Intro to String Theory - G. terHooft.pdf - String Theory - annam101

INTRODUCTIONTOSTRINGTHEORY ¤

version14-05-04

Gerard’tHooft

InstituteforTheoreticalPhysics

UtrechtUniversity,Leuvenlaan4

3584CCUtrecht,theNetherlands

and

SpinozaInstitute

Postbox80.195

3508TDUtrecht,theNetherlands

e-mail: g.thooft@phys.uu.nl

internet:http://www.phys.uu.nl/~thooft/

Contents

1StringsinQCD. 4

1.1Thelineartrajectories.............................. 4

1.2TheVenezianoformula.............................. 5

2Theclassicalstring. 7

3Openandclosedstrings. 11

3.1TheOpenstring.................................11

3.2Theclosedstring.................................12

3.3Solutions.....................................12

3.3.1Theopenstring. ............................12

3.3.2Theclosedstring.............................13

3.4Thelight-conegauge...............................14

3.5Constraints....................................15

3.5.1foropenstrings:.............................16

¤ Lecturenotes2003and2004

3.5.2forclosedstrings:............................16

3.6Energy,momentum,angularmomentum....................17

4Quantization. 18

4.1Commutationrules................................18

4.2Theconstraintsinthequantumtheory.....................19

4.3TheVirasoroAlgebra..............................20

4.4Quantizationoftheclosedstring.......................23

4.5Theclosedstringspectrum...........................24

5Lorentzinvariance. 25

6Interactionsandvertexoperators. 27

7BRSTquantization. 31

8ThePolyakovpathintegral.Interactionswithclosedstrings. 34

8.1Theenergy-momentumtensorfortheghostﬁelds...............36

9 T -Duality. 38

9.1Compactifyingclosedstringtheoryonacircle.................39

9.2 T -dualityofclosedstrings............................40

9.3 T -dualityforopenstrings............................41

9.4Multiplebranes..................................42

9.5Phasefactorsandnon-coinciding D -branes. .................42

10Complexcoordinates. 43

11Fermionsinstrings. 45

11.1Spinningpointparticles.............................45

11.2ThefermionicLagrangian............................46

11.3Boundaryconditions...............................49

11.4Anticommutationrules.............................51

11.5Spin........................................52

11.6Supersymmetry..................................53

11.7Thesupercurrent. ...............................54

11.8Thelight-conegaugeforfermions.......................56

12TheGSOProjection. 58

12.1Theopenstring. ................................58

12.2Computingthespectrumofstates. ......................61

12.3Stringtypes....................................63

13Zeromodes 65

13.1Fieldtheoriesassociatedtothezeromodes. .................68

13.2Tensorﬁeldsand D -branes. ..........................71

13.3 S -duality.....................................73

14MiscelaneousandOutlook. 75

14.1Stringdiagrams.................................75

14.2Zeroslopelimit.................................76

14.2.1TypeIItheories.............................76

14.2.2TypeItheory..............................77

14.2.3Theheterotictheories.........................77

14.3Stringsonbackgrounds.............................77

14.4Coordinateson D -branes.Matrixtheory....................78

14.5Orbifolds.....................................78

14.6Dualities.....................................79

14.7Blackholes...................................79

14.8Outlook.....................................79

1.StringsinQCD.

1.1.Thelineartrajectories.

Inthe’50’s,mesonsandbaryonswerefoundtohavemanyexcitedstates,calledres-

onances,andinthe’60’s,theirscatteringamplitudeswerefoundtoberelatedtothe

so-calledReggetrajectories: J = ® ( s ),where J istheangularmomentumand s = M 2 ,

thesquareoftheenergyinthecenterofmassframe.Aresonanceoccursatthose s values

where ® ( s )isanonnegativeinteger(mesons)oranonnegativeintegerplus 1 2 (baryons).

Thelargest J valuesatgiven s formedtheso-called‘leadingtrajectory’.Experimentally,

itwasdiscoveredthattheleadingtrajectorieswerealmostlinearin s :

® ( s )= ® (0)+ ® 0 s: (1.1)

Furthermore,therewere‘daughtertrajectories’:

® ( s )= ® (0) ¡n + ® 0 s: (1.2)

where n appearedtobeaninteger. ® (0)dependsonthequantumnumberssuchas

strangenessandbaryonnumber,but ® 0 appearedtobeuniversal,approximately1GeV ¡ 2 .

Ittooksometimebeforethesimplequestionwasasked:supposeamesonconsistsof

twoquarksrotatingaroundacenterofmass.Whatforcelawcouldreproducethesimple

behaviorofEq.(1.1)?Assumethatthequarksmovehighlyrelativistically(whichis

reasonable,becausemostoftheresonancesaremuchheavierthanthelightest,thepion).

Letthedistancebetweenthequarksbe r .Eachhasatransversemomentum p .Then,if

weallowourselvestoignoretheenergyoftheforceﬁeldsthemselves(andput c =1),

s = M 2 =(2 p ) 2 : (1.3)

Theangularmomentumis

J =2 p r

2 = pr: (1.4)

Thecentripetalforcemustbe

F = pc

r= 2 = 2 p

r : (1.5)

Fortheleadingtrajectory,atlarge s (sothat ® (0)canbeignored),weﬁnd:

r = 2 J p s =2 ® 0 p s ; F = s

2 J = 1

2 ® 0 ; (1.6)

or:theforceisaconstant,andthepotentialbetweentwoquarksisalinearlyrisingone.

Butitisnotquitecorrecttoignoretheenergyoftheforceﬁeld,and,furthermore,the

aboveargumentdoesnotexplainthedaughtertrajectories.Amoresatisfactorymodelof

themesonsisthe vortexmodel :anarrowtubeofﬁeldlinesconnectsthetwoquarks.This

linelikestructurecarriesalltheenergy.Itindeedgeneratesaforcethatisofauniversal,

constantstrength: F =d E= d r .Althoughthequarksmoverelativistically,wenowignore

theircontributiontotheenergy(asmall,negativevaluefor ® (0)willlaterbeattributed

tothequarks).Astationaryvortexcarriesanenergy T perunitoflength,andwetake

thisquantityasaconstantofNature.Assumethisvortex,withthequarksatitsend

points,torotatesuchthattheendpointsmovepracticallywiththespeedoflight, c .At

apoint x between ¡r= 2and r= 2,theangularvelocityis v ( x )= cx= ( r= 2).Thetotal

energyisthen(putting c =1):

E =

Z r= 2

p 1 ¡v 2 = Tr

T d x

Z 1

0 (1 ¡x 2 ) ¡ 1 = 2 d x = 1 2 ¼Tr; (1.7)

¡r= 2

whiletheangularmomentumis

J =

Z r= 2

p 1 ¡v 2 = 1 2 Tr 2 Z 1

p 1 ¡x 2 = Tr 2 ¼

x 2 d x

8 : (1.8)

¡r= 2

Thus,inthismodelalso,

E 2 = 1

2 ¼T = ® 0 ; ® (0)=0 ; (1.9)

buttheforce,or stringtension , T ,isafactor ¼ smallerthaninEq.(1.6).

1.2.TheVenezianoformula.

Considerelasticscatteringoftwomesons,(1)and(2),formingtwoothermesons(3)

and(4).Elasticheremeansthatnootherparticlesareformedintheprocess.Theingoing

4-momentaare p (1)

¹ and p (2)

¹ .Theoutgoing4-momentaare p (3)

¹ and p (4)

¹ .Thec.m.energy

squaredis

s = ¡ ( p (1)

¹ + p (2)

¹ ) 2 : (1.10)

Anindependentkinematicalvariableis

t = ¡ ( p (1)

¹ ¡p (4)

¹ ) 2 : (1.11)

Similarly,onedeﬁnes

u = ¡ ( p (1)

¹ ¡p (3)

¹ ) 2 ; (1.12)

Tvx d x

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