186

1211

rai éÄ

r icl~ dota®

ra

iwe inrel t

: "~ ;strls as

or

23

f our Universe ile t

t t

s

s l

pr clr t nd,

t fsts that t

sl

s res

i e slfiel t

r91

ies,

" �r.`,

s

e

sl

ts iitself.

t also istatistical sics

r at

s~x*f scltics .

0920-5632/89503-50 @ rElsevier Science PublishersQ-)rth-llolland Physics Publishing Division)

tici

i

of

School,

ri

which rlo

i t construction f four-i ss Bert

c lle e s. 1 y

e

to

re

ilei with and as a follow-up on notes

fished in Refs.[ 1 .21, which were the basis fort lectures. f.111 contains a detail

the old covariant and Ifithtti

strings, superstrings and ten dimer ic strings, and o their spectra. Other

s f l reviews string theory are o ifs .1 - 1 .

s a

introduction to modern methods

sstr

io s of string theories, the presenttes discuss the

tisa io

of oso °cstrings a

spinning strings, follo ii to ia

et

developed or

strst

y at i , r

i a

il ovis y(

) (for

review

references, see

ef.1

) .-

exercise,

first discuss theS

tisatio

of twi

f

relativistic c lti le (for references, s

s.1 ,101) . Thisst

already shares ,t i

c simplerr, most of the technicalities whic

r string theory.String theories however, possess

richer structures which

e shall describe

sr

through t discussion . y 1c o

Nuclear Physics (Proc. Suppl.) 11 (1969) 166-222North-Holland, Amsterdam

Ttrat®

t

t

rtric

s

dr st

rt ists o

e s°s 1sc tr,

Sl

t ce

,ell~e

articuls' at

e = ~c.

.

SC L.1 . La ra

fee syst~

i e

atr i

f

iti ,

s

v i t r1

r

ay ev rt less fi

ar

ai , t

ti

atiss

si

i

c

t~

i

-i

,

its s

1

ti

is si

s

e es e li 1

ai ts i

ert

r

t ~ in

~° i~clures

tt e

c

1

1

.

(it

i

1e s

c

tatia s as in ~ef.[ ]. I

pa ticl )

r

eter i

f

tr s ace-ti e

i

a s i

etric

ca

itia s, sa

( i) ~

~

(i 1, )re ~- + .. . + +),

e

us ally

ca e ue ce af

-f' i

i

. )

v ,,

st

v :.)

Si ~ il r cansi e

ti

f 1

r

t

'cti~

.)1- i e sian 1 scala "fiel s°°'

ar

e e ua ia

tia far

e

r s e

i[1,

t

ca ser~rati

th

articl

y-

s. ~° is

ra etrisati

I~ - Ifar ulati®

-linear actian

er

t e

article trajectary

(~) is

ive

yctians af the

arl -li

ara eter t, it isell

n®

[1, ] t at t e e uatian af

®ti

f®rly carres an s t l e canscrvatia af

e

®

~~) ~ ~®,2( )

,

~

u( ) ®

.

(

.

)

I

a

itia

®

a

av

t

i

f

Far an i

initesi

al capera

etrisatia

tains: e = .

( .11)_

h l

t is

ca trai t

f lla s

fr- ] ®

~

t ° i

+

re ar

etris tia

i vari

c

e

ee ,eit

r

y ca si eri _

v riatia

acti~~( . )

. ._d~r i- initesi - al repara - - ei~- isatic~i~s ( s ir~ctian

ei

invariant

un er

suc.~

( . ))® ®r

t

falla i

. I

r 1, fasfar

atians, ( . ) s a

s t at (

+

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1 fiel

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ry

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rat®r of

arl -li

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, ®. . . ~ ,1,. . ., - 1)arp is s, s® that in

is

escriptian af

~°it

t e

i st i - il e t at ia ,

e

av , ast e ca straint

c

e uenc

af re

r

tris ti

i vari

ce,t) ~

(~)+

] _

(

.S )t e

cavaria t

ca se v ti

af

t

al er

af

re ara etrisati®

e er y-

e tu te s®r

(~ ).ic

free ®

alla

us ta

i

i

tur

lea s ta

i

~ i 's e

ti ns:ra

sa e

au e-fi i

-

, ,

+

~

( .1 ~

.~. Govaerts/Strira~ theory constructBOns

IS7

~tia f r t

ein ei .

is is(t) is a

au iliary fiel

( .1

i

( .11~ .s

t

s

11 at isc

t

f r.

i

~f

sa ic tri

it

li

r acti

I , le

r

re t t i ilar

si er tio stat

trit _ (, =,1~

s in thaf caûm, are

- = -nse u

~f~

ea ar leic an

ey i varian

a t e

li

ar

ian.

ee , far

=2, { .1 ~ re

sa

=

. i

y i v i

re ires

t

e

a~~e = a

t

s T = .

strai

s for st i

t eory, i

ar a i trinsic

orl -s eet

led to

e ot_ e

e rees of

r

e

t

latiarb

tric is cf~r

fr

ron ta °°fi

per-a i

e

rac

the r se t s

,(2.14)

ta te

st it

~l ic e . ar 1, is

s

,~. ~iOVei~îÉSi S$P9Pflg t~180TY COI1StfüC$IOIDS

et isatia i varia

®f { .9) allows

e'° . o e~

le, wit t e

u e-fi~in

iti®ns

fer (~) is as in {2.7~, wit t e

allo

i_

ro

{ .1 I ~.cti

s ibe t e sa e(a

a

ally also u ntuca easily e seen y

li ea action f®r e~T) as iveg~

t ant~sationties ~ov riantly this constraineds first use t e general for ul ti®y

frac

( I

]

(f®r

a

r~s~

e,

seesi erin

t e e n-linear asti®n f 2. I ),rasa o fr e

(T) and theirta

{r),

it t e ®issu

-

r °

es

a e

n®t

at

t.

v t rl ary Str t

( . ).

areaver, the c ®nical

a il ®nian

v is s i

ic lly,

s

se en

®f

e

a etrisati®

i varia

(see ( .4~~ .

en

t e ta1

ilta i is iven y

e

( ~ is a

ar itra

f n

ian af ~, an

T)

is t e arely first-class

nstraint .

The °°~rreâpî~ din e

ati®ns ®f

ati®

s® f®!1®~, fra

the first-®r er acti~

s®1

ians

a (

.19)

~

e e

resse

in

te

s

f

i te rati®

ns ants relate

t®

a nda y

n itians . F~r

ple:

1

1Z . 1b~

T e

acti®n

~

.20~

in erits

fro

(2.1 ~

an

inv rian

un e traresf®r ations enerated by~ t r® h its

aissan

rac ets. F®r an

itesi

l f ncti®ra r(T) with r(Tfl~=0

(i= I ,2 D,

consi er the variations :

®r t2.20 i, we t en c;~tain :

1

it

the identifications

d fi s

in (2. ). it can easily

e seen that

t e variations (2.22 c®rres nn

t® world-line

re ara etrisa~cir~ns i t is for alisrn of thesyste , i.c. the constraint ' ~i gs the ~eneraior

far these 1®cal

u e transforrßzations.

tsf

ra

t

e

ti y

c

is

e

eic

~~ llerte l

!!ite

~

~~ its cfcr

e t

sf r

i

s

e tity t

sf

ti

.

sitive

ertic s,

e

ave f

r

r

tic

i

i

ec >

( ay

rticle~

scti®

f

c <

(

a ti articl l .eri

1 re

etrisati® s, i clu iwersin~ the ' rl =1i e crientstiûn, the

~

c 1

ar eters re u s c. ~

[

[ 1

) .free® i tec®i cfte rer

(T) t us

rres

s t® t e freec ci

cf

crl -li e

a etrisati®n .u e- ` i

cf t e

ste

is t

t

cu tfcr (T~. -c ever, ifferent ncti® s

ü~

t

t e s

- e value

f c lie

t ecr it, i.e. t ey es i e i entica~1

situati® , s seen in (2 . 1) .fu cti® s elcn~in tc t e su e ®

irize

y

v

ue c c

e

sra

etrise

~1~rrc,

s~a_®s r

T!ti

es11s c

ra~s 1 I;

ith

~f ( .2 ~,

e

ls

T. ~waerts/String the®ry c®nstr~cti®ns

ve

(2 .

( . )® in

t®

o er-ti eetrissticns in ( .6) is ~(~)_~~ ( ® >

).

®lvin f®r

i ( . ), cr~ r

v rsli e

i®

( . ~, whe e

iT~ is it

rl

-li e ein ein .

n ee

~ t ev riati®

(

.

),

wit

t

e

i entificsti®n

(

.

9),s ®n s tc the tr nsf®r

ti®n ®f suc a . .

fcr

1 ti~tity

ud e

re

r

etrisati®ns .

e

I t e

I (f r

ri fter c t us

essu es t e t®t

r® er-

se

e . 1 ),

ase®

ise ®f t e

icle.

f 11

°

i

t9c

° trI

is le t

s

e~ercise t®

ert®r

t

free c

( ~

j~

t t

(T ;

®us

lysis fcr t e line r acti®

(2. ),

( (T), ( ~~ ~ 1 ®

(2a

)res it t

t it eve tuslly !es s t® ~he

sit

t e first-clsss

°nStrR -nt

(~~~0 .

s~.~

iiscussi

as s ®ve

).

t

t

t c

str ° t

(~)~ ( )

(t~~

)®

e rs ' s ftt isi

st tes

> (res . I >

t( s . c et ~ rt

( .I >g I ~,

>= I ~

.~t~

-ti e v i

t

ti

stis r a ily st `

_ y c c 't t te er y-

-

t

tsatisf ~ th

.ysic

st tes

e

` e

y t

strI s >~

,

( .

i °es ® y( . ,tsttet ®f ~,

it

c l .t tve-fu ic

e

t ic

f ( . )®tic sp

s y:

ysic

s ates sre

eter

i

()® ®

( . )

is t e

ei:~-

r c e

ti

f r

sc°ticle .

t

c r® i e

tic,

I ;~>_

t i,ti

r 1 ticnsi i

):

Ign

J. (~®+~serts/S'trin~ t ~r~ c®nstructi®ns

ti-__ _

s

v ri

s

it

® ®

, ® ( . )

ct c ice f

, say :

ys i t~f ecti

1c ffl

_ ~®~m

i

f(x) s

fu

i

f ,e

~ .

( .s t~

f( )® t

ri

iv~

r t s ti ( . l ),r i®

(~, ,

,s t e s

ysica sit ati~ .te v

Siti® .

cSetef cti

( ), a

e c sen s c t at allz~

i ftes te c `e~taie .a!~

- i i

f

cti

e ist f r t e, ~ 1~

c rr s

s t~er-ti e

e i tr

ce

a

.

tl~erc ices re als

s i l [

. t

a eve , t

t

i

s

e su set

f

eic

~ ~lers a

e e

iti liciti s are c~u te

®itt

eir

~ ie

t

ti

]is

i

e

, l

.

~rf( )

(

>

) is s c

case-fi~~i fu cti

l .f d~

e-fi i

fu

cti

e ists

t ali,t e

r

1 ti

f t

syste

suffers fr®

ari ew

le

.

isist ec sef r srit e

y

® i

®

e s

al

i cuss.

illustrate~ e

er i

ic

e

y nevert elessr

ee

i

es

ce ®f

ri w

r

le

,1let u

c

si e

t e c

ice

ere ( ) is s

e functi

suc ~; at (ii)=ii(i=1, ),

, c are

ree

ara e ers . In t eerrB

i

e fective acti

a

a ilt® i

,iet us t

re efi e

-~an

-~

~ al

t e

t

e t e li

it

~

.

is lea s t®:°

~ °

x( ~

~eff = ~

t

-

( . 5)

learly, is r

e e r s

aui i

iti

les out t

i labelledc. The degrees of freedom n, q1 and P2 have

invariant system 1191ili r

variables. Solving in (2.46) ftities (this is consistent

it

2.4.

a ti

if r

ti

),obtains:

t

system

f

1 48)

is defined tLr

the (ant0commutatT relations

i ,

.

.

.-i fi;, (2.54)

t

~t ths

t

ittt

(2.22)t)

rr

(2.23) .(t) .

J. Govaerts/,string theory constructions

191

(2.49)o

motion forfollow fro

t_Wit

1171 . hrst presented

just` i~tr

io . This is the case not only ficl

19,101,

t

ofor string theory

bissen gack

1(t))

.

(2-5V~

it the boundary conditions (2.49), thsolutions in (2.21a) reproduced,are e

(44m 1(t), describing thus the ssituation

limit

-1,

, i.e.

"7Aghostcharges reduce o

(

+

) .

c=-i 21 .(2-51)

(2.50), generates9) and BRSTs leaving (2.48) invariant :

- . 1y~ + 2)( . )

.~t 1 ( -)h.(2-53)

orrre olvies rel tii ci le"

,

Ji f

ie lulstitute teternir

ts e

f

Ir

r Mi

0

licit representation

t

st

r(,

)--i is r

forfunctions

)of

rssmann variable 0. wit

ja .

) 9

voluti

s>2, it

(2.55)

(2.56)

e

(2-57)

-> :

()- 1 . I+> : ;p7)

(2.58b)

ysic states

iitio s:

I .t> . CI .1> - . (2.59)solved by

I

®t >_ IdDqT#(q ; t)I >

JdAP>01- >101 -

.

(2.60)2[3,m21 ,V(q; ,0-O. These states are thusto-one t vi

is ti

192

J. Govaerts/String theory constructions

d

(2.45) . T

t

i

eMe(s

t

s

(®t)

,Z) satisfies tt tisit st1

r

(2.61)

®~ L21

e s

-> , (2.64)

ti-gral representation of quan

) invtfixing fuclass 1 ,1function! f twell

ti

r

'

sA ~ nia .r.

1 f

s,

tion that interesting developmentsfly been achieved concerning this, i

*attemptt

f

a ,e t

role

lay

y

STtry in gauge invariant systems better.

I3.1 .

ra ian o

l tiothese notes, only t

non-linearG to action 1211 will be considered for toso °c string. A discussion similar to the onelow aoDfies to the linear action, wheretrinsic world-sheet metric is introduced . Letst

s consider :S1114 i t

,

°)

( .1 )T

s

1(® ®) _ ® ®Z , (_ .1®)

2r t

string trajectory Au, t) depends ot

world-sheet c

r i

tes t and 0 _-t

i w� andis the Regge slope.

e equations of motion simply corres o11,

to t

local conservation of the

orlet energy-m o

e t

current

(,n) :

ve the boundary conditions :

u ntities

and

are not allt.

e

satisfy the constraints:e

e

®_ ® 2(ei

= 0 ,

_

a

s

.

( .4)

e relations (3.) are

consequence ofr

etrisation invariance of the action.

s

c is au (i.e.tW21

r f.e-fi ii Fi

ra r r Util r s tral re t

( , ti) i(), i=1 . (3.3a)

i) open strings : (=0, ; t) = (3.3b)

ii) close strings: (a- m, t) (a=0, t). ( .3c)

in (23) aconsideri

itesi

d (M), this can be established by

ii) closed strings :the variation of (3 . 1 ) under an td(a + )i +0 0at woria-sneet reparametrisationV =V - 50 (3-5

ita) qO(a, 1i) = 0 .

i= 1,2

050b) i) open strings: qI(a-O,x; z) - 0

(3-5,.- )fi) closed strings: J00, xWqIWn, it (33d)

Such an analysis thus shows that theconstraints (3.4) are the generators of worlsheet rapacemetrisations . Actually, the pair ofconstraints in (3.4) are obtained from sacother by a transformation exchanging the roteof a and z (in a way preserving orl -seatorientation) .

Choosing to fix world-sheet para etrisatiothrough proper-time gauge conditions 191

+&2=0 , XO>O'0

(30We nion-Ifinear equationssimplify to linear Klein-Gorassless scalar fields iways

As is well knoequations of olio;simply D,21 :yen svkgs:

Aa, -0=42a'i qP+

J. GovaertsIString theory constructions

193

e" cos

(31)

here the integration constants mayterms of the boundary conditions

(3-3) 191 . Note that qF. at and % have beenormalised differently as in Refs.11,21 . Iparticular, the total momentum PP of the stri

ex ressea i

is given by cP,,=NF2;; PP for the open stri

by

cPo= h2a'P'I=&,, for the closed stri2

Z we denote an infinite summation f

with n=O omitted; I denotes the same

e"("G)

finite summation with n=O incluof

motion

(3.2)

These solutions h=ver, have to satisfy ton equations for

conformal gauge-fixing conditions (i ± x')2two dimensions . As A.

This

translates

11,21

into

the

Virasorultipliesi, let us restrict our choice in

constraints:(3.6) to the

or

al gauge, with A0 = 1 .

0 open strings : LWn= 0

(31)Such a gauge-fixing does not completely

determine the world-sheet coordinates 11,21 .

ii) closed strings : L'sP= 0 , E'O"= 0 .

(3-10)n nThe subgroup of (pseudo)co or altransformations leaves the

or al gauge

with the Virasoro generators:oconditions invariant A late gsine-fi ing of -. c mp x WI W I, A=li6jLA% O' yO-wALA AO "%,AJLAVVUU All ul w 117611I. WRIC saugul

which we shall not discuss in these notes 11,21 .n, the

solutions to the

As we shall see, these quantities generate iin the

or al gauge are

the

or al

gauge

(peso o)

or altransformations, which are the left-oversymmetries of the system after sine-ii iThis is the reason why the Virasoro generators

ust vanish for physical solut

elMvO$ (30

3.2 . Old covariant quantisatiLet us briefly review the so-called of

covariant quantisation of the bosonic striusing the canonical Hamiltonian methods

194

J. (FovaertsIString theory constructions

s

t

T

Is

loped by Dirac 1141. We have the degreesfreedom P and their conjurate moment

foil

tically, as a consequence oftion invariance (this can be s

(2,4) for the particle). Hence, the totalis:

A result s

ti

Rft,, 1 0 (3-14)

(a) WM 04 + A-0

(3-15)

(3-16)t 6. and 6- are first-class

s of motion obtained fr

+

(117a)

(a. %) are arbitraryt (3.15) consistently

tion of to systema

lished from treferences

from the first-order acti

MI3a)

Six, X, X',X-1- 1 72 d-T

do

ith the boundary conditions 191 :

)Ao, ,q) - x~(a)

i- 1,2

(3.1 9all

Oopen strings :

ji) closed strings: same values M a-0(3.19c)

The first-class constraints generate throutheir Poisson brackets symmetries of the actio(3.18). For infinitesimal functions c"la, 0 anr(o. )such that 191:

) (e+0(a . id-0,

i- 1 .2

(320a)) 0 open strings: (;L-e-X*r)(a=Ox,,[)=O (3.20b)ii) closed strinits : same values at

t

It us consider the quantity( )) F. . . a

[C 0 , +

(3.21)

to variations

(3.22b)2 aJ C.'4r,-t_4r- I

i~ - x+E"', + ;L 4.,e

(3.22c)

i~+ X-C-' - ;LA-(3 .22d)

It can then be shown, using the equation ofotion (3-17a) for IF, that these

transformations leave the action (118)invariant . With the identifications 191 :

C+_X+qo+ql ,r-1 -qO-q 1

(3.23)y bolving for 0 from (3-17a), it can be

seen that the variations (3.22) correspond toorld-sheet, reparametrisations as define

(15) . The constraints ~+ and ~_ are thus thegenerators of these local gauge transformationsin the Hamiltonian formalis

solving for 0 in (3.18). one obtainsr acti

C"dr. + C- dr-

(3.22a)

9

(

.I9b)

(3.20c)

S1

vtrte

s

fi

r

tthr

ar

r

in(vu)-11 do e $ .

-

~3.28b)

F

closed strings,

have:weorere go, Q is an arbitrary function. From Vriations (3.22c,d), it can be shown that &0

QN-Ne"' r _e7-in3 ,(129), ansforms indeed as a world-sheet metric

sor 191 . As in the case of the particle, the(S)

* (a)

(S)range multipliers A" and A- thus not only

$,-21JEnEn +cnL n I

(3.29b)

T

seA-. ItR

c

isametris

ameter caasece of

Allsidering t

uations of motiangian for

solutions (3.7)tris tins l

erc such that

or 0

VA-1 (ro)i

J. GovaertsIString theory constructions

(324)

(325)

trise the freedom in the choice of

rleet coordinates. but they also correspond t

onents of a world-sheet metric .2i (,-V) ( )n *is result allows us to discuss the gauge-

16 n

do e

_ ,

(3.29c)of the Nambu-Goto strin whic,g . . . . ma.

2in(vv) (")E(s)- I

I do e

(329d',t formalism is done by specifying A' and

n - 2e argued 191 that a complete gauge

the theory is obtained by the choice

The quantities just introduced can be seen t, 11) - AO - Ala, 1), 10 > 0,

(3.26)

reduce to those defined in (3-1).'This proves

ich corresponds to the proper-time gauges in

that the Virasoro Renerators inde(3 .6) .

particular, for configurations such that

rpseuaoxontormai transtormations. Note

thatInand

- are

it can easily be

in the

conformal gauge, for open strings weX*

A

independent of a,HTImseen from (3.22c,d) that the quantity 19,221

have

=L(0 ),

and

for

closed

strings

If2f, 2 difIx do

(3.27)1 1

Hr2(L(O0+E(5)), where it can also be sho.~ -11

0

2

0

0invariant A world-heetti s, i.e . the Teichmaller

erizing the gauge orbit of tse-space passing through 110

C,,-, .-,-uge AO-1, the(3-17) redueto thosfation for that gauge,and (3.8) . World-sheet

the Raupte conditionted by

te)L n

do [ein(vov)(O)+ein(r.-)(1~

that 2(000-

(®l ,, )) generates constant shifts iinally, from (3-16), we have the Poiss

rMets:'' "0 open strings: (L .,

Ms± a) 11,21 .

strings , we then havP (128a)=`5'cLC a., n

quantised system is defined by tThrouRh the c-nrresDondence orinciple. t

195

(330)G

G

,I = 0ii) closed strings :

(L*-', Ln M_

(L(e)

(a)L ff% )--i(n-m)Ls%

®-i(n-

)

~~ )

.(3-31)

06 J. Govaerts IString theory constructions

fttti I I

lation

IL"". L',il1 = Wml

W

+ î-2 Dn(e - 1 )jn M

n4m

8nthe I

Is

ssumea iraws

vralot ;^-ne 12 2RI Ag%f ;,nd% t ,%%h

U;,roenret aimoke_r .it .

with central charge c=D, each scalar field Aa, -T)contributing one unit, to this value.

ue to the existence of negative-norm statesin the covariant quantisation of the theory, one

s to check, as a necessary condition forunitarily, that none of the states which satisfies(3-36) or (3-37) has negative-norm . By the

st theorem 1231, we know that this isssible only if D _r. 26 for =1, or if D -4 2 5

for a< 1 .

i) open strings :(a)

0, L n 1 q> = 0

ii) closed strings:

00

here "a" is a subtraction constant followifrom normal ordering in the zero-moVirasoro generators.

Similarly. one expects a central extension forthe quantum, conformal algebra in (3.30) and(3311 Indeed, one finds (see the Appendix for

taus) :i) open strings:

d I d I

d I I

IL 'e%" , L f"n" 1 = (n-m)Lq%'.~ + 71- Dn(ne- 1 Mn.

ii) closed strings :

I lai lap

M

1L «, , L ,, 1 - Wm)L,,, .., + w Dn(n2 - 1 )â

%M 19

»`- 2alî W > - 0(3-37b)L +£".%

V > = 0

(3.37c)0 0

1 (3-36)

1-0 .(3-39)

(3.37a)

(338)

flfltu es, flt fls

~

t at the c rrecta

t

ir t-cl

s c

str i

~)=0larity s r c ure

e t® inter

e iate s

tes

-tai>~e

® ly if

=

].

ence, the values

ss

i te

t e c

i

-~_ ®,

- _ ,,

,ecessary ®r tr

-l v l a

® e-1

3~ -®

_ ~,

e

i tr

res

ctiv 1itarity are a=1 a

=

.

e

i i

r ss

a

v ri

1

,

)ut t is i

lies t at t e

ysical r~un -

( =1® ,~,a

eir

]

t

t

) , ]tes f~r t e ® e stri

a

f~r t e cl se

( =1, ,3,

(a ti- e

itia

ssare tac y~ns, a fact

ic

~pells

isaster v ria 1"rs], it

®st

ra

atic c~ se uence ®f t e e istence

(3. 1)ese tac y~ s is i

t e n® - ec®u li

®f

~°Ye

st c

r

is

fier®- ®r

hysical states i

i er 1an.

ilitudes [

,

) .

®r

le, i

t

c s

f

~ _

~ .

)t e ® en

s®nic stri

c®u led t~

r®

t

a

r

t s c

tre ~~ees ®f free ~

t r~u

C an- at®

fac

rs,

(a=l, , ,

],e

tai

~ e

e

r t~her~

c~

utin

®ne-1

a

litudes

i

_ ~

[ i - ~,

~

( , Abe ternal

ysical ® en stri

states,

it atleast ®ne ®f

er~- ®r

, ®ne fin s that t ese

-~ ~

~ +

a

,

( .

)a

lit es v is

u

t

t ® ty es ®

an t e

S i varitri uti®ns.

e firs

ntri uti®n v

is es

=

~ _

.

(3.)ly f®r

(

/ )=S

(13)

[

,

, 7),

but t ~

y

structti n,

il

t

f t~n

®ne, entirely

ue t® tac y~ns,

s

e~t

c ar e:

tri uti~n e ists an

it v

is

s

tr nsf~r

a i ns

f all varia 1 s c

'( 2).

is fa ®us result

as

e ® t in

[ , lea i

t a c i

fternal au e ®s~ns [

a

invari t ~u

y

iti® ,

°cbn-

itarity ®f

ac y

ic

s ec' y

el®

.

®te t at

e

ave:leea t e

re

®

n ®f t eir

( ~,

)(Tl

~

®

)

(T) ,

( .

), fl.e. fl

t e

r

au e, ~e ry.

ereiscussi®

n

in

the s e

a ®f

(T)

( )

(c)t ese stri

e®ries,

®t

ia the

®re li t-

e au e,

ay e f®

Sfl

isf~u

stStrlf~r tTe

t

fl

sev is [25,26]® ly t e firstf®r

( / )=seta lis e f®rravit® s [29) ®nly) .

stringy the® iau

J . Govaerts/String theory constructions

3.3 . F f®r ul ti®the

fir ulati®n a plie

t~ t e®sonic stri

(see ef.[ ) and referencest scam),

ass-s ace is e ten e as f 11

s.First, ~ e i tr®d~ces c n1u ate

®

~nt~

_~ , ~)

( , i) t®

-( , ~).

°( , i) res actively,

it

sa e

®iss

r ckets ( . ) s t e

s icc®nstr i ts ( ) an

( ) .e

t l uantities

,(c) a

(ç) ave t e

(T) and ( ) are t us t e

e

q t r

f

rl

198

J. G®vaerts/Strgng theory constructions

t

t

s effective,ti s

motion.

rf r~f . ® ti

sxt -o Pal and Fathen taking the limit

0. t~ti

,

(-1 ® )raints which are SOW

t r

erI

ff ctive II :a ' t

i

191

s

t

usfunction 1 :

>

taking

specified value and

ar.

One

mav

t

a i -

t

t'1

57, ,711

rft t tW 1,2). an

r

s

s

e

notation as before i

s

for thisquantity, although it is different . This shoul

t cause

y confusion, si

i

at follows

y t

restricted

operator is considerarticular, the boundary conditions (3.54)

ST invariant . and S,,ff in (3.51) isinvariant for transformations induced by

(3.55) (provided e uses t equationti

) . t may also be checked that

isff 1

il te t.

t

24064472 loll40A 1

(3-52)d the correct equations of motion are

reproduced

i

the Poisson brackets (3®®1,- )

, ), ( ® ) ®

® ,)® (3-53a)

(113(0,, )B 1( ® 9 O) - 40-01 ® (100-0- P2(0'-'9)) .(3-53b)

appropriate boundary conditions are 19):

(41)a

1 + 3a

11(® j)

. 1-1 .

-

-21(,Tl) -

, i®1 .

1140- i)

( ® d, (, d - ( ,Id, i l,22

) 1) open strings:

(

, ;

)0

closed stris : s

v

Solving

r

® qla (-1,) and Pa (-,)i

siste t with BRST transformations, which forthe r

degrees f r

a

taifrom the BRST charge in the gauge A'-AO-A- 191:

f®( (-)+

( ;)+®1 +®3a,

(3-54)

3

1 - aA 1

(3-55)

a

T

t~®te ~.

~Ct ®D+ ~~ t

)¢

mc_a c_

__ -~C+a C+

1[ c

__ + c~

~,

,

ra~kets:

, ~),

(

, T)) ~

i

-

) .

(

.

a)

~ . )~un ary ~~~ iti®ns s ecifie

in (3. 4) .at t .4 ) an

(3.

) lea

t~:

s s ® s that t e ®

n iti~ns at~c~t~

(~~ 1,

)

~c

~

~®lve

t

)

,a

ly

a

t

)(

, ~)~

i

Lt

le®~

t

~

"

(3.

ysic

s l ti®ns

®

®t

e en

®

Y s ecifi~~ l -s eet

ara

etrisati~n

hi~

is use .t is

s

ea

t®

t

(3.57),

ne re

vers t e a~t~an ® ta~ne

y

a ®a a t 0j i t eir ri in

treat e t ~fu

tisa i~n ~ t e

®s®nic strin

i~ i

j .1

® t e e eral result state

ter ( .

'i,a

ac c

e use t~ justifyS La ran i

a r® c . I

articul~:~r,e

ver'

t at t e

tfiN~ansf~r ati~t~s~

a

in u d by

lea

recisely ts~

J. fÂovaerts/String theory constructions

lgg

~ .5)

(3. a, ), it

~ a

r lar s

iv 1 .t

s t

eev-it

ra

tri ti

i v rit~T

cl , let

sis ti

t t e

i

f

ti

tr al

e, irs f r t

trI

si

e ®

iti

s ( .

) t~ ly,

e fi

st

f~

as i

( .

).

e

ave:

L~ ~

tTl ~

t~)~Lt~)

~ i °

t .e

urier

es f

s® t at

it L ive

i

t

.1 i ) a

Ltc)

et us s~ re ark t at s~ivi f®r ~~ ~ ~ [ L( )~ Lt ) , ( . )

t e f l

i

®iss®

r c ets:

early, e s ic,

st a t®ttri ti s t

e Viras r

erat rs eac

200

c'1

t)

~)-4

J.

ovaerts/String' theory constructions

iti

i

( . ),~ 1 hly t e

ic

iv ,it

ecisely t e

(3.73)~f1 .)r1 .1) rt

i

i

f t

s1ti ,l tly

y icit c

esv

ist e cl s

s i

is

it

efi

s:

Hg ~

t us

® txin

t

®

utin

i entical

s i

1 . ) ,13. 1 )

~~ies 1e

t f®r t

®s® ic

er®-

~ es) ®f t

stru ure

ic

a

ears f®r t e

13 . 1 )

®pe stri . As is ell knn~rn, tt~.is isc ara

e istic ®f cl®se

strin

t e®ries, an

is ~f

13.7 l c)

f

i

®rtan

i

the

nstructi~n ~ff® r

i

e si®nal

eter~ ic strin

t enries.

13.72 )

re

, c r

.ere,

~ ~,

i ) are ive

( . ), ~)i iv i 1 . )

() ast

t

e

ressi

as Lt l i

r

s ® t e lef -

s c

.

~rewer,

e

ave:

+ Z L~

c

¢ ILC

~ 2 L~

c~

,

t_ _ _

Fin

ly, i

®sink t e

® n ary

editions

t- °

i

-

) ®

(3.71

)

~~Ti

(i~ l ,

),

e

re

ver

precisely

the

sa

e

e i varia t physical solutions s in the- in1 ~ )

13. l )

a rx iar~

or

c®variant

a ilt®niana

r®ac es. All h®st e rees ®f free o

thent

s

~~ ti

i to ri t-

left-

vxnis , and the ®sonic s®luti®n has t® satisfyrs,

is

s

tai e

f r

t e

®sonic

t e usual

ir s~r® c®nstrai ts 13 .10).r e of free

lees 13.x)), extends also tost yste . T is is a c®nse uence of the

3.4. RST uantisations uc re f t e co or al

e r i

t ®

xvin

evelope t o

~ f®r ulati®n ofsi

s, ®r

is suc

se

rati

c rs

t e

~s~ ic strie i

t ~ c® nr al

au e, it islsee 13.31)) .

no

strai htfor and to pr®cee

t® its

RSTro 13.71),

® txi :

uantis tion. y the corres ®h ence principle,the uantu

syste

is defined th~~ou

t e)10®~) ~ Lt~le~~l (~ - ~) ® 13.72x) co utation an antic® utxtion relations

f®ll®

r®

the piss®n brackets 13.6 ).

These 1x ti)c®

utat®rs are e uivxflent, f®r the

os~nic

e rees of free o , t® the relations in

(3.33)the folio

p

(3-34) . n for t

ghost system ting

ti

ut tors :

' (c

(377)

zero-modes "

itb )_, . ., .+ ,®for open and for closed str nigs

its

one takes:,c_n and

=b-,,

similarlyand

for the left-

(3n

: cc

® :

® -

.

)ovi

ghost

odes of t

closed string . n

Having adopted these prescriptions, that follows, we explicitly write out the

quantum system is properly defined. Ilysis only for the open string or one of the

particular, we then ha--.r,two sectors of the closed string The

plate

( )

(c).

! )= L

+

- .®

..structure for the closed string isstraightforwardly obtained y Censoring the ( ) (c)quantum algebras and spaces of states with

-1:1

2 n cn

. (3.82)

themselves . except for the bosonic zero-modesq11 and an

hen pcom ared to the old cov iant __1qu

tisation of the system, the space of states

c =

(c~ ® -

®c®) + 11 c_.b. -

_

, (3-83)

J. Govaerts/,String theory constructions

201

In addition,

e also have -the ghost excitationsand bn . In this sector, let us introduce a

ground-state, also denoted by I f2 > , such that:n10>- , c,,Ifl>=0, nzt 1 . (3.79)

Therefore, the space of states of theuantised system is spanned by the oc vacua

I f); p >

I ± > and all their excitations in theos(,nic

odes

n (n1) and in the ghostodes c-, , b-n (n 1) . Note the doubling in

states due to the algebra of ghost zero-modes .e also need to specify a normal ordering

for all quantum operators . In the bosonic sector,the same prescription s in the old covariantapproach is obviously used . In the ghost sector,one again chooses to bring all the annihilation

ithi cl

is t us considerably enlarged In the bosonic.

00were LI I ®

are

ivysector, e have the ground-states I ß; >

nexpressions o ( .1pressi 1) and ( .)defined in Section 3.2, an

all their excitations.

ordered

subtraction constantIn addition, we now have ,a ghost sector . The

respectively, a

"a" is afollowing f

or

ordering

~ ).

s°algebra of ghost zero-modes cp and o isidentical to that of the

host ~Tsteforthe

techniques explained

Appendix, it iarticle. Let us thus introduce two states I - >toshow that :

(

.

) =(3-84)and I + > such that:

Ln and QB given in (3-8 1) and (3-82).C01_>=1+>, C01+>=0

(3.78a)b® I + >=1 - >, bn ll - >=0 .

(3.7gb)u it esa

esubtraction constant

.Due o this procedure f o

orderingcomposite operators, gauge invariance in th ,9quantum system is not guaranteed . This is awell known phenomenon, corresponding to ananomaly. As we know. this happens for thepresent system; a central extension

or

algebra generatedd by L(nO) was- foundin the old covariant quan-tisation (see (3.38) and(3.39)).

The question of gauge invariance mayequivalently be investigated by considering theal

a of the tot

° as o 0ec it

ilt

of ttypes of operators i e

symmetries of

system . i.e .rtransformations ithe

or

RauRe. and

yo

cr tors JBRST c1

tr t

202

J. Govaerts/String theory constructions

st

t

t

1

fiel

r

f1

t

ts

t

r.mr

tis4r

t

liz)fit

t of

re i ly t

conditions obti c ssio

f Section 3.2 . followinR frt theorem

the correct

l strs . However,

s e lsufficient for

mt

ity oo

. The value for "a" implies tif tac yo s n t

physical s

rspoiling

ita ity of t

ua ti amplitudes . valuesThe

re that gauge invariance 0.

rs

lulatis the

lati

oe ceven ftral e i"Tnly an

a.

rtsi

t

s

)is r1s .

s

)+(+ -) 1 . ,® .(3 . )

f t

t1

()1

( .t

o

Contributions (

)(c)

+ ~ (-

+

)

rt

s isis

ter

(3.84)).

t

first

s

o

finds (see t

r

t

r

ot

t

t o c

t

t

si

stss

t

or

,r l invariance of t o is

ol

ur

t

tcritical(3-85)

the of, i that ittio ost

rified by computing the

, )-

(- ) +(+ a-) +( - 1) ) c-

is clearly shows that

il oten

of the

Sr tijr is obtained

er the sa

conditions(3.88) which

sur t

cancellation of the1301 . This is to

i

L

re related, a., sho(3.84) . If t

charge is nilpotent, tfol o i

c

t

easily

® t i e°4 .1 , 1= ,

(3-90)1 ,L 1-(- )L $ .

(3-91)These results show that nil ote

of t(3-88)

BRST charge, or cancellation of the centralextension (with its two terms, one proportionalto

, t

other toA in the algebra of the totalVirasoro generators, are one and the saexpression o

or l invariance in tise

string theoIn the present formalism, physical states r

efi

y the

T conditionsI >= , c I >- 1 > . (3.92)

il o

of

ST charge isreserved

e quantum level, h firsitio ensures gauge invariance (in t

resent gauge:

or l invariance) forysical states. Physical states then correspond

to

o ol y classes of

, it the trivialclass containing all zero-norm states of the for

I

>. It can be shown 1311. when QB isnilpotent, that the general on-trivial solution(i.e. up to a null state

I

> ) to the firstitio

i

(3.92) is:I

> $ I

( )>

I®> + 1 ( )>

I + >a

(3.93)r I () > denotes states it no ghost

e citatio s.T er fore, these states fie entirely i

(1 ,

]-(- )L . ) . (3-89)

I turn,becomes:

ic thus leads to identically tysic conditions r I () > si the ol

t

uantisation of the theory (see (3.36)(3.37)). Note that if we had defined

,y ical states as having ghost charge (+1/2),Virasor conditionn involvinst 0.0 would notbeen o taiis shows

ositiv or ito-one correspophysical states I

tion of tI->

t1tr

tav~T

hostof zero-norm. Note that t

so belobutthat t

ttere

statesf t

I >=I 1-1> I-> .(3-94)

the first condition i (3.92) t

ft

io . Actually 1311, tto of strictly positiv,

transverse states.led The s(3.92). where the value (-1/2)or

ordering i

ct

implies-trivial o ol

class

t

f states, ss

of states i t

r as positive northe of covariantsuits as t

ever, is completehosts e t

ra

etris tion invariant

J. G®vaerts/String theory constructions

203

e .that physical states of strictlybot

approaches are ien . Moreover . zero-nor

in the old covariantstem correspond ere totrivial

o of

yclassI - > =

I

> )

hic

satisfyt physical conditions (3.92) .

tes I(i .e. IT invarithese states I c, >01 - > are free ofcitations, satisfy trivially (3.92) but

rstawsI ®> I+>

to the trivial o of y class ofey have ghost care (+ 1®2).

physical states areroach leads to

T approach.

ei that it inclues

e-fixing isyste

v

sa a

wi w

1

a

Y

XTIT,

. . ei cl t discussion f

tisati

of the bosonic string, witformulation f constrained

t

1

h ITT ues i

(3.88).

t

tipresented er t~b - t string,

o

kov string 1321, since t

rl -sheet metriche woit in the non-linear or linear formulation) doesnot

l

as a

y

ic

e

f f191. I

other words, t

f

ato the bosonic string, even with the linearaction, explicitly preserves the Weyl invarianceof t system 19). Only r

is iinvariance is spoiled by quantum anomalies,which vanish precisely for t

critical values(3-88).

This result is to

contrasted itolyakov's

ti tio of t

itrou path-integral methods 1321. I thatcase, the quantum system is defined such thatii, explicitly preserves reparametrisationinvariance . It i

e f

ever, that theor

mode becomes

y a icle precisely through t

s

or(

yl) anomaly,

it

the

or (D-26) 1321.that case,

eylinvariance (or

u li

of tor

mode) ca

rest

only

tini4 WO'

st

'ant

tis tL

40-92)tt s

s

®204

J. G®vaerts/String theory constructions

s

1341s

s

ti~

't tt

t

ofT

it " t

S .s

si t,

sit

tt

.tr

d

ste - tf view. t

,Cent(,c)f

0t

ttt

os c sect_ .described

re

r

t

s tir

filinst f

ltri

s.

stir

ttension o

en~~ls that of t

~atte sector6 . t

skost sector . Tinro degreesother iti

t on tzero-

l

t tcontribution o t

I;;I

r p cis,c) ghost syst °e-rm, . Hence,

t ct t

tri tlo c--

a precisel

t restriction)f freedom of ,,, rr, following fron

t extension,los i nv e of

r

c

tfiter

ro o

uri

stri

theory, determines the mass,

Otrle, for besonic strings considerednatter fields 0 are regarded asrdinates, the first condition fixes

sio of space-time to be D-26 (eac,scalar field contributes one unit to the centrFl,charge), and the second condition implies that

lowest physical statethe

i

tac yo .

.~

ofthe scalar fields however, may rdther beregarded as internal degrees of freedom, takingtheir values in some internal ("compactified")space, such

tor

for example (se

ef. I Ia

references therein).

this

y, it ispossible to construct consistent bosonic

theoriesstring

i

y space-time dimensionless t

Therefore, t

values (3.88) are not criticalval

for the dimension of

-ti

oft

physical ground-state,

u rather, they

et

iti

v

f

consistent quantumformulation o re

tri io (eyl)invariant bosonic string theories . This remark ist

sis o the recent constructions o fourdimensional stri

s.

se

or, sü

ace-fifl

r

w

5

t

t tt r®® f 1° its

.i .I ts s

st i ri te

I Si~

iscus e

t le

a

e,

s icetri ti

i v ria t

irl -

t,

r

ivale tly, travity t e rie c

ltter i l s.

e

a

siic

s

i v®l

a ®rl -stens r a

eass i t e c®varia tfr lti tr te r e

ulti liars

®f

e

c~ns ai .

Ga esy

etries

t

a t e

ies ca

e

ai tait t e ua tu

level e ly f®r s® e criticalva ues ®f t e nu

r ®f

alter fiei s a

ft a r u -state

ss ai e vai e .

t eer

1~ n

t en,

t ese

sy

etries

a eecessary f~r

u n u

c

siste

f

esaeries, a aly i a cwa ia t au e, f r t e

e a ive- an

er~- ®r

s ries1

litu es, a

i t e l°

t-c

eace-ti e c v ri

ce

er t e

ile ra ( ®r refera ces, see

efs.[ , )) .stri s

can easilyfr

a sï

ilar

i t ®f vie

.

urletries are e te

t® inclu e 1

alet y tra sf r ati

s, s

t

t t esere actually t

- i e si nal

a tuwity e®ries .

a ®rl -s eet alterre t e

n®~.

n y

~s nic scalar fiel s,ls

s i

t! fer i~nic fiei s, c u le

t~ aer ravity

ulti let, i.e . a

®rl -s eetwit

a

ravitin®. Gau e sy

etri s ®ft e

c rres

t

letry

t a s

ti® s

a

1

itrisati® s i

t e

rl -sheet, ®rcisely 1 al su ar- e ara etrisati ns i t er-

rl -s eet.

e

real° e

at t e

tu

l vel, t ey e s re t e c

sistery, in a c ari t er i a l° t-c

f®

l ti~

(f r refera ces, see

ef.[ l ]).cti

f r the

i

i

tri

is

= ,i al) su er ravity c i

f

alter

tntdfr

t t

ricti®

e ric, silt

c

1

ausst~suielu

sur

t e

syst~su arsyre ar

suae

ysice,f rScar al

ivst

eet syarsy®ries

r

J. Govaerts/String theory constructions

fial s ( ( ),

( ))t

s

r r vittii ( ~),

)

t° ,s

.lrfe c teei ).

r,

)tf

° i

sc i

es

f f

,

-' )t eir

e

i ic

=1

t e s,- e

i s,rass

ria l .

,

( ) i

ls

et

ei

ei

(

it

,l

=

,1r s ec ively

e t

si ices),

a

t )

i

vit°

=1

s er a

e ,1 -sc i e si ® ilass a

va i l s .

c°

ts eet

i r a e

,l

tf®ll ~

real r r~s ti

f ta ebr

-

0 ! $ Dl B

°

in

i

etric, re

e e

i

iv

y

/ 1 j- .

iS t

case f r L

is Str°t e i ar acti

s

iti al 1

ylsy

etry (i.e . l al

c °

t

s eet

etric), t

ctif r t

ri

as

al s

er- eyl sytry Qf r

ils,

see

ef.[ l ).

It~a

c

a i nt

ilt~

f®r l ir, t s

y

r ~

rea

eft e

r vits e

i 1 t

ec

le, s

t

t a ca

nicaiti

tie

t

syste

, eit er ie

i

rit

i ti~

r it e

f r ~1 ti

,ill trivi llye

yi

i varia t .

lys

er- e r

ri ti

invari ce

ay e r

t r u

aan ali~

f r t

s is tri ,

ecfi

i t

~ r

1s

rc

i vari

is

still

e iicit sy

etr ,r te

t e

urierf t

c st i ts f l al

a etrisati®a

s e syr

c~ _

2~5

200

t

Lg~

r ,I1

' I

~

~, v ,i1 " 1

,

1 ~

1 l

1

t -1 1 ` " 1

1 '1 +_

11 11 ~

~

' 1 !

1 1

" 1 I '1 ~ " '

I " ' 1

nTI I

1

Y.s

1

,1

! I

-

hl

I

~tl

Ids

Strt

rIs

i

t

w®s

- s

ett

tilti lie s

ts f tr vit

s

r

lti

t,transform as suc quantities

ers i

t

constraints . n tti_ ti ghost

is introduced i t sectori

the boson. ic constraints, and anutin

ghost system isrresp®nding to the

straints. Under localtr , all these

ost e rees oftransform as irreducible

lti lets of

-1 s

r r vity,

it

t

e-fi in

functionfound

the extended phase-space .existencethe

®f a s per- ri vfor the spinning string 191 . One may

3't

s,l

ts, generatingti .

ss iti

t

t

J. Govaerts/String theory constructions

Its3310ftsonic

ri ati sfer i ic

S

same rtheI te

S t

W '1

1 I

"

11

1

' 1

1 '1 " 1 1

Y.I

1

1

1

II " 1

""

I

' 1 Y'

1

1

1 -1 - "" 1 1 11 .

" II -1 1

1 '

" II 1 i .

" 1

"

"1 ï

' Ï

1 . 11

1

sl+0 1 ~

"

' o II

? " 1

Y "

~-

?'^1t

1 "

"1

Y!

" tl "II FJ -I

1 II

II 1

Y"

!1

" 1 "1 'II II '1 -

.1 II 1

- II ~II II -VI,

' 1

1

1

1

Y "

~1

Y

,,

`I1

3

l

Y"

1 II

I ~1

"

"1 ~1 !11

m a

! 1

~

~zY 1 .p1, 1

1

i

1 !

~

,.1

y

4.2. Old

vabiant

uantisation

efore considering

T quantisatinn itself,

let

s briefly rovie

the

iresults from the

old covariant ,tis ti®

®f the spinning

string .

the supers® ®r

l gauge, e havethe action :

" jd2t [a V* W A â aa&%p 1,

(4-3)

thaj®r

~4 . ~ t

Ia

lea

a~~r

sus

er

is t

~rl -s ss~

i

s i

tricia i-,¢~. an

~ ~ ~ ars

®rl -s et

s in®rs, wit :

Fir t et e sathe fer

wherec~nse

re r s ntati®

t . ~ ®f t

'ca s i

r

satisfies ~=

e a ire u

s t~

t® the e uati~ s ®f

®ti®

( ~-a ~

~

,

iii®:

Ther st

ercurrent :

= S

b and

uence ®f l al su

9

. ~.aOVÛ,~i$S/sc1~8°lll~ t~1~0I',j~ CO%dS$f~C$IOI~S

`~®â

( .4~

~4 . )

. s

erst' etry

su

r-

yl i

i

. t

ti

f

tir

iti

fi 1

1

s t

t

str ° ts :( ®~

i

° ar y, t

ati

f

ti

ti ei fiel

~ i

°

t e

str ° ts :

r t

er rrs ara st isa i

s

is iv

1

t e e

ati

f

ti

®t at t e ua titi s _ _ a

e

r e

licitly :

(4 . a)

currs ts .

®s®nic

e rses ®f frse ~

, we

ave

(4.7~

re

art

r a l

f

rl -s

et

e ®un ary c~n iti~ns as in ( . ~ . ®r

s ers~r

airy. T is

f

i~nic

s rees ®f free ®

, we

ave (see

Sc

arz s et®r ®

t

t f

sect

sf.[ 1 ] f~r

stails) :

his

rs

®f

s

ersy

etry

is

ls

i~ ~ en stri

s:

( =4,~) ®

( =0, ~)

rslats

i

t

rss

ce

r a se c

f tac

s

h ( = , t) _

(a= , ~¢) ,

( .7a)

in t s c®rres ®

i

ysic l s sctra 1 .

with r=+ 1 f~r t e

a

n

(

~ sect®r, an

~__ 1

®r

t s

® e

sari

,

t e

s~l ti® s

t

t

fnr t s

eveu-Sc

arz (N ) secter.e uati~ns®ti®

ar :

ü) chse

strin s:

it

~® - ll,

~ 5,

~- ~ an

~~.ll. res e

iv ly

f®r the

5- S,

-I~S,

-and

-se®rs f t

it

f ll

i

ati

cl®s

strin .susu

f®r cl®s

stris,

se

fr®

( .

) t at t e st u

re ®f th t e

is

iv n y~ a r~ u

f i t- an lsft- ®v rs .

sit

str ' ts (4.

( .1 )®

t)~( .lit

a

a

~

®a N v

-Scz

L

,

Jt) _

ac

s

r in

n ently.

t

.5)is

sn

invari

t

f

r

~l

t

~ = Lt

l +

t

l ,

(

.1

tr _ sf r

ati® s iu

t

erst l is

ivs

it .1

3o

®

( .13c~

(4.13 ~

208

J. Gova,erts/String theory constructions

r-

.)

( .

-i(-) (4.14b)

), ( , ), Mr, )

1®), respectively for -r

ses

-class constraintsy the use of ° c

f. 11).

i

so,tors :

ra

r

(4.13h)

(4.14c)

(n~) -,

,

,(4.15a)

(4.15b)

13) and ®()'

()

ressio s 'terms f the left-

c,rd

sItatfc

t .t

ts

t

f r ac

cA t

utclri

r

fitissicals o fr

(-3f fr

ich mustrackets If 41

tains the

y tel tio siss

(4.13e)

spaces ratherThe corresponding Fock

are°

to describe.

t

;sector of the open string, we have Fock vacu

(4.13f) 111 ; >

(),> is annihilated by1) . I

>

momentum

i e st t s,

(4-13g)

() is

representationn for the Cliffordf fermionic zero-modes do:

(d§, ®)

(4.17)suc t

(-) -

(4.18)

ere, f are usual

ir c matrices ii sio . hic satisfy :

finirator

of ( .13)rre pon _ini).

tly,

l states i

the

on

sector ofthe open spinning string are space-timefermions.

I the

even- c arz sector,

e simplyve

vacua I

; >,

it

I

> annihilatedy

,

(

-1, r a:1®). Consequently, all statesi this sector f the o

spinning string respace-time bosons .

For close

striais, the structure of the spacef states is obtained

y te suri

these spaces(except for the

sonic zero-modes).

e thustai

space-time bosons in the R-R and NS-NSsectors, and space-time fermions in the R-NS

- sectors.ormal

ordering

o

these

operators

isfine

s

fore. All creation operators to theleft of all annihilation operators. For fer ioniczero-modes i the a o

sector,

e choosea

t

tu

s e -Virasoroy t

or

ordered expressions(4.15), one

y compute tsuper__ or

al e r s (see theis leads to:

I L X)

X(n . ORPI - (n - m)Q

T is sattarse

reobt

is

lest

re

a atria tip,ajar

- eye fer1/2) W thebra (in unitse QI)) .hysical states iisatis

of the api

v

qu

R s

10

a

,r: V

(

s thatsector breassociated

etrisatio.tr

n

On'll - (n - M)L (X)o

+ I D n3 6n.M.0 ,nm 8

14)) - 2L (2 r

W]

( 12

+ I24

etth

FW -0

J. GovaertsISt-ring theory constructions

209

ï]- W -

)LMR n-WI. D, 1n3+n I 8 0-

(4 .21b)2 n-ra,

IS sector :

11)) - 2dris) + 1D (r2

qs'or 3

as 2

4

I L(PS G(X)I

n - r) G(X)n*r

W2 - 1) 8.. .n

(4.22b)

ere a. (resp . a-) is the subtractiin the Ramond (resp.

- c

r! SecThe corresponding no-ghost theorem 1371en imolies:t

sl these results, it is possible tscribe III the physical spectrum, both f,en and for closed strings in all their sectors .

In particular, the GSO projection 1381 of thesetheories leads to unitary and space-tisupersymmetric spectra, free of any tac s.For more details, see Ref.[ I].

I'o- a,-0 1 a--2

(125)

4.3 . BRST quantisatit us 59 rewrite t

osonic string in tsolving for A,0,

(3.24) and (3.57)) :

Lantum effects in thee symmetries of t

W world-sheet super-invariance. Indeed, wesions

due

to

quantum

S(bosonic) - Jd2k I -4

alptebras of world-sheetorld-sheet

articuler that ewtributes a vala to Vil dsoro,

re a scalar contributes

--- b =J

> - 0 = LT 19 >, n a 1 . T a In .th

hich is invariant under transformationsinduced by the BRST charge (see (3.58)) :

( YI) - 2. ) I W > - 0 -(4 .23c)0

(424a)

WXA) - a- ) I Q> - 0 -(4 .24b)

b.jdr, - a.) c' 1 . (426)) C + -1

ownic)-

do I -I- cli - x)2 +J R'ffino

a,,ebthe old covariant I 2 1+- -Q + 11 .+ c ..+ acing string are defined

(427)

or: F(1XP I w > - 0 - OP I w >, n 2 1 . (4.23a)

For convenience, let us intrtation (which is useful in t

(423b) butorld-sheet covariance of matter and ghost

21V3

s t

te :

~~ce :

a

~

. ( . )

a

a

J. Gavaerts/String' t eory canstructians

f' °

1 °

t l' t°

t .

° 0

r, t

str °

s ~ . ~tly e ress y

d

~

b~~a_cA e ,

t .

)

( .1s :

~ . 3)t rs

s er

tt

sf r ati s

e

fi

. c)e

t

e a va t

e ®f t e

l~

1

®1s

ersy

etry e

licit i

t

s

erc

®r al, let s i tr

ce t

real rass~aria 1 s - a

ß, a

f' e s

ersy

etrye

rat rs

_ a

4

ic

act ®

s

erfiel s :

a ~~L®i~t

t e

c®r~ s

i

s

~c®~ ri

te ivati~es:

a=a

~~

-

i

a

.

Let

s re

a

ere t at it is t e

®

let

r sf~r s as a =

a1®ra a s i ®

er 1 al ®re t tr sf r ati s . it

si er

ere

licitly

es i

s relatet® t is sy

et

f t e

te

, t 's

~tati ,i c~ v

ie ce is

t i

rta t.

efl

e acti® ( .3 )y t e

e re ritte ilici !

s

e

etric i v ia t f®r

® ~ + i_a_o t .3a)

~ i

t® the

attar fiel s ®f ts i i

stri ,

e t e i tr®

t

rea!sc

r s

erfiel

~~ , (4.3 ~

~f ~ti frte ui'ary i1e

y si

ly i nre it i

racti®

( .3 ~ is t e

t

t ® tstri

in tne su erc® ®r al

u e,is i variant ~ er t e su ersy

etrysfcr

ti~ s in uce

®n

( ,

)y t esu erchar es - an

¢ ( rovi e

ne uses te uati® s ~f

~ti®

si c

set

= ~.e c~nstraints (4.33) can als~

e e

re seter

s ®f the suprrfiel s:a

a

In ~r er t~ c®~f the s inni

,

uS Il®

ere t

re assr ther itia

ass

_a

~. (~®vaerts/String the®ry c~nstructi®ns

2~~

d

I ee ,

is

ity is e licitly °

ri tr

crl -s eet

s e sy

y,re

ces i

t e ( , c) s ct r t

t"

cst acti~t

b®s nic striT

t

tal acti

(s

~ '~

~

~ f

e s

e eve :

stri

i t s er

r al a e i tiven ytes

f ( . )

( .4 .e

ati®ns f

ti

e ( ,c~

st yare as i

t e

®s

ic st °

,

ly:

a $c =

,

_cs ~

,

(

f®r t

( , )

st y te ,

lve:

~a_c~

ese re ti®ns eter i e h®

the ®s~ ic a~~

a, __ ,a_ $~=fer i nic c®nstraints tra sf®r

u er

®rh

The ~un ry c

iti s f

t

( ,c~ ysts eet supersy

etry.

-re

as

in

(3.

),

ereas

t elate t e

T f~r ulati~~~

c~

iti~ s ®r t

( ® ~ yste

are i -

ti l t~strin

in t e su erc

~r - al

t ®se

®r t

fer

i® is

attar fi l s

s° ceintr® uce

h®st su erfiel s:

t ese t ® sets ®f

®

ary c

diti

s

re(e )a (~~i ® (), (4 . a~c®rrelated

t r®

®rl -s eetsupersy etry.

e

er

itian Grass

a n fiel s d~, ~, _

e

enerat®rs

®f

® l -s eatiate

t~ t e

ns~ ic c~nstraints (t ese

supersy

etry

a

crl -s eeth~sts ®f the

~s®nic stri

~, an

the

re ara

etrisati

i

t e

t sector are

iv~c~

uti

fiel s

±±,

are y :iated t® t e fer

i®nic c nstraints.

T+

_e acti~n fnr the nst sect®r ®f t

®°spinnin strin in t e su erc® ®r al au e is

(4. 4a)

,( .4c)

212

s(s

t

3

c

t

rators

racS4

es

ritities just introducedthe total spinning stre s

erco or

ring (see (3.62)), thit l

r i ic constraintst

)

s Riven

r v(3.63c)).

T+

( B ) )

. (4.46b)

i

the Ramond sector1/2 in the Neveu-Schwarz sect

) . (4.44e)

,,9 . Govaerts/String theory constructions

Dl -sheet

(4.45a)

rct%

t

t'

s for

iss

t

solutions t,+ a8

sly (see (3.7), (4.12b),r t

s

r

os s,

e-iq , (4.47b)

reviously . For the remaining quantities, we

D

c_

close spinning string, the solutionsequations of motion for the matter fields

the (,c) system re as previously (see(3.8), ( .,c),( .,c) a

(3.71 ,e)) . For the

superghost system, we have :

Y'(,) - 12=~ V,, e-2i

ß ,q

(, -T) - 2%F2q

e_'

®

(4.47a)

with as

fore (-n) e

or (-r) c

+t/correspondingly for

o

or

Neveu-Schwarz sector .

e then also obtai

(

)

T

(D e-2in _ ®

( ¢)

E(-k) e- 1

I

(n)+

(n) +

()+

() )+

Y_

(q) + ()) .

( .51)

e-2i

,c,Y (4.48a)

q q a (4.49)

(4.52a)

(4-52b)

( .52c)

(4-52d)

(4-53a)

2

9

k=a, X. `°'10

ef" e

s i the open stringy sector,

tical definitions in terms

q

,

. Here, L(k)n and J( )

al. G®llâh?ltSI .S$%'illg theory constructions

213

'

o`1

,

~

lf ~ 1 - ~" 1.

"' open 1 1 closedcorresponding 1 f'

II " 1

1

` .

i

' 1

1

1 ' specified

1

"

1

1 1 1'

s however remark that subtleties arise i1

~

" "

1

1'

r'

. 11 1

1 '

¢® do with the infinite de~e erac~r of the F

"

1

1 i

t " 1'TI' 1

1

"1

1

' 1 1

"11

14,51.

NorI~I s- ;

1

" 1

1-

in me YRlj,-p,

i;s c

a~~l

-

" 1 '1

1 'ÎI

~

1 ' ~1 11 1 1

1 1 "

1~ ~

" 1 1

~

1

1

' I -

I -

' 11 11 -1v ~1 '~

" tl

II ' !

w Iw o

ly" r l' " I,w 1

" 1

II

1

"

1I

"

-

II 1 :IS

I

I

il a . ~jF

,

11 1 1

I

1

1

1 ~ " 1 1

`(~ "

,1

I1

"

`

1

i t n

,,1 °

~1

1

1 -

~ 1 1

1

I-I1

~fi1%u

I_1

~ 1

1 ~ 1 "

I . 1 1

" 1

I 1

I1

1

' 1

1

1

°I iy

"i , ~1

1

'1

1

I I

` TI 1

1

1

"

,/~ 1 ^

~

~, ro

I

~ "11

`r "1

1

`

.1,

I71,

t II

"

- 1 '1'1 '

' II

` III 1

1 '

" 1

' 1

1 Ii

1 , `1

1

' II

1

,~vi

I

,

I1 '

" j II 11 ' t

1

1 I1

I I

1

' 1

" It

-1

1

~"1t °°~'1 '

~11 " / 1

"" 11 11

1 1

< 1 1

1

1

~1 I

" 1

1

1

" 11 " I 1

1

1

r.

';1 't 1

I '1 1

1

1 ' " 1 ' I1

i _" 11

1

,

! 1

!p

1

1

1

1

I

11

1 1

1 1

1

1

1 '

' 1 1 "

"

' " 1

II 1

l '

"

`1

II

1

" 1

1~

~~1

I I

1

I1

1

I I

11

~ 1

~

'IY

{1

1

1

I 'II 1

I 'II 1 1 '

. 1 1

1 1

I ~1 ' 1

1

' 1 1

II "

` 1 1

'

1 1

1 1 '1

II 1

" I 1

/ " " 1 '

1 71 '1

1 'II "1 '

"1

~ 1

"

" `

`1

(I "

. 1 1

1

1

1

I I

1

`1

I

,

"1

1 '1 -i

"1

,1

I Y

" _ 1

1 1 "1

l

~ f

°I. (x`®V~,~I°~S/5°$ï'9I1~ $~1POPy COIISÉI'11CtIOIIS

9

0

( . )

t ln ),

rt t

(4.6 )

tr~~ )

B

{~r), ~~

()a

~~~ _ {

- r)

let°

it

ur

evi

s resul s î~r

~~1,( ) a

L~ h

e ~ t ° î®r t e t~e erat rs (~

Ct~? +

tr)):

L ,

~ a(

- r)

n+r

r~~ .L =cn- )L

(4. 6)

e

° t e su er

~r

a ®

~~es ~ taveu- c

are sect®r ® ly vanis

f®~°:

10 , a_ ~

{4.67)

ese are precisely t e c~n iti®ns i4. )f®ll® i

Fr®

t

tt®-

®st t e~re~

i t

®1a:waria t a r~ac

~a

i

the li ht-c® e

It

ay als® e s ®

that

is il ®te tprecisely ~r t e sa e c®n iti®ns ®nly. In ee ,in t at c~se t e

eveu- c

ar

su~~erc®

®r

ala e ra

it

n® central e te si®ns can then

e® taine fr~ ( . 1) .

i

lly, let us si

ly say

ere that

ysicalstates i

t e

a tise

s i

i

stri

arefin

y t e

i varia L c~n~liti®

itio l condition restricting theirs er ost charges ( like i

the oso ic string). Here again,ho ol y classes defined y (4.68)

sponJ to strictly positive orsverse) physical states in the old covarianttisatio 1391, whereas e trivialol

class corresponds o zero-norstates.

e additional constraint on the

oser of these states is

ore subtle, as it isrelated to the problem of picture changing ithe s per

ost sector.

e 6.. aot consider thisquestion ere, s

e simply

ante

to presentthe

T

ua tisatio

of spinning strieories,

discuss o the critical values(4.64) a

(4.67) follow i

such a

approach, sothat the same spectrum of physical states isseen to e obtained as i the old covariantapproach.

t

a

ost(3.92) ftrivi

rr(r

4.4 . CommentsThe same comments as for the b®sonic stri

can e made ere. Without going into theBails of

the

at li -Fradkin-yilkoviskyformalism,

e derived the

S

q

tis tion ofthe spinnipg string in the s erco or algauge, by tai

advantage of the global

=1orl -s Bet supersy

etry preserve

y thisu e fixing .

e derived the conditions underis the gauge symmetries of the sYsteely super-rear etrisation and super-

eyl invariance on the world-sheet, can bepreserved at the quantum level. I thesuperco ormal gauge, these conditions reduceto conditions for the cancellation of the

or al ano apes in the =1al algebra .

these conditions are satisfied, thepreset uantis tio method of the theory ise uivalein to Polyakov's approach using pathintegral methods 14®1_ i that case, super-repara etrisatio invariance is explicitly

supersuper

e

J. Govaerts/Strlng theory constructions

<.tE ~ r

I

11

r " I 1

, .IF .I

1 '

" 11 '

I

i

I

" 1

"

I '

-j1

I 1 " 1

!

I

_

m _

,

, ,

- 11

I

I -

1

F,

!

,1

1 l ' o

, - -

- ,T

1 "

1

, '

! 1

1

1

1

1

I I

1 '

'1 "'

1

1

1

` "

' 1

I 1

- II

1

1

`1

.

" 1 v

1

! 11

1

11 11 l'

' = r1

1

"

1

1

1

1

1

1

1

.k '

l'~ . ~.

ti1

1 " 1

~- ,

" ,

I

systetisation of any Vtriaat~ and

is broken thn~ch preciselys as derived

res the

amis t

2115

. GovaertsIString theory constructions

ts

t

t

sS

r

tr

fr

t' e(f

references, s;string 191 and t

strifl'n% ,~i'Exe first considered i®i6s

f

tthe r s

t

,it

ts tiattention

I . , through

r t rrld-sheet

s t

for

C

r

ti

ss r t s

t

Ory

cc

t

r

strir

rres

cs

t_ tIt cl

t

1

11 . . l

, . 111' 1a_

s

a

t

et

t

ory 13

for lis

ic

s

~ erSul12a

Sulrief y

tt+

sr

v

{ II II -

-

. r ; :~ 1

t

II li

I

" 1

It AY

f?lo wefor

purposeic string)y i

®s t

foil

D tr

t

_... °

lesson to

1.

1

-1

ies in th

t

1 .1 , .1 1

II '

v 1

1 ~

1 W ,

, . .1 .

1

1

1 -1

"11 1

~

. 1

1

" 1 ,1 -

âFI1

-

'

I1

1

1

1 -1

1

I

-

I

11

-

-v1

1

1 1

1,

I '

"I I

1

1 'K,

T

1

~" I

11 1

1

.1 I 1 I1

-11 ,

' 1 "

"

i1 `1 '

" II

1 '

I

" 1

I

lo

.

- 1 1

~ -

7 .

~ 1

I 1

Yt

EMENTS

t is

pleasure to t

.r

o

is

invitation to l

this

j

tici a ts for their interest and theirstimulating questions . Part of this work waso

ilt t

str iUniversity (Canberra) n i Sydney. Thanksre also

e to Profs.

.A.

o so

a

Lj. Tass!for their warm hospitality, a

to

r. an

Mrs.Leroy for

most

e

ora le visit.

is

I

I1 1 ' 1 a

~W

1 ~" 1 II

iII

- 1

.1

1 1 " ,1' 1

" W

" II

1 -11I1 '1

1 1 `1 :1

I ~Is

tt~ et

r t® s,

t

i~

te tu i r i ist e

ec

ts (

) are is !er ~

®siti eears . is ~tati

ut.ee

cai te

! :Tf ) ~°f~)

)

f ) =

f f ) Tf )) ~ lTt

) Tt )

f

!e D

J.

®vaerts/String tiDea~ry c®nstr~c~imns

is re a i® , t

left-

d si e is

erstr i

~r e e

r

uct f t

t~

t . )e, ®

t e si

ul r ter

s

e

ave t e

i!

i

si

s:e , ai

u h a ' i iteers

®

( -

)

als

~

)f ) ®

L~

~z®

-

D

D

D cn Y n

f

.

)

! c® R~e ti

i!!e use

uivale ce et ee ( . ) a

( . )easil

be estabîis e

c® t~urrati

s (see

ef. 1

r

®re

et üs) .calcu ati

®

t e c®

®r

a!

e rasic

a

ear i

t e

a tisati

s

ic astri

s t

s re uces t® ® tai i

t eic

r!d-s eet ® erat rs. Let ustr

uce the ®I!® ire fie!

® er~t®rs, i~tati

s use

i

t e

ai

te t:

c(z) ~

c

~ t

,f

)

z~

_

~

(

.

)

~r

' itr :

tf)~ : fi z)) :,

f .i)

.)ccifz) ~ :cf )fz) ~ cf )

)

:

, f

.

)

i ~fz) ~ i :

)

tz) ~

z)

fz)

: ,

.i

)

z

~ ~ _ - 0 ®

a ®

n ( .i

ere ~ ~ f - ,~,~, )

c ) ( _ , ) are t

a tities i e i t e ai_

t.e

f t se

e t r ca e sil®ta°e us' ic' tere a te

c

e! ti

u cti s

a e t iiei s .

si

t e

ta i

~

) (~~) ~

~ t

ic

c tr c~.i

t e

® r re

c

( ) t )~n

t

if

)

~

: fi

fz))

z) :

,f

.i

)

t (z)~ :

( ~ ~~t ) : ,

( . i)

2~?

t

ae~e :

ere:

sea r:

) . c=

°Sc wetz Sect®r :

l(z) :

r th

c=~l1, =1 . ( .2)

f Jt )(z) ( ~=X, )

it' itself, we

T~

(y) ~ _

arz sect~r :

~(y) ~

I )( ) (

i .I

fi s:c = ,

t~-y,3 ~ c~-y)

l( ) = T( l(z) + TI I(z)

,

l~z) = T(c~~z) ~

( ~(z)

J. G®v~erts/String the®ry c®nstructi®ns

,c, )it

t e ara

ire si

ply have:

sin

the s

etechniques, nil ®tency ®l' the

char e

ay easily e c ec e . F®r t eb~s®nic strin ,efinin:

J

os®nàc)() _ : ctz) (

c )(z) ~ 2

lcl(z) - az-2 l : ,

(A .39)

~ ~~®i~}

dZ ~b~s®nic)_

~i~

(zi .

(A. o)

In this f®r , it is quite strai htf~rwar t~®btain the result ( .

).

( .1 )

, 16

i t ev~e - c a z sect~r :( . 1)

( .2 )

F lly, we als® have :

( )c )( ~ Jc )(~~ ®~~ ~~ ~(y)

. i~_~)L

(aty)yJ~ )(~) =, , , (A.3 )

i e

et is iirres active ® t a ®n r aveu-Sc warz

sector.( ). r® ese results, all c® ®r al and

superc® ®r al al e ras c®nsidere i t e ain

tent are easily derived. I ee , curresp~n in

t® ( . ) e eve t e iras~r® al e ra ( . ), and

, ( . )si ilarly fr ( . ) and ( .3 ), ®ne ® tains:

( .2 ~

1®fstress-e

()

e

S~

(s i

it

herratio,i .e . s ~

~c,t e®ry

T®r

resul s iT1 e

r aler

ly, f r t

e s

i

~

str~

~

s® ~~®,,~.

_

_iJ ~ ~

i

1(z)

,

(

.

1)

pis a rati~,anai®n

r its t

se ®f

®r al fiel

is als® equivalent t~

c i

il ®te

® t e

ver, r® t ese e ressi®ns f®r thee, it

e

es ve

ea

t®

e ive( .54) an

(4. 1) .( .3 ) is a~tually very eneral.

t®r

z) is s °

t® e a

ri ary fielei ht h (

] if its

ith t etens~r T(z) is:

J. Govaerts/Stringy theory coaastructloras

z)l~z $ia z)+h(n+l)z

z). . ( .44)

Let us assn e that z) obeys a brun ary~®

iti®

~f the type

being

- ri e integers, and t atal nu ber.

e then have the

® e

z-~- ,

(A.46a)s

the su

ati~n is

®ne ®ver alial nu

bers ®f the f®r

s = n+( /q - h~),( / - hr) ( h~ is h

inns its inte5er

art, i .e . the rati®nal part ®f the ~® ®r ali t ).

e ® es are iven y:

(ß.46b)

( . )ist

s

tt tt i

ti :

re, ( . )

( .s

t t

re

e ri

fi

-

v rat (i

( )),

(z). ( 1,

(z),

)

)11 -ri

fiel , v° re ive t

(z)

(z) e av

/ _ ,

f r

iel(z),

z)

( )e ave / ~®se r

/_ ° te

-se .

sai

ve, J( )(z) is

i

fi i

e t

t T( ) i ( .4) is

ri

fiel

it= an p/q=0

e t e

tr 1 e

i sis i entica y.

e

ay ener lise t e ( ,c) a

( , )

st

syste s t a syste

f

ti

ri~

ti

a

fiel s f ar i arati nal e

®r al

e' t

~

a

1-res e~tive y .

et us

efi e t e iel s :

a

b

s~

er t e su

ati n is

rf

v rra i®nal nu

ers

it

ein a rati

u er

t

-1 <

~ 1 .'s is t e

st e eral si

i

s c

t at t e

sal s

el

a

se se.

ly,t

ases ~ a

~ ~d2 re

ti ul

i i ns res e iv 1 .e

iel s

c(z) ® .

s~z-s~-i~- )

( .4 )s+

2i9

220

J. G®vaerts/Stria

theory constructions

t

ticc ~

(resp.

1) for commuting (resfing) fields

z)

(z).e

that c(z)

(z) satisfy(A.43) it -41-J)

ctively, and that the OPE (AA is obtainede following central extensions:

tin._ fields:J(J -1)

,

= (J -)(J +

-) (A-56)

sstems,Y

J=1/2 ' sir

ti fiels:J(J - 1)],--(J -

)( +

-1) . (A.57)

these general rt i e far

for tt

S® D

icesr stre r~a

r

tt[ig]st

so1

ts tw4

.mt

s ro~t t

fi

11

im-- -~

re tl4

i tr icalc l tictheories.am the re

s r

t

s ftdeo

d to ts

t

f

s

it

(su ea)

f strile tr

rrc

t

r

t

si

r

fie theories. Then,

(0,x)

o z

, it is Possible ttirely the formulation of these theories in

t

f t

ri

fields

ove only. This ist a

useful e

rcis

Su

r t c

f variables from (a,-) t

1 . . vaerts, tritro cti

School o

rttcl°are as, 19

(World Scient247-442.

a

Fields,

c -s. J.

.

ucio

.

e e aic, Singapore, 1987),

av

.sits lead t

t

s

Phys.

1 (1956)

3.far the (,c) a

(, y)J-3/2 respectively,

ter fermionsst

. J.

avaerts, An Introductory Guide to Strinerstri Theories,

reprint CERN-TH.4953/88(january 1988),to be published in : Proc . of theInternational Workshop on Mathematical

ysics, Bujumbura (Burundi),t

r28 - October 10, 1987.

M.B. Green, J.H. Schwarz

. Witten,erstring Theory (Cambridge University

ress, Cambridge, 1987).

Volumes.

Frie

, .

rtinec

. Senker,

cl.

5. S. Shenker, Introduction to Twoe sianal

ar

l an

u er

ar alField Theory, in : Unified String Theories,e s.

. Green

. Gross (World Scientific,Singapore, 1986). . 141-161 .

rie a . Notes an Strin T eaty n

Tension

ar

al Field Theory . in :ified String Theories, eds. M. Green andGross (World Scientific, Singapore, 1986),16-1 .

Theories:

. . es in, ntr ucti t® tri a d

1 . . l ~`Supe~ stri

e~ry II,

ie

~ui e , I

t.preprint

C-

-4251 ( arch 1957),

(1

)

.Lectures prese te

at the 3r

dvanceSL

y I stitute i

le

e~~tary Particle

19. J.

vaerts, un

lis d

.ysics, Santa Cruz, June

3 - July 1

,5 .

9. J.

r

vaerts, I

erc e,

. . c elleke ~ aner,

re rint C

-

.5

5/55(Au ust 1955), t® appear in

ysics e ®rts .

hysics ep~rts 1

(195 ) 1 .

t. J. I~~~

.ys.

(1959 ) 173.

10 . J . Gwaerts, preprint CORN-T .5010/55( arch 1955).

11 . A. . Schallakens. Self-dual Lattices in StringyThe®ry, in these

r®cee in s.

, Phys . Rev. I)25 (1952) 3159.

frac, Lectures ~n Quantanice (Yeshiva University,

- ®r , 1964 .

~T. ~ava~rts/~tring ihaary constr~ctiotas

16 . T.P .

illin back, C®

.at. Phys. 100(195 ) 67 .

.

. S®hv'ev, J

Lett. 44 (1956) 469.

1 .

.a

®

e

s aty

t, . er. y . (1 )

~veu a

.

est,

l.

y .

3(1957) .

. . .

er,

ys.

.

(1

) 1

(17) 3 .

5.

2 .

4. C. L®vel

,t/ ( t er1 )/ (

e ).

12 . C. Teitel ~i. J. ®vaerts, re '°i t

- Ii. (1 ~/13 . S,

ser an

. Zu

in~,

hys. Lett .

(aY 1

)®

e

°

y .t .(1976 ) 369.L.

rink,

.

i ye

hia an

.

® e,hys. Lett. 6

( l976) 47 l .. J.

®vaerts,

e ~ Pi

-

.

/

(

st 1

)® s~~

r

° t

u

~is

e

i

:

ra,~. ~v I

I

te

ti

1C 11 uiu

t°

e retic 1 et

sin

ysics,

te.

le (

ec,

)®June

- July

,

s. Lett. 3

(1 71)

1

. V.

. Gribw,

ucl. Phys . I3139 (1975) 1 .I .

.iner,

.

ath

hys. 60 (1975 ~ 7.

.

.eve

.

est,

ys.

tt. 1(l957) 0 .

. Gree

an

J. . Sc

arz,hys. e3t. 49 (1

) 11 ®ucl. ys. (1 ) .

17 . T.

~u~® and S. Ueh~ra, Nucl . Phys.

197(195 ) 375.

. .ean

. van

ieu

enhuizen,

.

i ®~

,ucl.

ys.

(

)

7 ~ucl.

hys.

2

(195 ) 317.

S i, .

®t , .

ra t

212

221

222

J. GovaertsIString theory constructions

33.

(1983)1)261

Y

M

2

37. P. Goddard, C. Rebbi and C.B . Thorn,72)425.

r and K.A . Fris.W. W (1973) 535 .

JJL Schwarz, Nucl. Phys . B46 (1972) 6 1 .

38 . P. Gliozzi, J . Scherk and D. Olive,,LM.B1220977053.

s . Lett. 179B (1986) 347;33(1986)1681 .

ux, Phys. Lett. 183B (1987) 59 .

41 WMakmM. LeU. 103D (1981) 211 .

1 . D. Friedan, E . Martinec and S. Shenker,". Lett. 160B (1985) 35.V .G. Knizhnik, Phys. Lett. 160B (1985) 403 .

42 . A.A. Belavin, A.M. Polyakov arJlodchikov, Nucl . Phys. B24-1

)333.