Table Of ContentPietro Fre' received his PhD from the University of Torino in 1974. He is currently
Professor of Theoretical Physics at the International School for Advanced Studies
(SISSA/ISAS), Trieste. He was Associate Professor of Theoretical Physics at the
University of Torino until 1990, and has also worked as Research Associate at the
University of Bielefeld, the California Institute of Technology, Torino University,
and at CERN. His research activities have been focused on particle physics and
statistical mechanics, especially supergravity, superstrings and topological field
theories. Prof Fre' has, in collaboration with Leonardo Castellani and Riccardo
D'Auria, written for World Scientific the three-volume textbook Supergravity and
Superstrings: A Geometric Perspective.
Paolo Soriani received his "laurea" degree from the University of Milano in 1987
and his PhD in particle theory from SISSA in 1992. He is currently a post-doctoral
fellow at the University of Milano. Dr Soriani has started his scientific career under
the supervision of Prof Luciano Girardello and Prof Fre'. His main research
interests have been in string theory, conformal field theories, topological models,
and applications of complex geometry to those subjects.
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From Calabi-Yau manifolds to
topological field theories
Pietro Fre'
SISSA-Trieste
Paolo Soriani
Universita degli Studi di Milano
V ^* World Scientific
wl SSininggaappoorere* •N NeewwJ eJresresye y 'LL ondon • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
THE N=2 WONDERLAND:
FROM CALABI-YAU MANIFOLDS TO TOPOLOGICAL FIELD THEORIES
Copyright © 1995 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, Massachusetts 01923, USA.
ISBN 981-02-2027-8
Printed in Singapore.
To Paola and Tiziana
and
to the memory
of our fathers
We can forgive a man for making a useful thing
as long as he does not admire it. The only excuse
for making a useless thing is that one admires it
intensely.
Oscar Wilde
from the Preface to
The Picture of Dorian Gray, 1890
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PREFACE
Tin's book is based on a series of lectures given by Pietro Fre' at SISSA, at DESY, at the
University of Torino and also at UCLA in tJie academic years 1991-92 and 1992-93.
Lecture notes were taken by Paolo Soriani and later the two authors elaborated and
considerably extended to the present book form the material, which includes also results
from Soriani's Ph.D. Thesis.
The aim was to present in a unitary perspective the logical development that uni
fies into a single, fascinating subject the topics related with N=2 supersymmetry in two
and four space-time dimensions. Beginning with the Kahler structure of low energy su-
pergravity lagrangians, through the analysis of string compactifications on Calabi-Yau
manifolds, one reaches the heart of the matter by considering the chiral ring structure
of N=2 superconformal models and of their parent N=2 field theories in two dimen
sions. The concept of topological twist relates in a profound way these theories with
d=2 topological field theories, deepening the understanding of mirror symmetry, of the
special Kahler geometry of Calabi-Yau moduli spaces and the analysis of the Picard-
Fuchs equations associated with the Griffiths period mapping. The relation between
the Landau-Ginzburg picture of N=2 superconformal models and the a-model picture
is elucidated by showing, as Witten recently did, that the two kinds of field theories
are effective low energy theories of the same spontaneously broken gauge model in two
different phases. Our emphasis, which is pedagogical, is on a self consistent presenta
tion of this beautiful subject that blends algebraic geometry with quantum field theory,
providing also new techniques to current research in various areas.
Suggestions to the Reader
The first chapter, "An Introduction to the Subject", is written in a style substantially
different from that of the other chapters, since it is a descriptive essay that covers, in
a simplified way, all the main ideas and the main results contained in the rest of the
book. The second chapter, "A Bit of Geometry and Topology", provides a summary
of the mathematics the reader should be familiar with in order to study the subsequent
chapters. From the point of view of physics, the logical development of our subject begins
with Chapter 3 "Supergravity and Kahler Geometry". Our suggestion is to read Chapters
3-8 in the given order since they are organized according to a consistent line of thought.
vn
Vlll
Chapter 1, instead, can be read independently as an introductory primer and it is mainly
directed to newcomers to the present field. Chapter 2 is a sort of reference chapter
where the reader can refresh his memory about mathematical definitions and theorems
utilized elsewhere. However, it is also conceived as a self-contained presentation of the
mathematical environment where the physical ideas are rooted.
Comments on Bibliography
As the reader will realize, we have supplied a list of references that, to the best of our
knowledge, should be a sufficient basis for further reading, although it is far from being
exhaustive. We apologise to all the authors of papers relevant to the subject that escaped
our attention.
A ckno wledgement s
We thank, for the many useful comments and illuminating discussions, our friends D.
Anselmi, M. Bianchi, M. Billo', L. Bonora, V. Bonservizi, R. D'Auria, B. Dubrovin, S.
Ferrara, M. Francaviglia, F. Fucito, L. Girardello, F. Gliozzi, R. Iengo, M. Martellini, A.
Van Proeyen, C. Reina, G. Rossi and W. Troost.
CONTENTS
1 AN INTRODUCTION TO THE SUBJECT 1
1.1 The Remarkable Interplay 2
1.1.1 Supergravity and Kahler geometry 2
1.1.2 Special Kahler geometry 4
1.2 Moduli and Criticality 4
1.2.1 Landau—Ginzburg critical models and the moduli 4
1.2.2 N=2 superconformal field theories 11
1.3 Moduli and Algebraic Varieties 12
1.3.1 The chiral ring in N=2 superconformal theories 14
1.3.2 The vanishing locus of the superpotential as a Calabi-Yau
manifold 15
1.3.3 The Griffiths residue map and the Hodge ring 15
1.4 The Art of Quantizing Zero 17
1.5 Mirror Maps 22
1.6 Bibliographical Note 23
2 A BIT OF GEOMETRY AND TOPOLOGY 25
2.1 Introduction 25
2.2 Fibre Bundles 25
2.2.1 Definition of a fibre bundle 25
2.2.2 Sheaves and Cech cohomology 27
2.2.3 Sections of a fibre bundle 30
2.2.4 Bundle maps 30
2.2.5 Equivalent bundles 31
2.2.6 Pull-back bundles 31
2.3 Vector Bundles, Connections and Curvatures 33
2.3.1 Fibre metrics 33
2.3.2 Product bundle 34
2.3.3 Whitney sum 34
2.3.4 Tensor product bundle 35
2.3.5 Principal fibre bundles 35
2.3.6 Connections on a vector bundle 36
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