An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces Second Edition (Theoretical and Mathematical Physics)

228 Pages·2007·1.99 MB·English

by  Martin Schlichenmaier

Theoretical and Mathematical Physics
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EditorialBoard
W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Germany
J.-P.Eckmann,DepartmentofTheoreticalPhysics,UniversityofGeneva,Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA,USA
S.Smirnov,MathematicsSection,UniversityofGeneva,Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,NY,USA
J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
Martin Schlichenmaier
An Introduction
to Riemann Surfaces,
Algebraic Curves
and Moduli Spaces
Second Edition
With 21 Figures
Prof.Dr.MartinSchlichenmaier
UniversityofLuxembourg
InstituteofMathematics
162A,avenuedelaFaiencerie
1511LuxembourgCity
Grand-DuchyofLuxembourg
Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and
Moduli Spaces, Theoretical and Mathematical Physics (Springer, Berlin Heidelberg
2007)DOI10.1007/b11501497
LibraryofCongressControlNumber:2007926181
ISSN0172-5998
ISBN978-3-540-71174-2 SpringerBerlinHeidelbergNewYork
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Preface to the Second Edition
During the second advent of string theory the first edition of this Springer
Lecture Notes volume appearedin 1990.There was anincreasing demand for
physiciststo learnmoreaboutthe modernaspectsofgeometry.Inparticular,
forafurther developmentofthe physicaltheorythe notionsmentionedinthe
title turned out to be of fundamental importance.
The first edition was based on lecture courses I gave at the Institute of
Theoretical Physics, University of Karlsruhe, Germany. During this time the
institute was headed by Prof. Julius Wess. Indeed, it was him who convinced
me to write them up. Instead of repeating everything let me refer to the
Introductiontothe1stEditionwhichcanbefoundessentiallywithoutchanges
in this volume.For the motivation coming fromthe physics ofthese yearssee
especially the introduction given by Prof. Ian McArthur.
Clearly,inthemeantimeTheoreticalPhysicsdevelopedfurther.Neverthe-
less, the geometric concepts introduced in the first edition (and even more)
remain to be the very basic knowledge expected to be known by everybody
workinginrelatedfields oftheoreticalphysics.Consequently,the book is still
in demand. But as it is out of print, the publisher suggested that I should
prepare a revised and enlarged version of it. This offer I gratefully accepted.
As far as the revised part is concerned the changes are mainly due to
removingtypographicalerrors,makingsomeunclearstatementsmoreprecise,
smoothening the presentation, etc. It was my clear aim to keep the structure
and the partly informal style of the first edition.
In the meantime even more advanced geometric techniques are needed by
physicists.FromthemanypossiblechoicesItookthefollowingtwoadditional
topics:(1)themodernlanguageandmodernviewofAlgebraicGeometryand
(2) Mirror Symmetry.
In Chaps.11 and 12 I will describe the modern point of view of Algebraic
Geometry. One of the basic ideas there is to replace the classical geometric
space by the algebra of functions on the space or even more generally by the
set of maps from this space to other spaces and vice versa, respectively. The
VI Preface to the Second Edition
geometry corresponds to some algebraic structure on the set of maps. These
concepts were established by Grothendieck in the 1960s.What does one gain
by this generalization? One of the advantages is that it is possible to extend
the techniques of geometry to more general objects which do not have an
obvious structure of a “classical space”. Moreover, from the problem of the
existenceofmodulispacesclassifyingclassicalgeometricobjectsoneisledim-
mediately outside of the category of “classicalmanifolds”. This point of view
was very fruitful in mathematics. See, e.g. the proof of the Weil conjectures
by Pierre Deligne, Faltings’ proof of Mordell’s conjecture, Faltings’ proof of
theVerlindeformula,Wiles’proofofFermat’sLastTheorem,....Butitisalso
ofimportanceinTheoreticalPhysics.Therenoncommutativespace,quantum
groups,and more involvedobjects show up. This is one of the reasonsfor the
increasinginterestinmodernalgebraicgeometryamongtheoreticalphysicists.
Furthermore, spaces with singularities are incorporated from the very begin-
ning ofthe theory.In Chap. 11 localaspects areconsidered.The spectrum of
aringisintroduced.Itisshownhowtheconceptofapointcanbereplacedby
the concept of a homomorphism.Also the noncommutative case,of relevance
forthequantumplaneandforquantumgroups,willbeconsidered.InSect.12
global aspects, i.e. affine schemes and general schemes, will be introduced.
Mirror symmetry is another example for the very fruitful ongoing inter-
action between Theoretical Physics, Mathematical Physics and Fundamental
Mathematics. This interaction works in both directions. On one hand, math-
ematics supplies the necessary tools and results for physicists. On the other
hand, based on the physical theory quite unexpected mathematical relations
areconjecturedwhichturnouttoberealchallengesformathematicianstobe
understood mathematically.
In very rough terms, the origin of mirror symmetry in physics might be
described as follows. There are five models for superstring theories. Their
common feature is that the dimension of spacetime equals ten. To obtain the
four-dimensional spacetime, which we experience, the remaining six dimen-
sions are thought to be very small and compact. This splitting is also called
string compactification. The most appealing candidates for the compactified
factors are some special compact complex three-dimensional manifold, the
so-called Calabi-Yau threefolds. The physical models obtained by different
compactifications show certain dualities. One of these (conjectured) dualities
(relatingtypeIIAstringtheoryononeCalabi-YauthreefoldtotypeIIBstring
theory on its conjectured mirror) is the mirror symmetry.
InChaps.13and14themathematicalbackgroundofmirrorsymmetryand
the mirror conjectures will be explained. To formulate mirror symmetry the
theory of Ka¨hler manifolds, Hodge decomposition and Calabi-Yau manifolds
will be developed. Some general remarks on the physical background can be
found in Sect. 13.1.
Indeed from the mathematical point of view there are different levels of
mirrorsymmetry,respectivelydifferentmirrorconjectures.Thefirstversionis
the Geometric Mirror Symmetry. It is the conjecture that for certain classes
Preface to theSecond Edition VII
of Calabi-Yau manifolds X there exist mirrors Y such that their Hodge dia-
monds, constituted by their individual Hodge numbers hp,q, are mirror sym-
metric. More precisely, we get hp,q(X) = hn−p,q(Y). The Numerical Mirror
Symmetry gives a relation between the generating function for the number of
rationalcurvesofdegreedonX anda functiondefined interms ofperiodin-
tegralsonthemirrorY.ThethirdvariantisKontsevich’s Homological Mirror
Conjecture. It says that for mirror pairs (X,Y) of Calabi-Yau manifolds the
derived Fukaya A∞-category of Lagrangian submanifolds of X is isomorphic
to the derived category of coherent sheaves of Y.
In Chap. 14 the geometric mirror conjecture is explained in detail. The
numerical mirror conjecture is sketched for the quintic hypersurface in four-
dimensional projective space. This is the classical example discovered by the
physicists.
As, comparedto the first edition, the total number of pages increasedthe
following Leitfaden might be useful for the reader.
The first part of the book (Chaps. 1–7) should be considered as the very
basicknowledge.ItgivesaconciseintroductiontothetheoryofRiemannsur-
faces,curvesandtheirmodulispaces.Itwasmygoaltokeepthepresentation
as elementary as possible. After some fundamental definitions, for instance,
the definitions of manifolds, differentials and other basic objects, we restrict
ourselvesto Riemannsurfaces.First,we study their topology,beginning with
theirfundamentalgroups.Asafundamentalexampleofhomologytheorieswe
discuss simplicial homology. A Riemann surface is a manifold which locally
looks like the complex numbers C. Hence we have available the theory of an-
alytic functions, differentials, and so on. If we restrict ourselves to compact
Riemann surfaces, we get some very useful results such as the theorem of
Riemann–Roch. Roughly speaking, this gives us informations concerning the
existence of functions with prescribed behaviour at certain points. The next
goal is to study the set of different analytic isomorphism classes of Riemann
surfaces.Equippedwith ageometricstructure representinghow the Riemann
surfaces “appear in nature” this set is called a moduli space. One important
step in doing this is to assign to every Riemann surface its Jacobian. This is
a higher dimensional torus with the crucial property that it is embeddable
into some projective space. Further, to study these moduli spaces we use the
language of algebraic geometry, which we must first develop. In the first part
of the book we use only the more classical concept of varieties.
ThesecondpartconsistsofChaps.8and9.Itcontainsthemoreadvanced
basics and introduces more advanced tools like line bundles, the relations
between line bundles and divisors, vector bundles, cohomology, the theorem
of Riemann–Roch for line bundles, etc.
Thefollowingpartsaredividedinblockswhichareessentiallyindependent
of each others. The reader can decide which parts he wants to study.
In Chap.10 the Grothendieck–Riemann–Roch–Hirzebruchtheoremis for-
mulated. It is shown how to deduce the Riemann–Roch theorem from it.
Furthermore, the Mumford isomorphism, of fundamental relevance for the
VIII Preface to the Second Edition
geometry of the moduli space of curves and for the string partition function,
is deduced. As prerequisites clearly the above two basic parts are needed.
The next block, consisting of Chaps. 11 and 12, deals with modern alge-
braicgeometry.Itscontentwasalreadyroughlydescribedabove.Toallowthat
these two chapters can be studied completely independent of the other chap-
ters I repeated in these chapters all definitions needed. Nevertheless, as the
objects there are rather abstract, an knowledge about the geometric objects
presented in Part 1 will be of advantage.
Chapters 13 and 14 on mirror symmetry depend again on the two basic
parts. A knowledge of Chap. 10 might be helpful.
The finalappendix onp-adicnumbers iscompletely independentfromthe
rest of the book.
This book addresses not only theoretical physicists, mathematical physi-
cists but also mathematicians. It should help the reader to enter the field of
modern geometry. It was my goal to keep the style of the first edition. In
severalrespects there will be mixture between different techniques and view-
points. In this way beautiful relations between different approaches can be
detected.
The book should be self-contained in the sense of the developed mathe-
maticalnotionsandunderstanding.Fortheproofofquiteanumberofcentral
theorems (like Riemann–Roch, etc.) I mainly refer to the mathematical lit-
erature. For other mathematical facts I supply proofs. The reason for these
proofs is either the fact that in these cases without the proof the mathemati-
calstatementcannotbefully understoodorthattheyshowhowtoworkwith
the developed mathematical theory and to draw consequences from it.
The first edition grew out from several lecture courses. This influenced
clearlythe styleofthepresentation.Severaltimes Ifirstmadeadefinitionfor
the particular important special case under consideration. The intention was
not to overload the reader with only definitions in the beginning. The most
general definition was given later on in those parts of the book where it is
really needed. At least to a certain extent it was my goal to keep the lecture
style also for the added chapters of the second edition. In fact the part on
modern algebraic geometry developed out from a write-up of lecture series
which I gave at an autumn school of the Graduiertenkolleg Mathematik im
Bereich Ihrer Wechselwirkungen mit der Physik of the Ludwig-Maximilians-
Universita¨t,Mu¨nchen.ItisapleasureformetothankProf.MartinSchotten-
loher and Prof. Julius Wess for the invitation to present the lectures there.
Luxembourg, Martin Schlichenmaier
September 2006
Preface to the First Edition
This is a write-up of a lecture course I gave in the fall term 1987/1988 at
theInstituteofTheoreticalPhysicsoftheUniversityofKarlsruhe.Ideasfrom
algebraicandanalyticgeometryarebecomingmoreandmoreusefulinTheo-
reticalPhysics,particularlyinrelationtostringtheoriesandrelatedsubjects,
andsoIwasaskedbyProf.JuliusWesstogiveanintroductorylecturecourse
for physicists on these exciting mathematical ideas.
Ofcourse,therearemanydifferentstrategiesavailableforputtingtogether
such a course. And there are as many arguments for as there are against the
choice of subjects made here and the way to proceed. Certainly my personal
preference playeda part in this choice,but, nevertheless,I hope the audience
of the lectures and now the readers of these notes, if interested, will be able
to study the appropriate mathematics books after the lecture in a particular
subject.
Let me give a brief review of the lectures. After some fundamental def-
initions, for instance, of manifolds, differentials and other basic objects, we
restrict ourselves to Riemann surfaces. First, we study their topology, begin-
ning with their fundamental groups. As a nice example of homology theories
we discuss simplicial homology. A Riemann surface is a manifold which lo-
cally looks like the complex numbers C. Hence we have available the theory
of analytic functions, differentials and so on. If we restrict ourselves to com-
pact Riemann surfaces, we get some very useful results such as the theorem
ofRiemann–Roch.Roughlyspeakingthisgivesusinformationconcerningthe
existence of functions with prescribed behaviour at certain points.
In Polyakovbosonic string theorythe partitionfunction is givenby “inte-
grating”overalltopologiesandmetricsonanorientablecompactsurface.By
gauge invariance the integration is eventually reduced to the set of different
analytic isomorphism classes of Riemann surfaces. Equipped with a certain
geometric structure this set is called a moduli space. As you see, it is not
enough to study one fixed (although arbitrary) Riemann surface, we have to
study all of them together. One important step in doing this is to assign to
every Riemann surface its Jacobian. This is a higher dimensional torus with
the crucial property that it is embeddable into some projective space. Fur-
ther,to studythese modulispacesweusethe languageofalgebraicgeometry,
X Preface to the First Edition
which we must first develop. Here we use only the more classical concept of
varieties. With algebraic geometry we have a very powerfulmachinery at our
disposal for studying the geometry of the moduli space.
We develop the theory of vector bundles, sheaves and sheaf cohomology
anddeduce the Mumfordisomorphismonthe moduli space starting fromthe
Grothendieck–Riemann–RochTheorem.TheMumfordIsomorphismallowsus
to formulate the integrand of the partition function of the Polyakov bosonic
string in holomorphic terms. We use the Krichever–Novikov construction of
“AlgebrasofVirasoroTypeonHigherGenusCurves”asanexampleofhowto
work with the Riemann–Roch theorem. Of course, there are other aspects in
the theory ofmoduli ofRiemann surfaces,suchas Teichmu¨ller theory,which,
because of limited time, could not be covered in the lecture course.
Letme addsomeremarksonthe style.Thelecture courseandthese notes
were aimed mainly at physicists, and not so much at mathematicians. But
this is nottosaythatthe resultsareimprecise,ratherthatthere is amixture
between different techniques and different viewpoints. In contrast, a classical
mathematical text would stick to one particular aspect, for example, to the
analytictheory.Therearegoodreasonsforthis,butIthinkthatinthecontext
oftheselectureswewouldlosealotofbeautifulrelationsbetweenthedifferent
approaches. Hence these lecture notes might also be useful for students of
mathematics who are looking for a short introduction to the various aspects
andtheirinterplay.Infact,therewerequiteanumberofmathematicsstudents
attending the lectures.
Inthelectures,youmightfinditparadoxicalthatsomecomparablysimple
propositionsareprovenwhilethereisnothingontheproofofsuchfundamen-
tal theorems as that of Riemann–Roch. But the reason for presenting proofs
was not to make a self-contained lecture, but rather to use them as examples
of how to work with the defined objects.
Inview ofp-adicorevenadelic stringtheory,I haveincluded anappendix
whereI giveashortintroductiontop-adicnumbers.Itshouldbe understand-
able even to someone without prior knowledge of advanced algebra.
Finally, let me thank Prof. Julius Wess for encouraging me to give these
lectures and for convincing me that there is a need for a write-up such as
this. I would also like to thank the audience for their active participation.
Theirquestions helpedme greatlyto makethe exposition(hopefully) clearer.
Let me thank Prof. Rainer Weissauer and Prof. Herbert Popp for valuable
suggestions and fruitful discussions concerning the content of Chaps. 7 and
10, Michael Schro¨der for doing some proof-reading and Gu¨nther Schwarz for
supportinpreparingthe finalprintoutofthe manuscriptonthe laserprinter.
I would especially like to thank Dr Ian McArthur for writing an introduction
ontheapplicationinphysicsofthemathematicalsubjectspresentedand,not
least, for correcting my English. However, I accept full responsibility for any
remaining errors.
Martin Schlichenmaier (1990)