String theory constructions: An introduction to modern methods

Nuclear Physics

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

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School, ri which r i t construction f fouri ss Bert c lle e s. 1 y e to re i lei with and as a follow-up on notes fished in Refs.[ 1 .21, which were the basis for t lectures. f.111 contains a detail the old covariant and Ifitht ti strings, superstrings and ten dime r ic strings, and o their spectra. Other s f l reviews string theory are o i fs.1 - 1. s a introduction to modern methods s str io s of string theories, the present tes discuss the tisa io of oso °c a strings spinning strings, follo i i to ia et developed or str st y at i , r i a il ovis y ( ) (for review references, see ef.1 ). exercise, first discuss the S tisatio of twi f relativistic c l ti le (for references, s s.1 ,101) . This st already shares , i c simpler r, most of the technicalities whic r string theory. String theories however, possess richer structures which e shall describe s r through t discussion . y 1c o

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

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re etrisati® s, i clu i the ' rl =1i e crientstiûn, the 1 ar eters re u s c [ 1 ).

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cr it, i.e. t ey es i e i entica es ~1 situati® , s seen in (2. 1). 11s c fu cti® s elcn~in tc t e su e ® i ra~ rize y v ue c c e sra etrise s 1 I; t ith ( . ) ® in t® o er-ti e ra etrissticns in ( .6) is ~(~)_~~ ( ® > ). ®lvin f®r i ( . ), cr~ r v rs li e i® ( . ~, whe e iT~ is i t rl -li e ein ein . n ee ~ t e v riati® ( . ), wit t e i entificsti®n ( . 9), s ®n s tc the tr nsf®r ti®n ®f suc a tity ud e re r etrisati®ns . e ter c t us es t e t®t r® eressu e ®f t e icle. c I is le t s e~ercise t® ert®r t ®us lysis fcr t e line r acti® (2. ), res it t t it eve tuslly !es s t® ~he iscussi as s ®ve ) .

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er-ti e e i tr ce a . tl~er c ices re als s i l [ . t a e ve , t t i s e su set f eic ~ ~ler s a e e iti liciti s are c~u te ®it is t eir ~ ie t ti ] i e ,l . c ~r t c ice f f( ) ) is s c case ( > -fi~~i fu cti , say: l . f d~ e-fi i fu cti e ists t ali, t e r 1 ti f t syste suffers fr® a ys i t~ ri ew le . isist ec sef r sri f ecti t e y ®i ® e s al i cuss. illustrate ~ e er i ic e y nevert eless r ee i es ce ®f ri w r le ,1 let u c si e t e c ice ( . 5) ere ( ) is s e functi suc ~; at (ii)=ii (i=1, ), , c are ree ara e ers . In t e errB i e fective acti a a ilt® i , an iet us t re efi e -~ ~ a -~ 1c ffl l t e t e t e li it ~ . is lea s t®: _

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J. Govaerts/,string theory constructions

les out t i labelled c. The degrees of freedom n, q1 and P2 have ili r variables. Solving in (2.46) f tities (this is consistent it f r ti ), obtains: t

1

48)

T

invariant system 1191 2.4.

a ti i system f is defined tLr the (ant0commutat relations t

i

-i o

191

,

.

.

fi;, (2.54)

.

voluti

(2.49)

motion for follow fro

(2.55) (2.56) M i

bissen gack 1(t))

.

r

(2-5V ~

it the boundary conditions (2.49), th solutions in (2.21a) are reproduced, e (44m 1(t), describing thus the s situation

s e 0 >2, it

Ir

limit 7 , i.e. " A ghost charges reduce o -1

,

( + ) . c=-i 2 1 . (2-51) 9) and (2 .50), generates BRST s leaving (2.48) invariant: - . 1y~ + 2 )( .

)

(2-53) )h . .~t 1 ( ~t t h it (2.22) (2 .23). orr s t t) (t). re olvi t ie lul t t es rel ti stitute t rr s e i ci le" eternir , Ji f _t f Wit 1171 . hr

st presented just` i~ tr io . This is the case not only f icl 19,101, t o for string theory

(2-57) r

licit representation t st is r (, )--i for functions )of rssmann variable 0. wit ja . -> :

®t >_

9

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

ysic states itio s: I .t> . solved by I

)

(2.58b) i

CI

IdDqT#(q ; t)I >

.1> -

.

(2 .59)

JdAP>01- >101 -

.

(2.60)

2[3,m21 ,V(q; ,0-O. These states are thus to-one t vi is ti

J. Govaerts/String theory constructions

192

t

(2 .45) .

(s

t

T

i

eMe (2.61)

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f

tion that interesting developments fly been achieved concerning this , i *attempt a ,e t

role

lay

y

try in gauge invariant systems better.

ST

I 3.1 . ra ian o l tio these notes, only t non-linear G to action 1211 will be considered for t oso °c string. A discussion similar to the one low aoDfies to the linear action, where trinsic world-sheet metric is introduced . Let st s consider: S1114T t i

L21

1

2

( .1 )

, °) (® ®)

®Z

_ ®

, (_ .1®)

r t string trajectory Au, t) depends o world-sheet c r i tes t and 0 _-t i w and is the Regge slope. e equations of motion simply corres o 11, to t local conservation of the orl ,n) s et energy -m o e t current ( : (®t) -> , (2.64) t

t ,Z)

satisfies t t

ve the boundary conditions: ( , ti) i(), i=1 .

ti

s it s t s r ti 1 ti -gral representation of quan c is au (i.e. ) invt t fixing fu W21 class 1 ,1 r r e-fi ii Fi function ra r r U ! ft til r s t well ' ral re t

i) open strings : s,

(ei

f.

d

ii) close strings:

sA ~ nia .r

.

e s

1f

(3.3a)

(=0, ; t) = (a-

m , t)

(3.3b) (a=0, t). ( .3c)

u ntities and are not all t. e satisfy the constraints: e 2 ® =0 , a ( .4) _ e

®_

s

.

e relations (3.) are consequence of r etrisation invariance of the action. s

J. GovaertsIString theory constructions

in (23) a d (M), this can be established by consideri the variation of (3. 1 ) under an itesi at woria-sneet reparametrisation

V = V - 50 (3-5 it a) qO(a, 1i) = 0 . i= 1,2 050 b) i) open strings: qI(a-O,x; z) - 0 (3-5, .- ) fi) closed strings: J00, xWqIWn, it (33d) Such an analysis thus shows that the constraints (3 .4) are the generators of worl sheet rapace metrisations . Actually, the pair of constraints in (3.4) are obtained from sac other by a transformation exchanging the rote of a and z (in a way preserving orl -seat orientation) . Choosing to fix world-sheet para etrisatio through proper-time gauge conditions 191 +&2=0 , XO>O' (30 0 We nion-Ifinear equations of motion (3.2) simplify to linear Klein-Gor on equations for assless scalar fields i two dimensions . As A. ways ultipliesi, let us restrict our choice in (3.6) to the or al gauge, with A0 = 1 . Such a gauge-fixing does not completely determine the world-sheet coordinates 11,21. The subgroup of (pseudo)co or al transformations leaves the or al gauge conditions invariant . A complate gsine-fixing of i6jLA%' O yOwALA AO "%,AJLAVVUU All ullw 117611I. WRIC saugu l which we shall not discuss in these notes 11,21. As is well kno n, the solutions to the equations of olio; in the or al gauge are simply D,21 : yen svkgs: Aa, -0=42a'i qP+

e" cos (31)

193

ii) closed strings:

(atd + )i + 0 0 e"("G)

elMvO$ (30

here the integration constants may ex ressea i terms of the boundary conditions (3-3) 191. Note that

qF .

at and % have bee

normalised differently as in Refs.11,21 . I particular, the total momentum PP of the stri is given by cP,,=NF2;; PP for the open stri by

cPo= h2a'P'I=&,, for the closed stri 2

Z we denote an infinite summation f

with n=O omitted; I denotes the same finite summation with n=O inclu solutions These h=ver, have to satisfy t conformal gauge-fixing conditions (i ± x')2 This translates 11,21 into the Virasor constraints: 0 open strings: L W n= 0 (31) ii) closed strings: L's n P= 0 , E'O"= n 0.

(3-10)

with the Virasoro generators: WI = ,

A

-W I

As we shall see, these quantities generate i the or al gauge (peso o) or al transformations, which are the left-over symmetries of the system after sine-ii i why This is the reason the Virasoro generators ust vanish for physical solut 3 .2. Old covariant quantisati Let us briefly review the so-called of covariant quantisation of the bosonic stri using the canonical Hamiltonian methods

J. (F ovaertsIString theory constructions

194

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

Six,

X, X', X-1- 1 72

d-T

do

ith the boundary conditions ) Ao,

191 :

,q) - x ~ (a) i- 1,2

(3.1 9all

Oopen strings: MI3a)

(

ji) closed strings: same values M a-0

.I 9b)

(3.19c)

The first-class constraints generate throu their Poisson brackets symmetries of the actio (3.18). For infinitesimal functions c"la, 0 an Rft,,

s

1

0

(3-14)

tically, as a consequence of tion invariance (this can be s (2,4) for the particle). Hence, the total is: (a) W M04 + A-0 (3-15) (a. %) are arbitrary t (3.15) consistently system tion of to a lis hed from t references

t (3

r(o. )such that 191: (320a) ) (e+0( a. id-0, i- 1 .2 ) 0 open strings: (;L-e-X*r)(a=Ox,,[)=O (3 .20b) ii) closed strinits : same values at 9

It us consider the quantity ) F. . . (a) [C 0 , +

(3.20c) (3 .21)

to variations C"dr. + C-dr-

(3.22a)

2 aJ C.'4r,-t_4r- I

(3 .22b)

i~ - x+E"', + ;L4., e

(3.22c)

i~+ X-C-' - ;LA-

.22d)

It can then be shown, using the equation of otion (3-17a) for these IF, that transformations leave the action (118) (3-16) invariant . With the identifications 191 : T A result s t 6. and 6- are first-class C+_X+ , r-1 -qO-q 1 (3.23) qo+ql for y bolving 0 from (3-17a), it can be ti s of motion obtained fr seen that the variations (3.22) correspond to + (117a) orld-sheet, reparametrisations as define (15) . The constraints ~+ and ~_ are thus the Is foil

from the first-order acti

generators of these local gauge transformations in the Hamiltonian formalis solving for 0 in (3.18). one obtains t r acti

J. GovaertsIString theory constructions

S1

(324) VA-

1 i (ro)

1~ do [ein(vov)(O)+ein(r.-)( -11 do e in(vu) $ . -

(325)

ere go, Q is an arbitrary function. From V v riations (3.22c,d), it can be shown that &0 tr sforms indeed as a world-sheet metric te sor 191. As in the case of the particle, the range multipliers A" and A- thus not only ar trise the freedom in the choice of rl s eet coordinates. but they also correspond t onents of a world-sheet metric. T is result allows us to discuss the gaugefi of the Nambu-Goto string, whic . ... ma. r se t formalism is done by specifying A' and A-. It c e argued 191 that a complete gauge R the theory is obtained by the choice , 11) - AO - Ala, 1), 10 > 0, (3.26) ich corresponds to the proper-time gauges in (3.6). In particular, for configurations such that X* and A - are independent of a, it can easily be seen from (3.22c,d) that the quantity 19,221 1 1 (3 .27) .~ 0 do 2 If2f,-112 difIx is r

te) Ln

195

~3 .28b)

For closed strings, we have:

QN-Ne"'

, r

_e7 3 ,(129) a

n -in

(S) * (a) (S) $,-21JEnEn +cn L n I

16 n

do e2in (,-V)*( _ ) ,

2in(vv) (") E(s)n -I 2 I do e

(3 .29b)

(3.29c) (329d',

The quantities just introduced can be seen t reduce to those defined in (3-1). This proves that the Virasoro Renerators inde ' r pseuaoxontormai transtormations. Note that in the conformal gauge, for open strings we have HTIm =L( 0 ), and for closed strings Hr2(L(O00+E(5)), 0 where it can also be sho

invariant A world-heet that 2(00(®l, )) generates constant shifts i 0 ametris ti s, i.e. the Teichmaller inally, from (3-16), we have the Poiss ameter c erizing the gauge orbit of t rMets: aasece of se-space passing through 110 0 open strings: (L ''" ., All si dering t C, -, .-,-uge AO-1, the (330) uations of moti (3-17) redueto thos G , ii) closed strings: (L*-', n L M_I = 0 t angian for fation for that gauge, th solutions (3.7) and (3.8). World-sheet (e) (a) (L L ff% )--i(n-m)Ls% r tris tins l the Raupte condition er ted by Ms c (3-31) such that ± a) 11,21. ®-i(n- ) ~~ ) . or 0 strings , we then hav ThrouRh the c-nrresDondence orinciple. t GP (128a) C =`5'cL a., quantised system is defined by t n

06

J. Govaerts IString theory constructions

i) open strings : 0,

(a) L n 1 q> = 0

1

(3-36)

ii) closed strings:

00

(3 .37a)

%M 19 (3-37b) L +£".%»`- 2alî W > - 0

0

0

(3 .37c)

V> = 0

here "a" is a subtraction constant followi from normal ordering in the zero-mo generators Virasoro . Similarly. one expects a central extension for the 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.

t t ti

f

ii) closed strings : IL

lation

the I

lai

«, ,

L ,,

lap

(338) M

1

I

I

1 - W m)L,,, . ., + w Dn(n2 - 1 )â

2 Dn(e - 1 )j8n n4m + î-

il W IL"" . L', M 1 = Wml

n

Is

(3-39) 1-0 .

ssumea i raws v r. it .

_

ralot ;^-ne 12 2RI Ag%f ;,nd% t h ,%% U;,roenret aimoke

with central charge c=D, each scalar field Aa, -

T)

contributing one unit, to this value. ue to the existence of negative-norm states the in covariant quantisation of the theory, one s to check, as a necessary condition for unitarily, that none of the states which satisfies (3-36) or (3-37) has negative-norm . By the st theorem 1231, we know that this is ssible only if D 26 for =1, or if D -4 2 5 for a< 1 . _r.

J. Govaerts/String theory constructions

flfltu es, flt fls ~ t at the c rrect a t ir t-cl s c str i ~)=0 larity s r c ure e t® inter e iate s tes ss i te t e c i -~_ ®, - _ ,, is tai>~e ® ly if = ]. ence, the values i tr res ctiv 1 f~ ,ecessary ®r tr -l v l a _ ~, e ® e-1 3~ -® e i i r ss a v u itarity are a=1 a ri 1 , ) = . ( eir ] =1® ,~, t t ) , ] ut t is i a lies t at t e ysical r~un ( =1, ,3, (a ti- e itia st tes f~r t e ® e stri a f~r t e cl se ss v ria Strl 1"rs], it are tac y~ns, a fact ic ~pells isaster f~r t Te ®st ra atic c~ se uence ®f t e e istence (3. 1) t ese tac y~ s is i t e n® - ec®u li ®f ~°Ye st c r is fi er®- ®r hysical states i i er 1 an. i litudes [ , ). ®r le, i t c s f ~ _ ~ . ) t e ® en s®nic stri c®u led t~ r® t a r t s c tr e ~~ees ®f free e ~ t r~u C an- at® fac rs, (a=l, , , ], tai ~ e e r t~ her~ c~ ( utin ®ne-1 a litudes i [ i ~, ~ ,A _~ b e ternal ysical ® en stri states, it at least ®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 vari tri uti®ns. ) e firs ntri uti®n v is es = ~ _ . (3. / )=S ( 13) ly f®r ( [ , , 7), but t ~ y structti n, il t ft se ~n ®ne, entirely ue t® tac y~ns, s e~t c ar e: v is [25,26] ® ly t e first tri uti~n e ists an it v is s tr nsf~r a i ns f all varia 1 s c ' f®r ( / )= ( 2). is fa ®us result as e ® t in [ , lea i t a c i f seta lis e f®r ternal au e ®s~ns [ a invari t ~u iti® , °c y ravit® s [29) ®nly) . bn- itarity ®f ac y ic s ec' y el® . ®te t at e ave : stringy the® i (Tl leea t e re ® n ®f t eir (T) , ) ~ ® ) ( ~, ( . ) au , fl.e. fl t e r au e, ~ e ry. ere iscussi® n (T) (c) in the s e a ®f ( ) t ese stri e®ries, ®t ia the ®r fl e li te au e, ay e f® Sfl

3.3. F f®r ul ti® the fir ulati®n a plie t~ t e ®sonic stri (see ef.[ ) and references t scam), ass-s ace is e ten e as f 11 s. e First, ~ e i tr®d~ces c n1u ate ® ~nt~ _~ , ~) (

,

i) t® -( , ~). °( , i) res actively,

it

,(c) a

(ç)

ave t e

s ic sa e ®iss r ckets ( . ) s t e ( ) ( ) t l uantities c®nstr i ts an . (T) and ( ) are t us t e e q t r f rl

J. G®vaerts/Strgng theory constructions

198

f

®( (-)+ ( ;) ®3a, +® 1 + 4 (3-52) 24064472 loll 0A 1 d the correct equations of motion are reproduced i the Poisson brackets (3®®1, - ) , ), (113(0,,

® ,)®

( ® ) ®

1( ® O) - 40-01 ® (100-0- P2(0'-'9)) . (3-53b) appropriate boundary conditions are 19): )B

(41)

9

a

1 +

11(® j)

3a

21(, Tl)

1140-

0

closed stri

Solving

t

us function >

r

:s

(

(,

d

, ; )

. 1-1 . -

, i®1 . (

,Id, i l,22

v

(3-54)

® qla (-1,) and Pa (-,)i

siste t with BRST transformations, which for

the r degrees f r a tai from the BRST charge in the gauge A'-AO-A- 191:

1:

taking

specified value and

r.

3

a

One mav t a i t r s effective ,ti s motion. rf r ft t ~f. ®ti s xt -o Pal and Fa W 1,2). 5,711 then taking the limit 0. t ~ti (-1 ® , ) raints which are SOW t t r er I s

t'1 t an s

7,

r s t ff 1

t

( ® d,

i)

) 1) open strings:

s

(3-53a)

ff ctive II:a

' t

i

191

1 -

aA

1

(3-55) e notation as before i s for this quantity, although it is different . This shoul t cause y confusion, si i at follows t restricted operator is consider y articular, the boundary conditions (3.54) ST invariant . and S,,ff in (3.51) is invariant for transformations induced by (3.55) (provided e uses t equation ti ). t may also be checked that is il

te t.

J. fÂovaerts/String theory constructions

a, ), it iv 1 .

(3. r s t

s I

t t )¢ ~C ® D+ ~~ m

c_a

1

lgg ~ a

it

st

r

eevtri ti

ra

la

i v ri

t~ T cl , let si ti t t e i f ti t r al e, irs f r t tr si e ® iti s ( . ) t ~ ly, e fi s t f~

c_ __ - ~ C+a C+

[c

__ + c~ ~,

~ .5)

,

ra~kets: , ~),

a

(

, T)) ~

i - ).

t~ ~un ary ~~~ iti®ns s ecif ie ®te ~. at t .4 ) an (3. ) lea t~:

( .

a)

~ . ) in (3. 4).

~ i °

t .e

as i ( .

urier

es f

e ave: tTl ~ t~)~Lt~) L~ ~ ).

it L

ive i t .1 i ) a Ltc)

s® t at

T s s ® s that t e ® n iti~ns at t ) ~c~t~ (~~ 1, ) a ~c ~ ~®lve , ly a ysic s l ti®ns ® ®t e en ® Y s ecifi~ ~ l -s eet ara etrisati~n hi~ is use . t et us s~ re ark t at s~ivi f®r ~~ ~ y a® (3.57), ne re vers t e a~t~an ® ta~ne a a t 0j i t eir ri in treat e t ~f i~ i u tisa i~n ~ t e ®s®nic strin j .1

~

e

® t e

e eral result state ter ( . 'i, a ac c e use t~ justify S La ran i a r® c . I articul~:~r, ver' t at t e tfiN~ansf~r ati~t~s recisely ts~ a in u d by lea

t is

)( , ~)~

s ea ~

t ef l

early, tri ti

i

Lt le®~ t

e s t

(3.

t

t®

[ L( )~

i

~ "

®iss®

Lt )

,

( .

r c ets:

s ic, e Viras r

st

t®t a erat rs eac

)

ovaerts/String' theory constructions

J.

200

ere, re , c r .

t

_

_

( .

iti

_

~)i

),

e

t

iv ressi

1 . ) as Lt l i

c

.

s

i ~

( .

),

i ) are ive ()

i

+ Z L~

h ly t e 1

~ ~,

r s ® t e lef -

~rewer, e ave: c ¢ ILC

ic

~ 2 L~

it iv , ecisely t e ~f1 .)r1 .1) rt

e fi

s: si 1 . ) , 13 . 1 ) 13 . 1 ) c'1

13 .7 l c) - ° i -

~)-4

t

s

is

r e of free s uc

st yste re

~~ ti

)10®

~) ~

in1

i to ri

s

tai e

) ® ~

)

(3 .71 ) 13 .

t-

l

)

left-

f r t e

®sonic

lees 13 .x)), extends also to

(3.73)

~ t us ® txin t ® utin i entical t f®r t ®s® ic er®- ~ es ~~ies 1e ) ®f t stru ure ic a ears f®r t e ®pe stri . As is ell knn~rn, tt~.is is c ara e istic ®f cl®se strin t e®ries, an is ~f nstructi~n ~f f i ®rtan i the f® r i e si®nal eter~ ic strin t enries. Fin ly, i ®sink t e ® n ary editions t ~~Ti (i~ l , ), e re ver precisely the sa e Hg

e i varia t physical solutions s in the a ilt®nian a rx iar~ or c®variant a r®ac es. All h®st e rees ®f free o then vxnis , and the ®sonic s®luti®n has t® satisfy t e usual ir s~r® c®nstrai ts 13 .10).

. T is is a c®nse uence of the

f t e co

si s, ®r lsee 13.31)). ro 13.71),

is

or

suc

al

se

e r

rati

i

t ®

c rs

® txi : Lt~l e~~l (~ - ~) ®

13 .72x)

3.4 . RST xvin

uantisation evelope

13 .72 )

t o

~ f®r ulati®n of

t e ~s~ ic strie i t ~ c® nr al au e, it is no strai htfor and to pr®cee t® its RST uantis tion. y the corres ®h ence principle, the co

uantu

utation f®ll®

t)

c~ ,

i f t s1ti , tly ici y tc es v is t e cl s s i is it

i l

rs,

ast

These 1x ti)c® os~nic

syste an

r®

is defined th~~ou

antic®

the

t e

utxtion relations

piss®n brackets 13 .6 ).

utat®rs are e uivxflent, f®r the

e rees of free o , t® the relations in

J. Govaerts/,String theory constructions

201

(3.33) (3-34) . n for t ghost system t the folio ing ti ut tors: zero-modes " (377) it. ' (c, . b )_ + ,® ., one takes: for open and for closed strigs n , its c_n and n=b-,, and similarly for the left: cc ® : ®(3. ) ovi ghost odes of t closed string . n Having adopted these prescriptions, t hat follows, we explicitly write out the quantum system is properly defined. I lysis only for the open string or one of the particular, we then ha--.r, two sectors of the closed string. The ( ) (c) plate !) =L + . . .® structure for the closed string is straightforwardly obtained y Censoring the ( ) (c) -1:1 quantum algebras and spaces of states with 2 n cn . (3 .82) themselves . except for the bosonic zero-modes q11 and an __ hen com p ared to the old cov iant 1 qu tisation of the system, the space of states , (3-83) c = (c~ ® - ®c®) + 11 c_.b . - _ is t us considerably enlarged . In the bosonic were LI I ® 00 ivy n are sector, e have the ground-states I ß; > ordered expressions pressi o ( .1 1) and ( .) defined in Section 3.2, an all their excitations. respectively, a "a" is a subtraction constant In addition, we now have ,a ghost sector . The following f or ordering ~ ). s° algebra of ghost zero-modes cp and o is techniques explained Appendix, it i identical to that of the host ~Tste for the to show that : p article. Let us thus introduce two states I - > ( . )= (3-84) and I + > such that: ith Ln and QB given in (3-8 1) and (3-82). (3.78a) C01_>=1+>, C01+>=0 u i cl i esa e subtraction constant . b® I + >=1t - >, bn l - >=0 . (3.7gb) Due o this procedure f o ordering In addition, e also have -the ghost excitations composite operators, gauge invariance in th,9 and bn . In this sector, let us introduce a quantum system is not guaranteed . This is a ground-state, also denoted by I f2 > , such that: n10>- , well known phenomenon, corresponding to an c,,Ifl>=0, nzt 1 . (3.79) anomaly. As we know. this happens for the Therefore, the space of states of the present system; a central extension uantised system is spanned by the oc vacua or algebra generatedd by L(On) was- found I f); p > I ± > and all their excitations in the in the old covariant quan-tisation (see (3.38) and os(,nic odes n (n 1) and in the ghost (3.39)). odes c-, , b-n (n 1). Note the doubling in The question of gauge invariance may states due to the algebra of ghost zero-modes . equivalently be investigated by considering the e also need to specify a normal ordering al a of the tot ° as o 0 r tors J for all quantum operators. In the bosonic sector, t il of t BRST c1 y c ec i the same prescription s in the old covariant o types of operators i e r t t approach is obviously used . In the ghost sector, r symmetries of system . i.e. one again chooses to bring all the annihilation transformations i or RauRe. and the

J. Govaerts/String theory constructions

202

t

(3.84)).

first

t ()

t

o

, (-

( .

+

-)

)

1 . ,® .(3. 1

) (1

o Contributions (

ft

)

t

(3-88) re i ly t conditions obt t i c ssio f Section 3.2. followinR fr t theorem the correct l str ur f1 s. However, se l t r t sufficient for mt ity o t .mr o . The value for "a" implies t r tis4 if tac yo s n t physical s r t r spoiling ita ity of t ua t t i amplitudes . The values s re that gauge invariance 0. )is r liz t of st t t 1 )fit t ot ss t r s. t s s r r t t or t oe c t ol lulati even f rt critical s the tral e i si (3-85) s is "Tnly an si the of lati is a. t , i that it ter r tio ost fiel s ,r l invariance of t o is rified by computing the s

finds (see t

c

) +(+

o

1

(c) + ~ ()

1

s

)-

+(+ ,

a-) ]-(-

+( - 1) ) c)L .

) . (3-89)

is clearly shows that il oten of the S r tijr is obtained er the sa conditions (3.88) which sur t cancellation of the 1301 . This is to i L re related, a., sho (3.84). If t charge is nilpotent, t fol 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 BRST charge, or cancellation of the central extension (with its two terms, one proportional to , t other to A in the algebra of the total Virasoro generators, are one and the sa expression o or l invariance in t ise string theo In the present formalism, physical states r efi y the T conditions I

>=

,

cI

>-

1

> .

(3.92)

il o of ST charge is reserved e quantum level, h firs itio ensures gauge invariance (in t resent gauge: or l invariance) for ysical states. Physical states then correspond to o ol y classes of , it the trivial class containing all zero-norm states of the for I >. It can be shown 1311. when QB is nilpotent, that the general on-trivial solution (i.e. up to a null state > ) to the first I itio 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

J. G®vaerts/String theory constructions

f states, s t s of states i t io . Actually 1311, t to of strictly positiv, led transverse states. The s (3.92). where the value (-1/2) or ordering t implies states -trivial o ol class ft (3-94) I >=I 1 1> I-> .

203

f

i

I turn, the first condition i becomes :

v sa

40-92) tt s s

a

a

Y

1

wi XT

w

'ant

t tis tL

IT,

c

(3.92) t

e i cl t discussion f tisati of the bosonic string, wit formulation f constrained t 1 .

.

t ic thus leads to identically t ysic 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), t Virasor conditionn involvinst 0.0 would not av~ been o tai e . T is shows that physical states of strictly ositiv or i bot approaches are i to-one correspo en . Moreover . zero-nor physical states I in the old covariant

tion of t stem correspond ere to I-> tes I trivial o of y class (i.e. I I->= I > ) hic satisfy T invari t physical conditions (3 .92). these states I 01 - > are free of host citations, satisfy trivially (3.92) but r of zero-norm. Note that t stawsI ®> I+> so belo to the trivial o of y class of butthat t ey have ghost care (+ 1®2). , c

t 1 t r

>

r as positive nor physical states are the of covariant roach leads to e suits as t T approach. tter ever, is complete i that it inclues e hosts e t e-fixing i ra etris tion invariant syste

tini4

WO'

ti ues i (3.88). t presented er t ~b - t string, s o kov string 1321, since t he worl -sheet metric it in the non-linear or linear formulation) does not l as a y ic e ff f a 191. I other words, t to the bosonic string, even with the linear action, explicitly preserves the Weyl invariance is i of t system 19). Only r invariance is spoiled by quantum anomalies, which vanish precisely for t critical values (3-88). contrasted it This result is to ti tio of t i olyakov's trou path-integral methods 1321. I that case, the quantum system is defined such that ii, explicitly preserves reparametrisation e f ever, that the invariance . It i mode becomes y a ic or s or le precisely through t or (D-26) 1321. ( yl) anomaly, it the u li of t that case, eyl invariance (or only rest or mode) ca h

ITT

J. G®vaerts/String theory constructions

®204

tt l tt r®® f 1 t ° its contribution o t I;;I r a precisel p cis e tension o ° Hence, ,c) ghost syste-rm,. n~~ls that of t t c ~atte sector tri tlo c-s t t c 6. t restriction t kost sector. Tin r ro degrees )f freedom of ,,, r t iti r, following fron other fi s. t extension, ter nv e of t los i stir t on se ur , ro zero- o i stri theory, determines the mass t

S.

s

s

s

s

si t,

sit

1341 it " t

0

,Cent (,c) f ste - t f view. t

't t

t

ti~

t .t t r

of T

t

Otr

r

or le, for besonic strings considered , sü natter fields 0 are regarded as 5 ace-fi rdinates, the first condition fixes fl w sio of space-time to be D-26 (eac, t scalar field contributes one unit to the centrFl, charge), and the second condition implies that of tac yo . .~ the lowest physical state i the scalar fields however, may rdther be regarded as internal degrees of freedom, taking their values in some internal ("compactified") tor for example (se ef. I I space, such y, it is d references therein). this a consistent bosonic possible to construct string theories i y space-time dimension less t Therefore, t values (3.88) are not critical of for the dimension of -ti val t physical ground-state, u rather, they e consistent quantum t iti v f teyl) formulation o re tri io ( filinst f invariant bosonic string theories. This remark is tt os c sect_ . t sis o the recent constructions o four t described lt dimensional stri s. re r t s ti ri r

J. Govaerts/String theory constructions

2~5

I t S fial s ( ( ), ( )) s r r vit i~ tii ( ~), t° ) ,s .l t le a e, s ic iscus e rfe c teei ). t r, ) r etri ti i v ria t i st i f °i sc i es f f , -' ) i te rl t, r ivale tly, t t eir e i ic =1 t e s, ravity t e rie c l - e i s, a si ri tter i l s. e rass ria l . , ( ) i l a ®rl -s cti® ic s i v®l s et ei ei ( it ,l = ,1 e ric, s c tens r a eass i t e c®varia t r s ec ively e t s lti tr te r e ilt fr i ices), a i vit° =1 t ) ulti liars ®f1 e c~ns ai . Ga e s er a e , -s sy etries t a t e ies ca e ai tai c i e si il ® t t e ua tu level e ly f®r s® e critical ass a va i l s. c° t f va ues ®f t e nu r ®f alter fiei s a l s eet i r a e , t r -state ss ai e vai e. t e t a u ti f t f®ll ~ real r r~s t er 1~ n t en, t ese sy etries a e a ebr n ecessary f~r u n u c siste f es t aeries, a aly i a cwa ia t au e, f r t e er~- ®r s ries d c~ _ e a ive- an i iv in i etric, re e e y i t e l° t-c e fr ysic 1 litu es, a j- . / 1 er t e il e,f rS ace-ti e c v ri ce 1 car al e ra ( ®r refera ces, see efs .[ , )). iS t case f r L is Str° i a stri s can easily yl t e i ar acti s iti al 1 i t ®f vie . url u vst fr a sï ilar c ° t sy etry (i.e. l al t® inclu e 1 al s eet sy etries are e te f cti r t ri s eet etric), t s arsy et y tra sf r ati s, s t t t ese ils, tryas al s er- eyl sy Qf r t~ ®ries re actually t - i e si nal a tu ilt~ a see ef.[ l ). It~ c a i nt su r wity e®ries. a ®rl -s eet alter r, t s y r ~ f®r l i iel re t e n®~. n y ~s nic scalar fiel s, e i 1 t t s ef e r vit rea u ls s i t! fer i~nic fiei s, c u le t~ a ti tie t ec le, s t t a ca nicai su er ravity ulti let, i.e. a ®rl -s eet e t i ri i ti~ syste , eit er i etri s ®f r wit a ravitin®. Gau e sy e yi 1 ill ti , trivi lly t ri e f r ~ t l t e syst~ t e c rres ri ti s ly er- e r i varia t . ti® s a 1 i su arsy etry t a s t r u a invari ce ay e r t e rl -sheet, ®r re ar trisati® s i an ali~ cisely 1 al su ar- e ara etrisati ns i t e ec f r t s is tri , su r- rl -s eet. e real° e at t e fi i t ~ r 1 a tu l vel, t ey e s re t e c siste i vari is s rc t e t ry, in a c ari t er i a l° t-c r te etr , still e iicit sy f® l ti~ (f r refera ces, see ef.[ l ]). c st i ts f l al f urier t t e tri is = , cti f r the i i r a s e sy a etrisati® f alter i al) su er ravity c i .i .I t s s

0

!

$

Dl

B

°

J. Govaerts/String theory constructions

200 I r, 1 ,

11

t

~

1

'

I

~

1

l

t

-

v

~, 1

` "

"1

,i1

1 1 +_'

1

same r W ' the I te

1 11

11

~

'

~

1

!

1

Str Lg~ t

1

1

1

"

1

I I I'

~

I

"'

T

"

'

S

n

1

1 I

"

11

' 1

1 1

1 '

"

Y.I

1 1

t

1

~tl

1

II

1

""

" 1

' 1

I

Y'

1

hl 1 1

1 " II

1 Y.s 1 I

,

Ids

+0

t

S

" 1

i .

"" 1 1 1

"

11

.1

'

"1 ï

" II 1 . 11

' Ï

?

r

Its

3310

"

1 ~

sl

1t

Is

1

-

1

!I -

1 II

1 -

"

Y"

1 1

" 1 II

' ?'^

~Y!

" II

o II

1

"

tl

"I

FJ -

Y"

" 1 " ' II ' ft II VI, .1 II 1 - II ~ II ' 1 t sonic t ' 1 Y " ~1 1 1 ri ati s s, ®s w I1 Y ,, ` l fer i ic 3 et ts, generating - s l t ti ti . ss i ti lti lie s 1 Y" " 1 II I ~ t ts f t r vit s r lti t, 11 ! 1 er " ~ transform as suc quantities s i t constraints. n t Y 1 1 i p1, m a ! 1 1. ti_ ti ghost ~ ~z is introduced i t sector t i he boson.ic constraints, and an y ~ ,. 1 utin 1 ! ghost system is rresp®nding to the straints. Under local vab iant uantisation 4.2 . Old tr , all these ost e rees of efore considering T quantisatinn itself, results transform as irreducible let s briefly rovie the i from the lti lets of -1 s r r vity, it t the spinning tis old covariant , ti® ®f l gauge, e have string . the supers® ®r e-fi in function the action : found the extended phase-space. a the existence ®f a s per- ri v " jd2t [ V* W A â aa&%p 1, (4-3) for the spinning string 191. One may

3

II 1

i

!

~ .aOVÛ,~i$S/sc 1~8°lll~

.

is t

er

~rl -s ss~ i an ~ ~ ~ ars

ia i-,¢~.

a~~r

s i tric ®rl -s et

a

su rr

s in®rs, wit :

th re r s ntati® c ' t . ~ ®f t

aj®r . ~4 ~ t

a s i re u

r s t~

`~®â

yl i iti

i . t fi 1 1 st t

ti

f ti str ° ts: ( ®~

(

I

t~1~0I',j~ CO%dS$f~C$IOI~S

satisfies

t

ati

~ i

°

r t er rs ara st isa i s

e a i

~=

i ° ar y, i ei fiel

.4~

r

t e

f

ti str ° ts:

t

is iv

1

lea

t® the e uati~ s ®f ~ , ( ~-a ~

t e e ati f t at t e ua titi s _ _ a

®ti®

(4. a)

Fir t e ®s®nic e rses ®f frse ~ , t e sa e ®un ary c~n iti~ns as in the fer i~nic s rees ®f free ® , we sf.[ 1 ] f~r stails): i~ ~ en stri s: ( =4,~) ® ( =0, ~) h ( = , t) _ (a= , ~¢)

we

ave ( . ~. ®r ave (see

,

( .7a)

r=+ 1

ü) chse

it

~- ~ an

~ 5,

~~ .ll. res e

rl -s et art r a l f (4 .7~ re airy. T is f s ers~r sect Sc arz s et®r® t t f etry is ls his rs ®f s ersy s rss ce r a se c f tac rslats i t ysic l s sctra 1 . in t s c®rres ® i ®r t s ® e sari , t e s~l ti® s t t

s, stri cl®s s strin . usu f®r cl®s ( .

) t at t e st u

t y~ a r~ u f

i

( z a it a a ~

iii®:

it

ac

r in

s

re ®f th t- an

t e

b

-Sc

c~nse uence

®f

l

al

L

n ently.

and

~l

t

ers

etry

t)~

,

Jt

( .1

)_

)® .l

( .1

t ~ = Lt l + t l , .1 3o l is ivs i ®

~4. ) .

su erst'

ati

str ' ts (4.

si

®v rs .

9

= S

i

is

lsft-

N v

®

f ll

se

sn invari t f r is The t .5 ) st t u tr _ sf r ati® s i su r s ercurrent :

where

ar :

iv ly

f®rand the se®rs 5- S, -I~S, ft

iv n

e

strin s:

~® - ll,

fr®

®

currs ts .

f~r t e a n ( ~ sect®r, an ~__ 1 ®ti® e uati~ns fnr t s eveu-Sc arz (N ) secter.

with

ti

( .13c~ (4.13 ~

s re

licitly :

J. Gova,erts/String theory constructions

208

(4.13e) (4.13f) > r-

(4-13g) (4.13h) ( .

.) -i(-)

The corresponding Fock spaces are rather ° to describe. t ; sector of the open string, we have Fock vacu is annihilated by 111; > (), 1). I > momentum i e st t s, () is

suc

representation n for the Clifford f fermionic zero-modes do: (d§, ®) (4.17) t (-)

(4.14b)

-

(4 .18)

ere, f are usual ir c matrices i i sio . hic satisfy:

(4.14c) ), (

,

), Mr,

)

1®), respectively for r

-

(4 ( n~) -, , , .15a) (4.15b) 13) and ®()' () terms ressio s ' f the lefts Itat c, fc rd t . t

fi y te r l tio s ut ti cl ssical iss t c ri s o fr s A t r s a (-3 t f r ac r es f fr -class constraints ich must y the use of ° c rackets If 41 f. 11). i so, tains the tors:

tly, l states i the on sector of the open spinning string are space-time fermions. I the even- c arz sector, e simply ve vacua I ; >, it I > annihilated y , ( -1, r a:1®). Consequently, all states i this sector f the o spinning string re space-time bosons . For close striais, the structure of the space f states is obtained y te suri these spaces (except for the sonic zero-modes). e thus tai space-time bosons in the R-R and NS-NS sectors, and space-time fermions in the R-NS - sectors. or mal ordering o these operators is fine s fore. All creation operators to the left of all annihilation operators. For fer ionic zero-modes i the a o sector, e choose a fini t tu s e -Virasoro rator y t or ordered expressions of ( .13) (4.15), one y compute t rre pon _i super__ or al e r s (see the ni). is leads to:

J. GovaertsISt-ring theory constructions 14)) (4 2 - 2Lr(

,r: V

W] ( 1

(

2

( YI) 0 - 2. ) I W > - 0 -

ï]-

S sector:

r

)L W W - MR nI . D, 2 n3+n I 8n-ra, (4.21b) 0-

1

I (r2 11)) as) + 12 D 3 - 2dris

I L(PS G(X)I I

4

qs'o

n - r) G(X) n*r

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

is + I

24

s that T is s attar sector bre se associated re a etrisatio. obt tr et les

th

W2 - 1) 8...n

Rs

(4.22b)

antum effects in the e symmetries of t W world-sheet superinvariance. Indeed, we sions due to quantum alptebras of world-sheet orld-sheet articuler that ew tributes a val a to Vil dsoro, re a scalar contributes

t re a atria tip, ajar - eye fer 1/2) W the bra (in units v e QI)). hysical states i qu isatis of the api

the old covariant ing string are defined

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

FW 0 -

.23c)

> - 0 = LT th 19 >, n a 1. T a In . (424a)

n

+ I D n3 6n.M.0 , On'll - (n - M)Lnm (X) o 8 (4

10

209

(423b)

WXA ) - a- ) I Q> - 0 -

.24b)

ere a. (resp. a-) is the subtracti - c in the Ramond (resp . r! Sec The corresponding no-ghost theorem 1371 t en imolies: I (125) 'o- a,-0 1 a--2 these results, it is possible t sl scribe III the physical spectrum, both f, en and for closed strings in all their sectors . In particular, the GSO projection 1381 of these theories leads to unitary and space-ti s. supersymmetric spectra, free of any tac For more details, see Ref.[ I]. 4.3. BRST quantisati L t us 59 rewrite t osonic string in t solving for A,0, (3.24) and (3.57)):

S(bosonic) - Jd2k I -4 ) C + - b.jdr, - a.) c' 1 . (426) 1 invariant hich is under transformations induced by the BRST charge (see (3 .58)): --- b =J

ownic)+

I

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

Q + 112 +

c- ac-

+a,,eb .. 1 . (427)

For convenience, let us intr tation (which is useful in t but orld-sheet covariance of matter and ghost

J. Gavaerts/String' t eory canstructians

21V3 f' °

1 °

t

l'

te :

s

t° t .

.

c) e a va t e ®f t e l~ 1 ®1 e licit i t s erc ®r al tr ce t real rass ß, a f' e s ersy etry ic act ® s erfiel s:

e

t s ersy etry , let s i ~aria 1 s - a e rat rs _ a

t °

0

~~ce:

a

~

®

.

( .

)

4

a

~

+

t .3a)

i_a_o

a a

d

~

t .

b~~a_cA e ,

®i~t t e e ivati~es: ) s re

er t t® t is i c~ v

str ° s ~ . ~ ress y

( .1

s: ~ . 3)

t rs s er t t sf r ati s e fi y

a

a

~~

i -

s i

a

~c®~ ri

t

ere t at it is t e ®

.

let

sf~r s as a = a1®ra a s i ® 1 al ®re t tr sf r ati s. i si er ere licitly es i s relate sy et f t e te , t 's ~tati , ie ce is ti rta t.

s i i sc r s

r

~ i stri , erfiel

e acti® ( .3 ) lici ! s e efl

~~L

a

=

Let

r, t tly e

c®r~ s

t® the e t e

attar fiel s ®f t i tr® t rea!

t e e re ritte i etric i v ia t f®r

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

2~~ d

_a

I

~~

~f

~a_c~

,

(4.3 ~

~ti

frte ui'ary i1 e r y si ly i nre it i acti® ( .3 ~ is t e t t ® t stri in tne su erc® ®r al u e, is i variant ~ er t e su ersy etry a sfcr ti~ y s in uce ®n ( , ) t e su erchar es - an ¢ ( rovi e ne uses t e uati® s ~f ~ti® si c set = ~. e c~nstraints (4.33) can als~ e e re se a ter s ®f the suprrfiel s: ese re ti®ns eter i e h® the ®s~ ic a~~ fer i nic c®nstraints tra sf®r u er ®rh s eet supersy etry. In ~r er t~ c® late t e T f~r ulati~~~ ~f the s inni strin in t e su erc ~r - al , uS Il® intr® uce h®st su erfiel s: (e )a (~~i ® (), (4. a~ ere t e

er itian Grass a n fiel s d~, ~, _ re ass iate t~ t e ns~ ic c~nstraints (t ese r th h~sts ®f the ~s®nic stri ~, an the er itia c~ uti fiel s ±±, are ass

iated t® t e fer i®nic c nstraints. e acti~n fnr the nst sect®r ®f t spinnin strin in t e su erc® ®r al au e

ee , is ity is e licitly ° ri r crl -s eet s e sy y, re ces i t e ( , c) s ct r t t" cst acti~ t b®s nic stri T t tal acti ~ f e s (s ~ '~ ~

t

stri i t s er r al a e i t iven ytes f ( . ) ( .4 . f e ati®ns ti e ( ,c~ st y are as i t e ®s ic st ° , ly: a$c =

f®r t

,

_cs ~

( , )

,

(

st y te ,

l

ve:

The

~un

a, __ ry c

,a_ $~= iti

-re as in (3. ), c~ iti~ s ®r t ( ® ~ t ®se ®r t fer i® is t ese t ® sets ®f ® c®rrelated t r® supersy etry. enerat®rs e supersy etry re ara etrisati y: T+ _ ®°

is

s f t ( ,c~ yst ereas t e yste are i - ti l t~ attar fi l s s° ce ary c diti s re ®rl -s eet

a i t e

®f

® l -s eat crl -s eet t sector are iv~

(4 . 4a)

e eve : ,(

.4c)

,,9 .

212

Govaerts/String theory constructions

) .

,c,Y (4.48a)

(4.44e) D

l -sheet

rators

(4.45a)

reviously . For the remaining quantities, we

q q

a

(4.49)

D

c_ I

(n) +

( )+ n

(q) +

Y_

T+

s t

(s

()+

() )+

()) .

( .51)

close spinning string, the solutions equations of motion for the matter fields the (,c) system re as previously (see t r t' ritities just introduced ,c), ,c) ( . a (3 .71 ,e)) . For the (3.8), ( . the total spinning str ct% e s erco or s for superghost system, we have : (B ) )

. (4.46b)

ring (see (3.62)), th it l r i ic constraints

S4

rac es c

)

t 8

t

iss t

e-2i

1

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

solutions t, + a

e-iq

1/2

®

(4.47a) ,

(4.47b)

i the Ramond sector in the Neveu-Schwarz sect

(4-52b)

ß ,

q

( .52c)

s Riven r v sly (see (3.7), (4.12b), r t s 3 (3.63c)). r os s, e_'

(4.52a)

(, -T) -

2%F2

(4-52d)

q fore (-n) e

with as

or (-r) c

correspondingly for o or Schwarz sector. e then also obtai ( )

T

(D e -2in

_ ®

( ¢)

+t/ Neveu-

E(-k) e- 1 (4-53a)

al.

G®llâh?ltS I .S$%'illg theory

constructions

213

2 9

q

k=a, X.

`°'1

ef" e

1

. Here, L(k) and J( ) n the open stringy sector, ,

0

s i

tical definitions in terms

1

i 1

I~I s- ;

1 '

1v

I

1I

"

-

" 1

~1 11

-

w

"

`(~ "

1 1

,

~ 1

1 1 ' 1 1 1

1

I 1 1 ~ " 1

1

°I iy

~ 1 "

"

~1 1

1 -

1

. 1 1

I

I

-

'

o

Iw

11

' 1 1 1 Ii " j II

`

I

1 1 ' ' 1 1 " "

';1

II

1

!

'

" 1

1

t n

,

~fi I

1

1 "

,/~ 1 ^

~ ~, ro

I

t

I

I

1

I

~vi

, I

,

""

'

11

1

'

11

1 1

1

< 1 1

1

{

'IY ~~

1 " 1 II

~1

-

~"

o`

1.

f'

1 71

1

`

specified

1 1 1 1 1 1 '

,1

1

'

.

1

1'

1 s however remark that subtleties arise i 1

1

r'

1

"

II

. 11 1

1

'

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

"

"

1

1

" `

1

1 1

II 1

' 1 ' / "

"

"

~ . 1 1

(I "

1

`

1

I Y

1

I ~

" I 1

`

1

II

1

1

'

_

" 1

1 . 1 1

`

1 1

I

1 1 '

II

1 1 II

`

"

1

i

1 1

II 1

1

" 11 " I 1

" 1 ' I

1

I '

1

1

1 '

I

l '

1 ' i

1

"

1

'

1

1

1 1

' 1

-

" 1

1 1 '

1

1

1

1 I '

'f l

~

,

"' open closed 1 II 1 corresponding 1

1 II

1~

1

1

1

1

" 1

1

~1 I

" 11 !

~

1 "

1

I ' ! 1

" I

' 1

1

II 1 II

" "

tl

1

il a . ~jF

1

1 ` TI 1

1 '

' II

II

,

1 1 ~1 1

~

"

1

,

.1,

1

" / 1

1 p

1

~

t

1 1 t 1 r.

-

'

" I,w

I

u

1 II

1

`

I

1 I

1 ~"1t °°~' 1 ' ~1

'

~1

l'

" 1 ~

1

11 '

t 1 1 1

1 1

-

r

i

'

1

1 , `

a~~l

c

1~ ~

I

71,

1 '

11

ly"

I-I

1

`r "

`

" 1

1

Nor

i; s

1 1 "

:IS

I1

"i , ~

" 1 - II 1 1 ' ' ' ' II " 1

j,-p, YRl

1 1

II 1

,1 °1 ~ 1 1 1% I_

14,51.

I

1 ~" 1 1 1

"1

1-

in me

1 1

11

1

"11

1

~ '

1

1

' 1 1

" 1 '

1 ÎI '

II

t " 1'T I'

I

,

"

1 '

1

1 -i

"_

1 '

"

1

1 1 " l

~ f

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

t r~~ ) {

~

(

) r ,

B

~~

~~~ _ {

)a

- r)

let° it ur ( ) a L~ h e erat rs ( Ct~?

0

~ (

.

t

rt

t

evi s resul s î~r ~~1, e ~ t ° î®r t e t~ +

tr)):

)

l ), n

L ,

~

a

(

-

r)

=cn-

r~~ .L

n+r

)L (4. 6)

9

e ° t e su er ~r a ® ~~es ~ aveu- c are sect®r ® ly vanis f®~°: 10 , a_ ~

(4.6 )

t

{4.67)

ese are precisely t e c~n iti®ns i4. ) f®ll® i Fr® t tt®®st t e~re~ i t ®1 a:waria t a r~ac ~a i the li ht-c® e It

ay als®

e s ®

that

is

il ®te t

precisely ~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 al a e ra it n® central e te si®ns can then e ® taine fr~ ( . 1). i lly, let us si ly say ere that ysical states i t e a tise s i i stri ar efin i varia L c~n~liti® yt e

J. Govaerts/Strlng theory constructions

itio l condition restricting their ost s er ost charges ( like i .92) f (3 the oso ic string) . Here again, trivi ho ol y classes defined y (4.68) rr sponJ to strictly positive or (r sverse) physical states in the old covariant tisatio 1391, whereas e trivial ol class corresponds o zero-nor states. e additional constraint on the os er of these states is ore subtle, as it is related to the problem of picture changing i the s per ost sector. e 6.. aot consider this question ere, s e simply ante to present the T ua tisatio of spinning stri t eories, discuss o the critical values (4.64) a (4.67) follow i such a approach, so that the same spectrum of physical states is seen to e obtained as i the old covariant approach.

res

the

~r

<.tE

I

I I

"

m

1

"

-

1

' !

,

!

1

II

,

-

,

, '

!

1

1

1

1

1

`

1

!

1

11

"

'

11

o

1

,

1

"

'

I

-

-

II

1

I

.

1

1

1

~-

,

1

1

"

1

" ,

I

-

1 '

1

1

1

l

`

"

ti1

i

I

,

,

.I

-j1

1

1

' = r1

, .IF

1

I

I

I

_

1

amis t is broken thn~ ch precisely s as derived syste tisation of any V triaat~ and

'

"1

F

1

11

"

1

1

k

'

" 1

_

r "

11

1

T 1

4.4. Comments The same comments as for the b®sonic stri can e made ere. Without going into the Bails of the at li -Fradkin-yilkovisky formalism, e derived the S q tis tion of the spinnipg string in the s erco or al gauge, by tai advantage of the global =1 orl -s Bet supersy etry preserve y this u e fixing . e derived the conditions under is the gauge symmetries of the sYste a ely super-rear etrisation and supereyl invariance on the world-sheet, can be preserved at the quantum level. I the superco ormal gauge, these conditions reduce to conditions for the cancellation of the super or al ano apes in the =1 super al algebra. e these conditions are satisfied, the preset uantis tio method of the theory is e uivalein to Polyakov's approach using path integral methods 14®1_ i that case, superrepara etrisatio invariance is explicitly

2115

I

-

,

'

"'

1

1

v

11

11

1 .

'

l'

l'~ . ~.

1

. GovaertsIString theory constructions 1

,

1 .

1 1

1 ~

1 11

"

1 1

I

1 W , 1

â FI

-

K,

1 .

.

~

' I

1 1

1 T1

.

, ..

1,

1

1

1

1 I

I '

-

I 1

"

1

1 -

-

I 1

is 1 1

-1

1

t

' 1 "

t ' e(f references, s t t ;string 191 and t s strifl'n% ~, i'Exe first considered i s t_ t ® i6s f t It cl the r s t ,it t r s ti attention tr I . , through r t r rld-sheet s t t for

"

i

`

r

t

t,

.

1

11 . . l

111'

1a_

{

-

II

II

C ss r

t

s

. r ; :~ 1

t

r

-

" II

1 '

I1

I

1

~ -

7 .

I

~ 1

1

t et

EMENTS t is

ic

s

t ory 13

lis

invitation

pleasure to

l

to

t

. this

o

tici a ts for their interest and their stimulating questions. Part of this work was o il t str i University (Canberra) n i Sydney. Thanks re 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.

1~

II

11 ",

1

D

is

j

t+

"

1

I 1

purpose t ic string sr i )y v t ~ 1 ®s W I I 1' 1a t li s II Ory t " 1 II i S ~ er c rres r Sul f 1 c -1 . 12a ?lo we 11' 1 " W " II 1 for 1 Is I1 ' 1 1` : stri t Sul r c r rief y t s foil fr ti I

1

,

-

'

.

lo

for

It AY

.1 I

"1

a

-

~" I 1 Y

1

1 11

- v1 1

1 '

_. . ° 1. lesson to th ies in

t

1

-

" 1 ,

1 -1

v

II '

I~

J.

!e

®vaerts/String tiDea~ry c®nstr~c~imns

D

f

c(z) ~

c

~r

'

i~

te

tu

i

r i

ist

2~?

tr :

e

~t ,

tf)~ :

fi

i ~fz) ~ i :

t s (

(

f .) ccif z ) ~ : ) fz) cf

t is re a i® , t leftd si e is erst f t~ et r i ~r e e r uct f t t~ r t® s, ! : Tf ) ~°f~) ) f ) = f f ) Tf )) ~ lTt ) Tt )

) are er ~ ®siti e ers ® ( - ) als ears. ise ~tati ! c® ti i!! use ut. ee uivale ce et ee ( . ) a ( . ) ca easil be estabîis e c® t~ur i te rati s (see ef. 1 r ®re et üs) . calcu ati ® t e c® ®r a! e ras ic a ear i t e a tisati s ic a stri s t s re uces t® ® tai i t e ec ic r!d-s eet ® erat rs. Let us tr uce the ®I!® ire fie! ® er~t®rs, i ~tati s use i t e ai te t:

e

ave t e ~ )f ) ®

: fi

i! L~ ~

z)) :,

f .i)

~ ~~t ) : ,

( .

i)

) : , f .

)

~ cf )

) tz) ~

t if ) ~

t . ) ® t e si ul r ter s e, is ! e , ai u h a ' i ite

~ ( . )

i

t (z)~ : t

~ z _

)

z)

fz) : ,

fz))

.i )

z) : ,

i

si

z®

-

D

D

.i )

D

s:

cn Y

n

f .

)

R~e

z

~

~

_

-

0

®

a

®

c ) ( _ , ~ ~ f - ,~,~, ) t. a tities i e i t e ai_ e t r ca e f t se us' ic' tere ®ta°e u cti s c e! ti s. si t e ta i ~ ) (~~) iei t e ® r re ic c tr c~.i

ere

(

)

t )~n

n

(

.i

) are t

e sil te a a e t i ~ ~ t c

J. G®v~erts/String the®ry c®nstructi®ns (y) ~ _

( .1

)

16

, i t

( . 1)

ev~e - c

a z sect~r :

arz sect~r : ~(y) ~

( .2 ) F ( .

lly, we als® have : c )( ~ Jc )(~~ ~ ®~ ~~ ~(y) i~_~)L yJ~ )(~) , = , (aty)

)

I )( ) ( it i e ,c, ) t

i fi

s: c=

.I

t e

ara et is

,

( .

i

irres active ® t )

a ®n

r®

ese

t® ( . ) c=

,

( .

)

aveu -Sc warz

®r

results,

si

e

all

c®

®r

al

al al e ras c®nsidere i t e

tent are easily derived. I ) .

(A .3 )

sector. superc®

sea r:

r

,

eve t e

ilarly fr

( .

and

ain

ee , curresp~n in

iras~r® al e ra ( . ), and

) and ( .3 ), ®ne ® tains:

( .2 ~ °Sc wetz Sect®r :

l(z) :

c=~l1,

=1 .

( .2)

sin the s

e techniques, nil ®tency ®l' the

char e ay easily e c ec e . F®r t e refinin: th b~s®nic strin , f Jt )(z) ( ~=X, ) it' itself, we os®nàc)( ae~e : lcl( z ) - az-2 l : , ) _ : ctz) ( c )(z) ~ J

2

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

ire si

ply have:

~ ~~®i~} _

ere: l( ) = T( l(z) + TI I(z) T~ l~z) = T(c~~z) ~ ( ~(z)

,

(A .39)

In this f®r

dZ

~i

~

~b~s®nic)

(zi

.

(A.

, it is quite strai htf~rwar

®btain the result ( .

).

o) t~

J. Govaerts/Stringy theory coaastructloras

S~

ly, f r t e s i _

(s i

_i

~

J ~ ~

str~ ~ i

1(z)

s® ~ ~®,,~.

,

( . t i

( . 1)

r

its t

)ist ti

re, ( .

s

~c, t e®ry T ®r

2i9

se ®f ®r al fiel c i il ®te ® t e

r® t ese e ressi®ns f®r the es ve ea t® e ive e, it e resul s i ( .54)e an (4. 1). T ( .3 ) is a~tually very eneral. 1 1 e t®r z) is s ° t® e a ri ary fiel

:

s

)

tt

t

( .

t

re e ri fi at (i ( )), (z) . ( 1, (z), ) 11 - ri fiel , v° re

t

v r ) ive

t

ver,

®f r al ei ht h ( ] if its stress-e er tens~r T(z) is:

ith t e

(z) (z), se

(z)

.

z) se

r

/ _ , f r ave / ~® /_ ° te -

e av ( )

ve, J( )(z) is

sai

i

iel

fi i

() is als® equivalent t~ z)l~z $ia

z)+h(n+l)z

Let us assn e that ~® iti® ~f the type

e t t T( ) i ( .4) is ri fiel it p/q=0 e t e tr 1 e i s an a is i entica y. st e ay ener lise t e ( ,c) a ( , ) f ti r syste s t a syste ti a fiel s f ar i a i~ rati nal e ®r al e' t ~ a 1res e~tive y. et us efi e t e iel s: =

z). . ( .44)

z) obeys a brun ary

p being - ri e integers, and t at is a rati~, al nu ber. e then have the ® e e anai®n it

s

z-~-

,

(A.46a)

the su ati~n is ®ne ®ver ali ratio, al nu bers ®f the f®r s = n+( /q - h~), i.e. s ~ ( / - h r ) ( h~ is h inns its inte5er her

art, i.e. the rati®nal part ®f the ~® ®r al 's i t ). e ® es are iven y: (ß.46b) e

c(z) ® . s+

s~

z-s~-i~- )

(

.4 )

s~

er t e su ati n is ra i®nal nu ers

rf

a rati is t e

v r

it ein < ~ 1 . -1 s c t at t e ases ~ t

u er st e eral si a se se. sal s el ~ ti ul a ~d2 re

b

i i ns res e iv 1 .

i

t ly,

iel s

J. G®vaerts/Stria theory constructions

220

si

r t c

r

fie theories. Then, f variables from (a,-) t

o z (0,x) , it is Possible t tirely the formulation of these theories in t ft ri fields ove only. This is t a useful e rcis

1. .

vaerts, tri Theories: tro cti School o rttcl ° a Fields, av c are as, 19 s. J. . ucio . e e a (World Scient ic, Singapore, 1987), 247-442.

t S®

D

(resp. 1) for commuting (res fing) fields z) (z). e that c(z) (z) satisfy it 41-J) (A.43) ctively, and that the OPE (AA is obtained e following central extensions: ._ fields: tin J(J -1) , tic ~ c

= (J -)(J +

-) (A-56)

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

-1) . (A.57)

these general r sits lead t t s s t i e far far the (,c) a (, y) stems, J-3/2 respectively, for t ter fermions Y it J=1/2 ' si t rr 11 st t r r so ts tw4 s c r im 1 .mt s ro~ - -~ re tl ices tt t t fi tr i f r str [ig] calc l tic s ft r e r~ s f stri theories. deo s s a t le t am the re d to t t (su ea) r t t 4

i

. J. avaerts, An Introductory Guide to Strin erstri Theories, reprint CERN-TH.4953/88(january 1988), to be published in: Proc . of the International Workshop on Mathematical ysics, Bujumbura (Burundi), t r28 - October 10, 1987. M.B. Green, J.H. Schwarz . Witten, Su erstring Theory (Cambridge University ress, Cambridge, 1987). Volumes. .

Frie Phys.

, . rtinec 1 (1956) 3.

. Senker,

cl.

5. S. Shenker, Introduction to Two e sianal ar l an u er ar al Field Theory, in: Unified String Theories, e s. . Green . Gross (World Scientific, Singapore, 1986). . 141-161 . rie a . Notes an Strin T eaty n T ension ar al Fiel d Theory. in: ified String Theories, eds. M. Green and Gross (World Scientific, Singapore, 1986), 16-1 .

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

. . es in, ntr ucti t® tri a d Supe~ stri e~ry II, preprint C-4251 ( arch 1957), Lectures prese te at the 3r dvance SL y I stitute i le e~~tary Particle .ysics, Santa5 Cruz, June 3 - July 1 ,

1 .

.

erc e, . . c elleke ~ an r er, re rint C - .5 5/55 (A u ust 1955), t® appear in ysics e ®rts. a hysics ep~rts 1 9. J . ys vaerts, I t. J. I~~~ . .

(195 ) 1 . (1959 ) 173.

10 . J . Gwaerts, preprint CORN-T .5010/55 ( arch 1955). 11 . A. . Schallakens. Self-dual Lattices in Stringy The®ry, in these r®cee in s. 12. C. Teitel ~i , Phys . Rev. I)25 (1952) 3159.

221

. l ~` ie ~ui e , I t. (1 ) .

19. J.

vaerts, un

lis / /

1. . y

. . .

er,

er. .

y.

est,

ys .

frac, Lectures ~n Quanta nice (Yeshiva University, - ®r , 1964 .

(1 l.

.

).

)

y.

3

) 1

(1

(17) 3 . 4. C. L®vel

. J . ®vaerts, st 1 (

(1 71)

s. Lett. 3

- Ii. (1 ~/ . ° y.

. J. ®vaerts, re '°i t t 1 )® e ( aY . Zu in~, hys. Lett .

13. S, ser an (1976 ) 369. L. rink, . i ye hia an . ® e, hys. Lett. 6 ( l 976) 47 l .

)

s at

.

~veu a (1957) .

,t

( t er1 ( e

® e t,

d

e )® s~~ ~

Pi

-

r

.

/

° t

u ~is e i : ra,~. ~v I I te ti 1 s C 11 uiu e retic 1 et le ( ec, )® in ysics, te. June - July , t°

1 . V. . Gribw, ucl. Phys. I3139 (1975) 1 . . est, ys . tt. 1 eve . . in I. . er, . ath hys. 60 (1975 ~ 7 . (l957) 0 . 16. T.P.at. illin back, C® . Phys. 100 . Gree an J . . Sc arz, 5. (195 ) 67 . ) 11 ® hys. e3t. 49 (1 . . S®hv'ev, J Lett. 44 (1956) 469. (1 ) . ucl. ys. 17. T. ~u~® and S. Ueh~ra, Nucl. Phys. 197 ra ®t , . 2 . S i, . (195 ) 375. ys. ( . i ®~ , e . . an ucl. . van ieu enhuizen, ucl. hys. 2 (195 ) 317.

t ) 7 ~ 212

222

J. GovaertsIString theory constructions

37. P. Goddard, C. Rebbi and C.B. Thorn, 72)425. r and K.A. Fri s. W . W (1973) 535. JJL Schwarz, Nucl. Phys. B46 (1972) 6 1.

M

38. P. Gliozzi, J. Scherk and D. Olive,, L M.B1220977053 .

33.

s. Lett. 179B (1986) 347; 33(1986)1681 . ux, Phys. Lett. 183B (1987) 59. 41 W Makm M. LeU. 103D (1981) 211 . (1983)1 )261 Y

2

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 arJ lodchikov, Nucl. Phys. B24-1 )333.