Seiberg-Witten Gauge Theory

205 Pages·1999·0.815 MB·English

by  Matilde Marcolli

Seiberg-Witten Gauge Theory
Matilde Marcolli
1
Contents
1 Introduction 5
I Seiberg{Witten on four-manifolds 12
2 Preliminary Notions 13
2.1 Cli(cid:11)ord Algebras and Dirac Operators . . . . . . . . . . . . . . . 13
2.2 Spin and Spinc Structures . . . . . . . . . . . . . . . . . . . . . 15
2.3 Spinor Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Topology of the gauge group . . . . . . . . . . . . . . . . . . . . 17
2.5 Symplectic and K(cid:127)ahler Manifolds . . . . . . . . . . . . . . . . . . 18
2.6 The index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 Sobolev norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.9 Fredholm properties . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 The Functional and the Equations 30
3.1 The Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 The Gauge Group . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 The Seiberg{Witten Functional and the Variational Problem . . 31
3.4 Analytic properties of the Seiberg{Witten functional . . . . . . . 34
3.4.1 The Palais{Smalecondition: sequential compactness . . . 35
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Seiberg{Witten invariants of 4-manifolds 40
4.1 The Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.1 Computation of the Dimension . . . . . . . . . . . . . . . 40
4.1.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.3 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 The Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 A Cobordism Argument . . . . . . . . . . . . . . . . . . . . . . . 50
II Seiberg{Witten on three-manifolds 55
5 Three-manifolds 56
6 A three-manifold invariant 57
6.1 Dimensional reduction . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 The moduli space and the invariant. . . . . . . . . . . . . . . . . 60
6.3 Cobordism and wall crossingformulae . . . . . . . . . . . . . . . 63
2
6.4 Casson invariant and Alexander polynomial . . . . . . . . . . . . 65
7 Seiberg{Witten Floer homology 72
7.1 The Chern-Simons-Dirac functional . . . . . . . . . . . . . . . . . 73
7.2 Hessian and relative index . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Flow lines: asymptotics . . . . . . . . . . . . . . . . . . . . . . . 78
7.4 Flow lines: moduli spaces . . . . . . . . . . . . . . . . . . . . . . 80
7.5 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.6 The cobordism argument . . . . . . . . . . . . . . . . . . . . . . 88
7.7 Equivariant Floer homology and wall crossing . . . . . . . . . . . 91
7.8 Exact triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.9 The relation to instanton Floer homology . . . . . . . . . . . . . 99
7.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
III Topology and Geometry 105
8 Computing Seiberg{Witten invariants 106
8.1 Connected Sum theorem . . . . . . . . . . . . . . . . . . . . . . . 106
8.2 The blowup formula . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.3 K(cid:127)ahler Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.4 Symplectic Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . 112
8.5 Pseudo{holomorphiccurves . . . . . . . . . . . . . . . . . . . . . 116
8.6 Beyond the symplectic world . . . . . . . . . . . . . . . . . . . . 118
8.7 Algebraic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9 Topology of embedded surfaces 123
9.1 The Thom Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 124
9.2 Contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.3 Three-manifolds: Thurston norm . . . . . . . . . . . . . . . . . . 130
10 Further applications 131
10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
IV Seiberg{Witten and Physics 137
11 Mathai-Quillen formalism and Euler numbers 138
11.1 The (cid:12)nite dimensional case . . . . . . . . . . . . . . . . . . . . . 138
11.1.1 The Mathai-Quillen form . . . . . . . . . . . . . . . . . . 138
11.1.2 Intersection theoretic approach . . . . . . . . . . . . . . . 141
11.2 The in(cid:12)nite dimensional case . . . . . . . . . . . . . . . . . . . . 142
11.2.1 The localised homologicalEuler class. . . . . . . . . . . . 143
11.2.2 Equivariant homology and the Atiyah-Je(cid:11)rey formalism . 145
3
11.3 Euler numbers in Seiberg{Witten theory . . . . . . . . . . . . . . 146
11.3.1 Atiyah-Je(cid:11)rey description . . . . . . . . . . . . . . . . . . 148
11.3.2 Seiberg{Witten Invariants Revisited . . . . . . . . . . . . 150
11.3.3 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.4 N =2 symmetry and the Euler characteristic . . . . . . . . . . . 153
11.4.1 Three dimensional Atiyah-Je(cid:11)rey formalism . . . . . . . . 155
11.5 Quantum Field Theory and Floer homology . . . . . . . . . . . . 156
11.5.1 Relative Seiberg{Witten invariants . . . . . . . . . . . . . 158
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
12 Seiberg{Witten and Donaldson theory 163
12.1 The Physics way: S-duality . . . . . . . . . . . . . . . . . . . . . 165
12.1.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . 165
12.1.2 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . 166
12.1.3 Weak and strong coupling . . . . . . . . . . . . . . . . . . 168
12.1.4 The u-plane . . . . . . . . . . . . . . . . . . . . . . . . . . 169
12.1.5 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2 The Mathematics way . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2.1 Non{abelian Monopoles . . . . . . . . . . . . . . . . . . . 173
V Appendix: a bibliographical guide 180
4
1 Introduction
Where the (cid:12)re is churned, where the wind wafts, where the Soma juice (cid:13)ows
over |there the mind is born.
S(cid:19)vet(cid:22)a(cid:19)svataraUpanis.ad 2.61
Inthefallof1994E.Wittenannounceda\newgaugetheoryof4-manifolds",
capableofgivingresultsanalogoustotheearliertheoryofDonaldson,butwhere
thecomputationsinvolvedare\atleastathousandtimeseasier"(Taubes). The
dawnofthenewtheorybeganwiththeintroductionofthemonopoleequations,
whose roots lie in the depths of the still rather mysterious notion of S-duality
2
in N =2 supersymmetric Yang-Mills theory (IX: [59], [60]) .
The equations are in terms of a section of a spinor bundle and a U(1) con-
nectiononalinebundleL. The(cid:12)rstequationjustsaysthatthespinorsection
has to be in the kernel of the Dirac operator twisted with a connection A. The
secondequationdescribesarelationbetweenthe self-dualpartof thecurvature
of the connection A and the section in terms of the Cli(cid:11)ord action.
The mathematical setting for Witten’s gauge theory is considerably sim-
plerthan Donaldson’sanalogue: (cid:12)rstofallit dealswith U(1)-principalbundles
(hermitianlinebundles)ratherthanwithSU(2)-bundles,andtheabelianstruc-
turegroupallowssimplercalculations;moreovertheequation,whichplaysarole
somehowanalogoustothepreviousanti-self-dualequationforSU(2)-instantons,
involves Dirac operators and Spinc-structures, which are well known and long
developed mathematical tools.
The main di(cid:11)erences between the two theories arise when it comes to the
properties of the moduli space of solutions of the monopole equation up to
gauge transformations. As was immediately observed, the moduli space of
Seiberg{Witten equations turns out to be always compact, essentially due to
the Weitzenb(cid:127)ock formula for the Dirac operator: a fact that avoids the com-
plicated analytic techniques that were needed for the compacti(cid:12)cation of the
moduli space of SU(2)-instantons.
InWitten’s seminalpaper (I:[19]) themonopoleequationisintroduced, and
the main properties of the moduli space of solutions are deduced. The dimen-
sion of the moduli space is computed by an index theory technique, following
an analogous proof for Donaldson’s theory; and the Seiberg{Witten invariant
(whichdependsontheChernclassofthelinebundleL)isde(cid:12)nedasthenumber
of points, counted with orientation, in a zero-dimensional moduli space. The
circumstancesunder whichthe Seiberg{Witten invariants provideatopological
invariant of the four{manifold are illustrated in a similar way to the analo-
1
The quotes from the Upanis.ads are from the Oxford University Press edition, in the
EnglishtranslationofP.Olivelle.
2
ReferenceslistedintheIntroduction refertothebibliographicalappendix.
5
gous result regarding the Donaldson polynomials. Again as a consequence of
the Weitzenb(cid:127)ock formula, the moduli space turns out to be empty for all but
(cid:12)nitely many choices of the line bundle L. Another advantage of this theory
is that the singularities of the moduli space only appear at the trivial section
0,sinceelsewheretheactionofthegaugegroupisfree. Thus,byperturbing
(cid:17)
the equation, it is possible to get a smooth moduli space.
Shortly after the new equations were introduced and their properties anal-
ysed,animpressiveseriesofnewresultswereproveninthespanofafewmonths.
As pointed out many times, the experts in the (cid:12)eld had all the right questions
to ask, questions that were too di(cid:14)cult to attack with the tools provided by
Donaldsontheory,andsuddenlythenewtoolofSeiberg{Wittentheorybecame
available. A very rapid advancement in the (cid:12)eld was thus possible.
The invariants were (cid:12)rst computed for K(cid:127)ahler manifolds, and then, in ex-
tendingthetechniquetotheworldofsymplecticmanifolds,Taubesuncovereda
deeprelationbetweenSeiberg{Witteninvariantandpseudo-holomorphiccurves.
The non-vanishing results for Seiberg{Witten invariants of symplectic mani-
folds ledto conjecturethat these constitute the most \basic"(indecomposable)
kind of four-manifolds. The conjecture was later disproved, again by means of
Seiberg{Witten theory, and there is at present no reasonable conjecture that
would identify the \building blocks" of smooth four-manifolds. Thus, although
in a way Seiberg{Witten theory helped proving several new results in four-
manifold topology, it also showed that our understanding of the structure of
four-manifolds is still very limited.
Among the very (cid:12)rst applications of the new gauge theory was the proof of
theThomconjectureforembeddedsurfacesinCIP2,in(II:[21]),latergeneralised
toobtainvariousestimatesontheminimalgenusofembeddedsurfacesrealising
agivenhomologyclassinafourorthree-dimensionalmanifold. Thisapplication
leddirectlytothedevelopmentofthethreedimensionalSeiberg{Witten theory
and of an associated Floer homology which is in many ways analogous to the
instanton homology constructed by Floer within the context of Donaldson the-
ory. The Seiberg{Witten Floer homology, however, presents some interesting
phenomena of metric dependence, due to non-trivialspectral (cid:13)ows of the Dirac
operator.
In the original string-theoretic context, in which Seiberg{Witten theory
arose, this can be seen as equivalent to Donaldson theory, the two being two
di(cid:11)erentasymptoticlimitsofonecommontheory,whichgetinterchangedunder
the symmetry of S-duality. Unfortunately, at present there is no clear math-
ematical understanding of this picture, but the equivalence of Donaldson and
Seiberg{Witten theoryis the object of current studies that areleading towards
a proof, based on a slightly di(cid:11)erent perspective. The main idea is to realise
both the Donaldson and the Seiberg{Witten moduli spaces as singular strata
withinalargermodulispaceofsolutionsoftwistedPU(2)-Seiberg{Wittenequa-
tions. The corresponding relation between the Donaldson polynomial and the
Seiberg{Witten invariants explained and justi(cid:12)ed (I: [19]) only at the level of
6
physical intuition, is that the Seiberg{Witten invariants should coincide with
the invariantsderivedbyKronheimerandMrowkain the structure theorem for
Donaldson polynomials.
After this brief recollection of the exciting days that saw the \change of
paradigm"ingaugetheory,wecandiscussbrie(cid:13)yhowthematerialinthisbook
is organised.
The (cid:12)rst part of these notes, Seiberg{Witten on four-manifolds, collects
some preliminary notions that are needed through the rest of the book, and
then proceeds to give an introduction to Seiberg{Witten gauge theory on four-
manifolds. ItfollowscloselytheoriginalpaperbyWitten(I:[19]). Weintroduce
the Seiberg{Witten equations as a variational principle, and discuss the ana-
lytic propertiesof the correspondingfunctional. Then, weproceed to introduce
the moduli space of solutions. We describe reducibles and how to avoid them
with a suitable perturbation. We discuss the compactness of the moduli space,
the transversality result, and the orientation. Then we introduce the Seiberg{
WitteninvariantsofasmoothfourmanifoldX. Wegivea(cid:12)nitenessresult,and
+
discusstheindependenceofthemetricandperturbationforthecaseb2(X)>1,
+
and the wall crossing formula for b2(X)=1.
The second part, Seiberg{Witten on three-manifolds, deals with the dimen-
sional reduction of the gauge theory. We (cid:12)rst describe how to derive Seiberg{
Witten equations on a three-manifold Y and construct a moduli space and a
correspondinginvariant,asin the four-dimensionalcase. We discussthe metric
dependence of the invariant in the case of rational homology spheres, and the
dependence on the cohomology class of the perturbation in the case of man-
ifolds with b1(Y) = 1. We then illustrate the relation between the invariant
and classical invariants such as the Casson invariant, Milnor torsion, and the
Alexanderpolynomial. Weproceedtoreinterpretthemodulispaceasthesetof
critical points of the Chern-Simons-Dirac functional, and the four-dimensional
equations as its downward gradient (cid:13)ow. We describe the Hessian and the rel-
ative index of critical points, and give a gluing theorem for the moduli spaces
of (cid:13)ow lines. This allows us to de(cid:12)ne compacti(cid:12)cation by lower dimensional
strata of the spaces of (cid:13)ow lines. This is a crucial technical step in the con-
struction of a boundary operator that connects critical points of relative index
one. The corresponding homology is the Seiberg{Witten Floer homology, and
the numerical invariant previously introduced can be reinterpreted as its Euler
characteristic. WeillustratetheconstructionoftheequivariantFloerhomology
that bypassesthe problem of metric dependence and reprovesthe wall crossing
formula for the invariant. We (cid:12)nally discuss the problem of the exact triangles
and surgery formulae.
The third part, Topology and Geometry, is an attempt to summarise in a
somewhat coherent picture the current literature on the subject of topological
and geometric results obtained via Seiberg{Witten theory. The results covered
in thesechaptersspana wide rangestaringfromK(cid:127)ahlermanifolds and the Van
de Ven conjectures foralgebraicsurfaces,to Taubes resultson symplectic man-
7
ifolds and the relation to pseudo-holomorphic curves and Gromov invariants,
continuing with the estimates on the genus of embedded surfaces in four and
three-manifolds,withrelatedresultsonthree-manifoldswithcontactstructures.
With the purpose of presenting the largest possible spectrum of results, in this
part we often give only a brief sketch of the proofs, and we refer the reader
to the appropriate sources in the literature, where she/he can (cid:12)nd the detailed
arguments. Giventhatthisisstilla(cid:12)eldineverincreasingandrapidexpansion,
itisimpossibletopresentallthecontributionswith auniformlevelofattention
and detail. Therefore a choice has to be made. This is forced upon by the
speci(cid:12)c researchinterests and taste of the author.
Through all these parts of the book, Seiberg{Witten gauge theory is con-
sidered as a completely self{contained subject and no a priori knowledge of
Donaldson theory is assumed. In fact, all the sections that refer to Donaldson
theory can be skipped by the reader who is not familiar with the non{abelian
gauge theory, and this will not a(cid:11)ect the comprehension of the remaining sec-
tions.
Itisthe author’sbelief, supported in thisbymoreauthoritativesourceslike
(I: [19]) or (VIII: [2]), that it is inappropriate and in some sense misleading to
presentSeiberg{Wittentheorywithoutmentioningthecontextofphysicaltheo-
riesthatareresponsiblefortheveryexistenceofthisnewpieceofmathematics.
It is a hard and ambitious task to present quantum (cid:12)eld theoretic results in a
form that may be comprehensible and useful to a mathematician.
Thus, we have decided to enclose a fourth and last part in the book, ded-
icated to Seiberg{Witten and Physics. In this a fairly detailed description of
the Mathai-Quillen formalism and the mathematically rigorous formulation of
the regularised Euler class of Fredholm bundles have been included precisely
to the purpose of making the reader familiar with the mathematical meaning
of certain quantum (cid:12)eld theoretic statements. This construction behaves par-
ticularly nicely when applied to Seiberg{Witten theory, since the compactness
of the moduli space reduces some of the technicalities involved in the rigorous
de(cid:12)nition of the regularisedEuler class.
This (cid:12)rst step does not lead us close to the realheart of the matter. There-
fore, we patiently proceed to attempt a brief exposition of some of the ideas
of the theory of S-duality. This is especially di(cid:14)cult, since at the moment no
clear mathematical understanding of the concept of S-duality is available and
similarly for many of the physical concepts involved. However, this is by itself
a good reason to include a brief discussion of what seems to be a promising
direction for future studies and research.
Wehavealsoincludedachapterwherethemathematicalapproachtoproving
the equivalence of Seiberg{Witten and Donaldson theory is outlined. Some
knowledgeofSU(2)-gaugetheoryis needed in orderto approachthe contentof
this section.
The reader that approaches this fourth part of the book must keep in mind
that the author is not a physicist, therefore the content of these chapters will
8
inevitably appear to the eyes of a Physics-oriented reader as overly simpli(cid:12)ed.
In particular, due to the fact that the (cid:12)eld is expanding so fast, the au-
thor should apologise to all the colleagues whose work might not have received
the necessary attention in these notes. This especially applies to the Physics
literature, where it is well known that the expansion rate of the set of new
contributions to the subject is some orders of magnitude faster.
Because of all these and other limitations, these notes do not intend to be
a comprehensive introduction to the Seiberg{Witten gauge theory. An initial
versionofthesenoteswaspreparedforaseriesofseminarsthattheauthorgave
at the University of Milano in the summer 1995, intended mainly for an audi-
ence of (cid:12)rst or second yeardoctorate students, and of advanced undergraduate
studentsinMathematicsorPhysicswhowerepreparingtheirlaureadissertation
in topology, di(cid:11)erential geometry, or quantum (cid:12)eld theory. The purpose of the
set of notes was to help the students in reading the references available on the
subject: strictly speaking, the purpose of the notes was to provide a guide on
\howtoapproachthestudyofSeiberg{Wittengaugetheory". Asprerequisites,
in fact, we assumed only some knowledge of the topology of (cid:12)bre bundles, and
notionslike that of connectionand curvature, that areusuallycoveredin a Eu-
ropean undergraduate course of algebraic topology or di(cid:11)erential geometry; as
well as of theoretical or mathematical physics. Some other concepts are brie(cid:13)y
introduced, and the reader who is not familiar with the subject is addressed to
references, where she/he can (cid:12)nd more detailed information.
This initial version of the notes, roughly covering part one and part three,
with a draft of part four, is still available on the electronic preprints as dg-
ga/9509005, and it had the short-lived privilege of being the (cid:12)rst available
expository work that collected a good part of the existent literature on the
subject. Many valuable references had since then appeared, which ful(cid:12)ll the
same purpose, with particular attention to di(cid:11)erent mathematical orientations
of the reader. Among these we recommend (I: [5]), but the readershould check
the appropriate section of the bibliographical guide at the end of this volume
formorereferences. Thus, itbecameintheleastadvisabletoreviseandupdate
these notes. A (cid:12)rst revision was made in the spring 1996, when the Interna-
tional Press Lecture Series took place at Irvine and many experts presented
new results (among these Taubes’ results on holomorphic curves, a (cid:12)rst outline
of Kronheimer and Mrowka’s work on the interplay between three-dimensional
Seiberg{Witten theory and contact geometry, and Furuta’s progress in the di-
rectionofthe11=8conjecture: thelatteriscertainlyasin ofomissionfromthis
book). Shamelessly,Iletothertwoyearspassbeforeattemptinga(cid:12)nalrevision
of the book.
Acknowledgements are due to many people. First of all to T.R. Ramadas
forhiscarefuleditingofthesenotesandforo(cid:11)eringmetheopportunityofpub-
lishing them in the Texts and Readings in Mathematics series. Many thanks
are due to Paolode Bartolomeis, for carefully reading the manuscript and sug-
gesting several corrections and improvements, and to Paolo Alu(cid:14) and Arthur
9
Greenspoon for spotting innumerable typos and providing useful feedback.
I especially thank Tom Mrowka, for his patient help and guidance in my
research work on Seiberg{Witten Floer theory, and for the many stimulating
conversations during my staying at MIT that were of invaluable help to my
understanding of gauge theory. Many thanks are due to Erion J. Clark for col-
lecting an extensive and updated bibliography. The content of the section on
thequantum(cid:12)eldtheoryformalismandtheEulerclassistheproductofconver-
sationswithRogierBrusseeinthe courseoftwovisits ofminetotheUniversity
of Bielefeld. Thanks are due to him and to Stefan Bauer for the interesting
conversationsand for making those visits possible. I thank PaulFeehan for ex-
plaining to me the PU(2) monopoles and the relation between Seiberg{Witten
and Donaldson invariants, and for many other useful conversations. I am very
grateful to Bai-Ling Wang for the many discussions, the continuous exchange
of ideas, and the productive and pleasant research collaboration.
The(cid:12)rstversionandsubsequent(cid:12)rstrevisionofthisbookwerewritteninthe
courseofmyshortstayingattheUniversityofChicagoasagraduatestudent. I
bene(cid:12)ted in those days of conversations with Kevin Corlette, Victor Ginzburg,
Je(cid:11) Harvey, J.Peter May, Dan Pollack, and, of course, with my advisor Mel
Rothenberg, who helpfully and patiently supervised my study of gauge theory.
I worked on the (cid:12)nal revision of the notes during my staying at the Max
Planck Institut fu(cid:127)r Mathematik in Bonn, in the summer 1998, and during a
subsequent visit to the Tata Institute of Fundamental Research in Mumbai
in January 1999. I thank these institutions for the kind hospitality and for
support. I also acknowledge support from two research grants of the CNR
(Italian National Council of Research), during the academic years 1995-96 and
1996-97, as well as from the NSF (National Science Foundation, USA), grant
DMS-9802480.
I am grateful to a number of people whose lectures, in the course of confer-
ences,seminars,andsummerschools,greatlyimprovedmyunderstandingofthe
subject. Amongthese, SimonSalamonfromwhosesummercourseinCortonaI
(cid:12)rstlearntDonaldsontheory,andthenStefanBauer,DietmarSalamon,Cli(cid:11)ord
Taubes, for their lectures and talks on Seiberg{Witten theory. For comments,
remarks,andsuggestions,IshouldthankRalphCohen,KevinIga,EricKlassen,
Anatoly Libgober, Gordana Mati(cid:19)c, Rafe Mazzeo, and Ron Wang.
After I completed the (cid:12)rst revision of these notes, a draft of the manuscript
was submitted to the Springer Lecture Notes for consideration. After a (cid:12)rst
refereeing, I was expected to revise the draft and resubmit it for a second ref-
ereeing and a (cid:12)nal decision of the editor. Some textbooks on Seiberg{Witten
theoryhadalreadyappearedinthemeanwhile,soIfelt thisvolumewouldhave
a better use if published elsewhere, in such a way as to serve a di(cid:11)erent audi-
ence. Nonetheless, many thanks are due to Albrecht Dold for considering the
previous draft, for his encouraging comments, and for submitting the draft to
a refereeing process. Thanks are due to the referee as well. Her/his valuable
comments and suggestions have been taken into consideration in preparing the
10