Basic Concepts of String Theory

786 Pages·2013·6.017 MB·English

by  Ralph Blumenhagen, Dieter Lüst, Stefan Theisen (auth.)

Basic Concepts of String Theory
Theoretical and Mathematical Physics
The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs
in Physics (TMP) publishes high-level monographs in theoretical and mathematical
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EditorialBoard
W.Beiglbo¨ck,InstituteofAppliedMathematics,UniversityofHeidelberg,Germany
P.Chrusciel,HertfordCollege,UniversityofOxford,UK
J.-P.Eckmann,DepartmentofTheoreticalPhysics,UniversityofGeneva,Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
A.Kupiainen,UniversityofHelsinki,Finland
H.Lo¨wen,Heinrich-Heine-University,Du¨sseldorf,Germany
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA,USA
N.A.Nekrasov,InstitutdesHautesE´tudesScientifiques,Bures-sur-Yvette,France
M.Ohya,TokyoUniversityofScience,Noda,Japan
M.Salmhofer,InstituteforTheoreticalPhysics,UniversityofHeidelberg,Germany
S.Smirnov,MathematicsSection,UniversityofGeneva,Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,NY,USA
J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
Forfurthervolumes:
http://www.springer.com/series/720
Ralph Blumenhagen
Dieter Lu¨st
Stefan Theisen
Basic Concepts
of String Theory
123
RalphBlumenhagen StefanTheisen
Werner-Heisenberg-Institut Albert-Einstein-Institut
Max-Planck-Institutfu¨rPhysik Max-Planck-Institutfu¨rGravitationsphysik
Mu¨nchen Golm
Germany Germany
DieterLu¨st
Ludwig-MaximiliansUniversita¨tMu¨nchen
Arnold-SommerfeldZentrumfu¨r
TheoretischePhysik
Mu¨nchen
Germany
ISSN1864-5879 ISSN1864-5887(electronic)
ISBN978-3-642-29496-9 ISBN978-3-642-29497-6(eBook)
DOI10.1007/978-3-642-29497-6
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012945175
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
ThisisarevisionofVol.346ofSpringerLectureNotesinPhysicswrittenbyD.Lu¨st
and S. Theisen in 1989.The notes have long been outof print, butover the years
wehadcontinuouspositivefeedbackfromstudents,whofoundthebookhelpfulin
their attemptto enterstring theory.To make the bookusefulfor a new generation
ofstringtheoristsrequiredarevisionandasubstantialextension.Unfortunatelythis
meansthatitbecameintimidatinglyvoluminous.
Thepurposeofthisneweditionisthesameasoftheoldone:topreparethereader
forresearchinstringtheory.Itisnotacompendiumofresultsbutisintendedtobe
a textbookin the sensethat, atleastin mostparts,thereaderisnotreferredto the
originalliteratureforderivations.Wetrytobepedagogical,avoidingexcessiveuse
ofphrasessuchas “itis wellknown,”“onecan show,”whichare oftenfrustrating
(notonly)forthebeginner.
Aimingatapedagogicalintroductiontoavastsubjectsuchasstringtheoryalso
meansthatwe had to makea selection oftopics. Majoromissionsare blackholes
instringtheory,stringsatfinitetemperature,stringcosmology,anomaliesinstring
theory,modelbuilding,matrixmodeldescriptionofM-theory,stringfieldtheory,to
nameafew.
We give references at the end of each chapter. We restrict ourselves to some
basic papers and reviews from which we have profited and also some additional
referenceswhichcovermaterialwhich goesbeyondwhatwe couldcover.Almost
allreferencesareeasilyavailable.Withfewexceptionstheyareeitherpublishedin
journals, available as preprints on the arXiv or scanned at: http://www-lib.kek.jp/
KISS/kiss prepri.html.
Theinfluenceoftheclassic stringmonographsbyGreen,Schwarz,Witten,and
Polchinskicanbefeltthroughoutmostchapters.
Mu¨nchen,Germany RalphBlumenhagen
Mu¨nchen,Germany DieterLu¨st
Golm,Germany StefanTheisen
v
•
Acknowledgement
R.B.thankstheAlbertEinsteinInstituteinPotsdam,theKITPinSantaBarbaraand
the KITPC in Beijing for hospitality duringpart of this work. D.L. acknowledges
the hospitality of the theory group at CERN. S.T. is grateful to the Heisenberg
Institute in Munich, to the Departamento de F´ısica of the Universidad Central de
VenezuelaandtotheInstituteforTheoreticalPhysicsattheUniversityofHeidelberg
for extended hospitality while writing parts of this book. I. Adam, A. Font,
S. Fo¨rste, S. Fredenhagen, M. Gaberdiel, D. Ghoshal, M. Haack, A. Hebecker,
S.Hosono,A.Kleinschmidt,O.Lechtenfeld,I.Melnikov,R.Minasian,O.Schlot-
terer, A. Schwimmer, S. Stieberger, D. Tsimpis, T. Weigand were very patient
answeringquestionsonvarioustopicsofstringtheory.WealsothankN.Akerblom,
S. Moster and E. Plauschinn for their help in recovering the 1989 edition of the
bookandfor preparingpartof the figures. Moreover,we thank A. Deser, X. Gao,
B.Jurke,T.Rahn,F.RenneckeandH.Roschyforcommentsonthemanuscript.All
theirhelpisverymuchappreciated.
vii
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Contents
1 Introduction................................................................. 1
2 TheClassicalBosonicString .............................................. 7
2.1 TheRelativisticParticle.............................................. 7
2.2 TheNambu-GotoAction............................................. 10
2.3 ThePolyakovActionandItsSymmetries........................... 12
2.4 OscillatorExpansions................................................ 23
2.5 ExamplesofClassicalStringSolutions............................. 31
3 TheQuantizedBosonicString ............................................ 35
3.1 CanonicalQuantizationoftheBosonicString...................... 35
3.2 Light-ConeQuantizationoftheBosonicString .................... 42
3.3 SpectrumoftheBosonicString ..................................... 46
3.4 CovariantPathIntegralQuantization................................ 53
3.5 Appendix:TheVirasoroAlgebra.................................... 59
4 IntroductiontoConformalFieldTheory ................................ 63
4.1 GeneralIntroduction................................................. 63
4.2 ApplicationtoClosedStringTheory................................ 85
4.3 BoundaryConformalFieldTheory.................................. 93
4.4 FreeBosonBoundaryStates......................................... 101
4.5 CrosscapStatesfortheFreeBoson.................................. 103
5 ParametrizationGhostsandBRSTQuantization ...................... 107
5.1 TheGhostSystemasaConformalFieldTheory................... 107
5.2 BRSTQuantization .................................................. 110
6 StringPerturbationTheoryandOne-LoopAmplitudes............... 121
6.1 StringPerturbationExpansion....................................... 121
6.2 ThePolyakovPathIntegralfortheClosedBosonicString......... 126
6.3 TheTorusPartitionFunction ........................................ 146
6.4 TorusPartitionFunctionsforRationalCFTs ....................... 152
6.5 TheCylinderPartitionFunction..................................... 157
ix
x Contents
6.6 BoundaryStatesandCylinderAmplitudeforRCFTs.............. 163
6.7 CrosscapStates,KleinBottleandMo¨biusStripAmplitudes ...... 166
6.8 Appendix:D-braneTension ......................................... 171
7 TheClassicalFermionicString ........................................... 175
7.1 MotivationfortheFermionicString................................. 175
7.2 SuperstringActionandItsSymmetries ............................. 176
7.3 SuperconformalGauge............................................... 180
7.4 OscillatorExpansions................................................ 189
7.5 Appendix:SpinorAlgebrainTwoDimensions..................... 192
8 TheQuantizedFermionicString ......................................... 195
8.1 CanonicalQuantization.............................................. 195
8.2 Light-ConeQuantization............................................. 200
8.3 SpectrumoftheFermionicString,GSOProjection................ 202
8.4 PathIntegralQuantization........................................... 208
8.5 Appendix:DiracMatricesandSpinorsind Dimensions .......... 210
9 Superstrings................................................................. 223
9.1 SpinStructuresandSuperstringPartitionFunction ................ 223
9.2 BoundaryStatesforFermions....................................... 232
9.3 D-branes.............................................................. 235
9.4 TheTypeIString..................................................... 241
9.5 StableNon-BPSBranes.............................................. 251
9.6 Appendix:Theta-FunctionsandTwistedFermionic
PartitionFunctions................................................... 254
10 ToroidalCompactifications:10-DimensionalHeteroticString........ 263
10.1 Motivation............................................................ 263
10.2 ToroidalCompactificationoftheClosedBosonicString........... 264
10.3 ToroidalPartitionFunctions......................................... 280
10.4 TheE (cid:2)E andSO.32/HeteroticStringTheories............... 284
8 8
10.5 ToroidalOrbifolds.................................................... 294
10.6 D-branesonToroidalCompactifications............................ 308
11 ConformalFieldTheoryII:LatticesandKacˇ-MoodyAlgebras...... 321
11.1 Kacˇ-MoodyAlgebras ................................................ 321
11.2 LatticesandLieAlgebras............................................ 327
11.3 Frenkel-Kacˇ-SegalConstruction..................................... 337
11.4 FermionicConstructionoftheCurrentAlgebra:Bosonization.... 340
11.5 UnitaryRepresentationsandCharactersofKacˇ-Moody
Algebras .............................................................. 343
11.6 HighestWeightRepresentationsofsbu.2/ ......................... 349
k
12 ConformalFieldTheoryIII:SuperconformalFieldTheory .......... 355
12.1 N D1SuperconformalSymmetry.................................. 355
12.2 N D2SuperconformalSymmetry.................................. 370
12.3 ChiralRingandTopologicalConformalFieldTheory............. 382