SU(2), x U(1)-symmetry, was finally acknowledged to have the support
of experimental evidence. The quantum chromodynamics of quarks and gluons,
based upon strict gauge SU(3)c-symmetry, was also able gradually to acquire the
status of an accepted theory of strong interactions, despite the lack of a theoreti-
cal explanation of confinement. The work of Pollitzer and Gross-Wilczek in 1974,
which showed that quarks are asymptotically free at small distances, contributed
to this acceptance of the theory.
At the same time, the success of these two theories was always regarded as
a provisional situation· until a unified theory could incorporate all interactions,
including gravity. A totally unexpected step in this direction was taken in the
last ten years with the discovery of supersymmetry, which intermingles b080ns and
fermions, and the observation that the localization of supersymrnetry inevitably
leads to curved space-time and gravity.
It is natural that cooperation between physicists and mathematicians, which
is as firmly rooted in tradition as is their difficulty understanding one another,
received a fresh impetus during these years.
What was probably of the greatest importance from a technical point of view
was the discovery of new methods of solving nonlinear partial differential equa-
tions. The famous inverse scattering problem method is effective in one- and
vm Foreword

two-dimensional models. But in realistie quantum field theory, whieh works with
more eomplicated Lagrangians (the Lagrangian of the unified SU(5}-model eon-
tains more than five hundred vertices), the role ofnonperturbational effects beeame
clearer. The inclusion of these effects in the quasiclassical approximation is eon-
nected with the existenee of loealized solutions to dynamieal equations of monopole
type, solitons, and instantons. The solutions are studied by means of topologie al
and algebra-geometrie deviees.
From a philosophical point of view, one ean speak of the geometrization of
physieal thought; more precisely, of a new wave of geometrization whieh for the
first time is sweeping far beyond the boundaries of general relativity. Tables of
the homotopy groups of spheres and Cech eoeycles have started to appear in
physies journals, and nilpotents in the structure sheaf of a seheme or analytie
spaee, whieh in the 1950s might have seemed little more than a caprice of Alexan-
der Grothendieck's genius, have acquired a physical interpretation as the supports
of the external degrees of freedom of the fundamental fields in supersymmetric
models: the statistics of Fermi induces the anti-commuting coordinates of super-
space.
This book is intended for mathematicians. It is a modest attempt to intro-
duce the reader to certain types of problems which are motivated by quantum
field theory. But we shall keep to the level of classical fields and dynamical equa-
tions, without going into secondary quantization. The reader can leam about
quantization from the classical monograph Introduction to Gauge Field Theory by
N. N. Bogolyubov and D. V. Shirkov and the excellent book Introduction to the
Quantum Theory oE Gauge Fields by A. A. Slavnov and L. D. Faddeev.
Following this foreword we shall devote a few pages to helping the mathemati-
cian reader translate the physicist's terminology into the geometrical language of
this book, which is the standard jargon of the theory of complex manifolds and
sheaf cohomology.
Part of the material presented here was taken from lectures given by the au-
thor at the Mechanico-Mathematics Faculty of Moscow State University and at
various mathematics and physics meetings. I am deeply grateful to many peo-
pIe whose intluence is retlected in one way or another in the pages of this book:
my teacher I. R. Shafarevich; I. M. Gel'fand; L. D. Faddeev; M. F. Atiyah; and
my friends, colleagues and coauthors A. A. Beilinson, A. A. Belavin, V. G. Drin-
fel'd, S. I. Gel'fand, S. G. Gindikin, G. M. Henkin, I. Yu. Kobzarev, D. A. Leites,
V. I. Ogievetskii, I. B. Penkov, A. M. Polyakov, M. V. Savel'ev, A. S. Svarc,
Ya. A. Smorodinskii, I. T. Todorov, V. E. Zakharov.

Yu. I. Manin
FOREWORD TO THE ENGLISH EDITION

Qnly three years have gone by since the publication of the Russian edition of
this book. But in this short time complex-analytic methods have taken center stage
in quantum field theory. This is connected with aseries of papers by E. Witten,
J. Schwarz, M. Green, A. Polyakov, A. Belavin and their collaborators, in which
one begins to see an amazing picture of the world at high (Planck) energies:
(a) Space-time is ten-dimensionalj six of these dimensions are compactified,
and perhaps form a complex Calabi-Yau manifold.
(b) The elementary constituents of matter are one-dimensional objects called
strings (or superstrings). The mathematical theory of these objects is based upon
the classical Riemann moduli spaces of algebraic curvesj the fundamental quantities
in the theory are complex-analytic.
(c) Supergeometry replaces ordinary geometry wherever fundamental interac-
tions are described in aGrand Unified fashion.
I would like to express my sincere thanks to Profs. N. Koblitz and J. King for
their long and difficult labor in translating this book.

Yu. I. Manin
TABLE OF CONTENTS

Introduction. Geometrical Structures in Field Theory 1

Chapter 1. Grassmannians, Connections, and Integrability ........... 7
§ 1. Grassmannians and Flag Spaces ........................... 7
§ 2. Cohomology of Flag Spaces ............................... 17
§ 3. The Klein Quadric and Minkowski Space ................... 23
§ 4. Distributions and Connections ............................. 35
§ 5. Integrability and Curvature ............................... 41
§ 6. Conic Structures and Conic Connections .................... 45
§ 7. Grassmannian Spinors and Generalized Self-Duality Equations 51
References for Chapter 1 .................................. 59
Chapter 2. The Radon-Penrose Transform ......................... 61
§ 1. Complex Space-Time ..................................... 61
§ 2. The Self-Duality Diagram and the Radon-Penrose Transform .. 72
§ 3. The Theory of Instantons ................................. 81
§ 4. Instantons and Modules over a Grassmannian Algebra ........ 99
§ 5. The Diagram of Null-Geodesics ............................ 106
§ 6. Extensions and Obstructions .............................. 113
§ 7. Curvature on the Space of Null-Geodesics ................... 124
§ 8. Cohomological Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
§ 9. The Flow of a Yang-Mills Field on the Space of Null-Geodesics 133
§ 10. Extension Problems and Dynamical Equations .............. 141
§ 11. The Green's Function of the Laplace Operator ............... 146
References for Chapter 2 .................................. 150
Chapter 3. Introduction to Superalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 153
§ 1. The Rule of Signs ........................................ 153
§ 2. The Tensor Algebra over a Supercommutative Ring .......... 159
§ 3. The Supertrace and Superdeterminant ...................... 164
§ 4. Some Complexes in Superalgebra .......................... 168
§ 5. Scalar Products ......................................... 173
§ 6. Real Structures .......................................... 175
References for Chapter 3 .................................. 180
Chapter 4. Introduction to Supergeometry ......................... 181
§ 1. Superspaces and Supermanifolds ........................... 181
§ 2. The Elementary Structure Theory of Supermanifolds ......... 188
XII Table of Contents

§ 3. Supergrassmannians and Flag Superspaces .................. 192

§ 4. The Frobenius Theorem and Connections ................... 204
§ 5. Right Connections and Integral Forms ...................... 206
§ 6. The Berezin Integral ..................................... 213
§ 7. Densities ............................................... 216
§ 8. The Stokes Formula and the Cohomology of Integral Forms ... 220
§ 9. Supermanifolds with Distinguished Volume Form.
Pseudodifferential and Pseudointegral Forms ................ 222
§ 10. Lie Superalgebras of Vector Fields and Finite-dimensional
Simple Lie Superalgebras ................................. 225
References for Chapter 4 .................................. 232
Chapter 5. Geometrie Structures of Supersymmetry and Gravitation .. 233
§ 1. Supertwistors and Minkowski Superspace ................... 233
§ 2. Scalar Superfields and Component Analysis ................. 241
§ 3. Yang-Mills Fields and Integrability Equations
along Light Supergeodesies ................................ 243
§ 4. Monads on Superspaces and Y M-sheaves ................... 253
§ 5. Some Coordinate Computations ........................... 263
§ 6. Flag Superspaces of Classical Type and
Exotie Minkowski Superspaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . 267
§ 7. The Geometry of Simple Supergravity ...................... 277
References for Chapter 5 .................................. 286
Bibliography ................................................... 287

Recent Developments (by Sergei A. Merkulov) ...................... 297

Chapter A. New Developments in Twistor Theory ................. 299
Chapter B. Geometry on Supermanifolds ......................... 329
Notes........................................................ 338
Bibliography ................................................. 339

Index 345