“Twisted” strings and higher level Kač-moody reprsentations

Volume 229, number 4

PHYSICS LETTERS B

19 October 1989

' q ~ ' I S T E D " STRINGS AND I t l G H E R L E V E L K A ( ~ - M O O D Y R E P R E S E N T A T I O N S Z. H O R V A T H Ir~'tttute for lheoretical Physws, Roland EOtvrs I.'mversit),,, Puskm u. 5-7. 11-1088 Budapest, llungar3'

and L. PALLA ' Department of Mathernatwal Sctences, Untversttv of Durham. South Road. Durham Dill 3LE. UK

Received 17 June 1989

Using an orbifold-like construction the twisted sector of a closed string moving on G × G (with G simply laced ) is determined. A level-two (3 current operating there is constructed explicitly. The decomposition of the twisted sector into products between appropriate con formal and level-tv,o ~ representations is given if 2 rank G - 2 dtm G/( 2 + g) < 1.

According to current belief hctcrotic string theories [ I 1 offer a way to unify all intcractions. In particular heterotic strings obtained by the covariant lattice construction [2 ] provide us with a large n u m b e r o f thcories having phcnomenologically interesting gauge groups and chiral fermions. However, if we want to connect directly the gauge groups o f these thcories (7#~,r,°g) with that o f effective low energy models (f~¢fr) then one must face the problem that the ranks and d i m e n s i o n s o f the two are rather different. If :~,ri,,~ contains some identical factors an effective way to reduce its rank is to apply an orbifold-like construction based on an external a u t o m o r p h i s m exchanging the identical factors. This procedure yields twisted strings o f a rather special sort. The simplest example where this procedure can be studied is the case o f the 10 dimensional heterotic E8 string built from the Es × Eg one [ 3 ]. In a recent paper we found [4] that the rank reduction using the outer a u t o m o r p h i s m applied to the E s x E8 model leads to the appearancc o f level-two representations o f the 1~s KaY-Moody algebra ( KM ) in the " u n t w i s t e d " internal space. The difference between the central charges o f the level-one and levelPermanent address: Institute for Theoretical Physics, Roland Etitvrs University, H-1088 Budapest. Hungary. 368

two representations is carried by a critical lsing model. Exploiting this we successfully decomposed the "untwisted" internal space into products between level two ~8 and lsing representations. Mathematically this d e c o m p o s i t i o n is equivalent to the d e c o m p o s i t i o n o f symmetric and a n t i s y m m c t r i c tensor products o f the basic level one fZ8 irrep. In a recent paper [5] the problem o f decomposing the tensor products o f level one representations o f Ka(z-Moody algebras into products o f higher level ones and conformal systems was explored. In particular all KaY-Moody algebras were d e t e r m i n e d when the central charge o f the conformal system is less than one. In these cases the dec o m p o s i t i o n s contain only a finite n u m b e r o f terms and the I~8 example coincides with the results o f rel: 141. The purposc o f this papcr is to show that for all examples o f ref. [ 5 ], that belong to a simply laced aigcbra, G, the analogue o f the " t w i s t e d " sector can bc dcfincd using thc orbifold-likc construction; moreo~.er, they also a d m i t a decomposition into new c o m b m a t i o n s bctwcen the same higher level KM and conformal reprcscntations. We achieve this in two steps. First we show how a lcvcl-two (~ current can bc constructed from the ingredients o f twisted sector i.e. from I integrally and / half-integrally m o d e d oscillators with l being the rank o f G together with vectors

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on the " h a l f " of the weight lattice of G. The novelty of this sector is in the simultaneous presence o f these tu, o types of oscillators. Then, in the second step, wc repeat the "coset space construction" based on the Sugawara e n e r g y - m o m e n t u m tensor associated to this current and on the e n e r g y - m o m e n t u m tensor built directly from the oscillators. The fact that the level-two (3 operates also in the twisted sector gives us hope that this orbifold construction, built on the transformation exchanging the two :',e's (with G being the Lie algebra of ~) in ~,-,,8, can really bc used for rank reduction in string theory. To construct actual models is beyond the scope of the present investigation. We start by describing the orbifold construction we use in some detail. All orbifold constructions consist of basically two steps. In the first one we truncate the Hilbert space of the original string theory by keeping only those states that arc invariant under the action of a suitably chosen discrete group, P. the so-called point group o f the orbifold. In the second step of the construction we enlarge this restricted Hilbert space by adding the twisted sectors, i.e. Hilbert spaces where the boundary conditions of the string arc modified. The nature of these modifications and the number o f twisted sectors depend on P. The necessity of introducing these twisted sectors follows from the requirement of modular invariancc [6]. The discrete group, P, used in our construction starting with a string theory having f¢s,~,,s. . . . '.¢x ~¢... is a Z2 group consisting of the identity element and o f g = C cxp(2ztiJ~2), where C is the transformation exchanging the two f¢'s and the second factor describes a 2rr rotation of spacetime. Since this group is abelian and has only two elements there is just one twisted sector. With this P only the bosonic (fermionic) states being symmetric (antisymmelrie) under the exchange o f the two :4"s survive the group invariant projection. From now on wc shall concentrate only on the internal space and will not kccp track of the spacctime degrees of freedom i.e. wc consider the degrees of freedom of a purely left moving (say) closed string on G × G . However, to describe the untwisted internal space in details we have to give the operator C representing C. Because o f the form of .~,n,g all the states in the Hilbert space of our initial string are characterized by a pair o f G quantum num-

19 October 1989

bers l at, a2 ) thus we can define C as ¢~1a~, a2 ) = l a2, al~>. In the twisted sector the string is closed only up to a transformation generated by an element of P. In our case this transformation exchanges the internal degrees of freedom corresponding to the two G factors. Let X' denote the I coordinates of the first G factor of G × G while Y' the coordinates ofthe second G. Then the twisted boundary conditions have the form X ' ( o ) = Y ' ( o + I )+ rtW'~ , Y'(tr)=X'(a+ I )+rtW~,

(1)

where a is the space-like worldsheel parameter and W]. W~ are arbitrary vectors from the weight lattice of G. As eq. ( 1 ) implies that X ' + Y' satisfies periodic and X ' - Y' antiperiodic boundary conditions in the mode expansion of both X ' ( t - o ) and Y ' ( t - o ) there arc /integrally (or') and / half-integrally (fl') moded (bosonic) oscillators. However, eq. ( 1 ) is even more stringent as it restricts the constant and linear terms in X' and Y'. If we write (t is the time-like worldsheet parameter) X'( t - a ) = X~, + ztM'~( t - n ) +oscillators. Y'(t-o)=

Y~+nM'v(t-a)+oscillalors,

(2)

then eq. ( 1 ) implies M ' x = M ' v = M ' = !( W] + 14"'2), Y'o = XI, + ~M' + ~rW~ .

(3)

The presence of the half-integrally moded oscillators and this c o m m o n M' "'half weight lattice" vector in X' and Y' modifies the internal contribution to the masses: ,¥~ + NB+ M 2 + 76l, where N,, Np denote the occupation numbers for integrally and half-integrall3, moded internal oscillators, and the last term is the zero mode contribution. Having described the orbifold like construction we now analyse how the higher level (3 representations appear in the internal space. For completeness wc consider the untwisted sector first. The relevant part o f the internal space of our initial string, -,~,,,,, constitutes some (or all ) of the possible leve...~lone representations of the affine Lie algebra G × G generated by the currents J~(z)®l], l]®J~(z), a, h= 1..... d i m G ( z = e x p [ 2 i z t ( t - o ) ] ) . Through the Sugawara con369

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19 October 1989

struction we can associate to these currents Virasoro generators. (Jim. forming a Virasoro algebra (VA) with central charge c ~= 2/, the rank of G × G. It is easy to see that the subspaces o f ~ , , being symmetric ( . ~ ) or antisymmctric ( ~ ) under the exchange of the two G's are in fact invariant under this VA since the exchange J~(z),--,J~(z) does not alter ~,,,. ~ t As subspaces they are also invariant under the algebra o f the diagonal, level-two G currents

stands for an irrep o f t h e (3 KM algebra with (affine) highest weight (h.w.) vector A = Y,=o '= n,A, (where n, are nonnegative integers and A, are the fundamental weights of(3 ) and 1'~.7 ) denotes a Virasoro representation with c = c ( m ) = l - 6 / l ( m + 2 ) ( m + 3 ) ] and highest weight h ~ ' ) = { [ ( m + 3 ) r - ( m + 2 ) s ] 2 1}/14(m+2)(m+3)]. Using this notation the E8 example (which coincides with the results of ref. [4] ) looks like

T"(z)=J~(z)®ll+l]®J'~(z)

S2L(Ao) = L ( 2 A o ) ® l"l.~j) + L ( A T ) ~

~

.....

'

V~, ) ,

m

A2L(Ao) =L(AI )® I"~.12 . since T"(z) is invariant under the exchange o f the two G's. Using thesc level-two currents in the Sugawara construction we can get another VA, f~,, leaving both .~ and .~¢~ invariant, and having a central charge c 2 = 2 d i m G / ( 2 + g ) with g denoting the dual Coxeter number o f G. The important thing is to realise that the "coset space argument" or ref. [7] based on ~t,,, and ~( , 2, having the same commutation relation with T ~ , [ ~ ' ~ , Tan] = [ ~ ,

T~] = -nT~+,,,,

implies that

,~,,=.~'. _.~. also forms a VA with central charge c=c ~- c 2. Thus we can split ~,,, ( J into two commuting pieces: .~,~ _,:2 +.~,,, r~,7 - -

~ e r l

--

(4)

with .~;, describing a "pure" conformal VA. Since the conformal generators, Jg,, and the level-two currents, T,",. commute and both .~ and . ~ are invariant subspaces under both of them one can try to decompose both ~ and Y¢~,into direct product representations of the level-two (3 and the conformal system. In ref. [5] it was proved that i f c = c ~ - c 2 < I then these decompositions contain only a finite number of terms. There are only a few possibilities satisfying c < 1 with G being simply laced, they appear for 1= 1,2,6,7,8 with the following G. c pairs respectively: A~, ~; A2. 5. E6, 6. ET. ~o; Es, ~. Ref. [5] also gave explicitly for all these cases the decompositions o f all tensor products between the various level-one G representations. For later references we give here just two of lhese decompositions following the notational convention of ref. [5]. This means that in what follows L(A) 4.

370

(5)

with S 2 and A 2 denoting the symmetric and anlisymmetric products respectively. For the A2 case ref. [ 5 ] gives S2L(A,)) = L(2A,,)® V~.3,' + L(A, + A 2 ) ® V[.3,' , A2L(Ao) =L( 2Ao)® V[~ + L(A, + A 2 ) ® i/~3],

L(A~)®L(A2) =L(2Ao)®V~:]+L(A, + A 2 ) ® V[33) .

(6)

Since the extended Dynkin diagram of A2 has an automorphism Ao--,A~, A2--.Ao, A~ --.A2 thc dccomposilions obtained from (6) by these substitutions also hold true. Betbre passing to the discussion of the twisted sector we want to make a few remarks. The first concerns the form o f the level-two current. T ~ , in the Cartan-Weyl basis, using the vertex operator construction [8] for J'~.2. From the X'(z) and Y'(z) coordinates (which here, in the untwisted sector, are both integrally m o d e d ) we can define two commuting canonical free fields,

O'(z)= ~I [X'(z)+Y'(z)l. ~'(z) = ~

1

[X'(z)- Y'(z)] .

(7)

in terms of these fields the components of T"(z) belonging to the Carl.an subalgebra, ll'(z), or to the various roots, o~, V'~(z) have the form (in the latter one we drop the sign correcting cocycle factors)

lt'(z) =i~,~ O 0 ' ( z ) ,

(8)

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19 October 1989

define two conformal fields:

V,~( z ) = z,2/2:exp( i Ct'rP( z )'].

,f'2 /"

O'(z)=q'-ip'lnz+i ~_, °t'z-", n~O

X [ : e x p ( i °t'tP(z)']-~T/:+ : e x p ( - i °t'W(z,']'l,fr} f..j. ( 8 cont'd ) Our second remark is about the level-two G and conformal representation content of the various tcnsor products in (5), (6). Once we know that the dccomposition contains only a finite number of terms (which basically follows from the fact that with c< 1 the VA has only a finite number of unitary h.w. representations) wc can proceed as follows: recalling eq. (4) first wc find those combinations for which f{] + ~o have the same cigenvalues as ~6. Then to determinc explicitly which of these appear in ~, or in .~,, we analyse the multiplicity of stales belonging to the first few lowest lying cigenvalues of .L¢~ in thesc subspaccs. Next we turn to thc decomposition of the internal TW space in the twisted sector, Y{,,t • Let us recall that the twisted sector is described in terms of/integrally (a~,) and I half-intcgrally fl~ model oscillators satisfying

[ a ' , a'.l=m,~'J,~.+,.. 1#',,8',l=r&"'&+, (with m. ne/7. m ¢: 0 and r, s e e + I ) together with the "half lattice" vector M'e ~AG. This means that the structure o¢_~,,,,~¢lavis of the form .~® [Ac; with .~ denoting the oscillator Fock space built on the Fock vacuum 10> annihilated by all a',,,. [~, with positive lower indices. The novelty of this twisted sector compared to the ones considered earlier in connection with twisted string emission vertices [9] is the simultaneous presence of both "'untwisted" (ot~) and "twisted" (rid oscillators. Since we also have an almost unchanged momentum (the M"s) the construction of cocyclc factors will be simplified considerably. In constructing the level-two current in .~,r]v we follow the spirit of the vertex operator construction [8] by first setting up operators which satisfy "the almost commutation relations" of(3 [8 ]. and determining the cocycle factors afterwards. Motivated by cq. (7) and the way the twisted oscillators emerged from the twisted boundary conditions, cq. ( 1 ), we

~,'(z)=i

Z t~t+|/2

gl

fl;_--',

(9)

g

where [q', # ] = i O 'a and M' is related to p' by M'=p'/,v/-}. Comparing to (7) we see that in .,~rw,°, it is the ~u'(z) field that has become twisted. Clearly to construct a level-two (3 current with the aid o f # ( z ) and ~,'(z) we make an ansatz as close to (8) as possible. With the Caftan subalgebra generators there is no problem, they have an analogous form

h'(z)=ix/~ OO'(z) .

(10a)

However, with v'~(z) [the analogue of V~'(z)] we have to cope with the following: the generators of the KM algebra are obtained by taking various contour in tcgrals of u" (z), and these make sense only ifv '~(z) is periodic (i.e. contains only integer powers of z). Now although ~,' (z) contains only half-integer modes, its normal ordered exponential contains all modes. integer and half-integer. Also since M ' = p'/x/'2 ~Ac; the factor :cxp[ia.O(z)/x/~]: contains halfinteger power o f z i f 2 M is on the "odd" part Of AG. After some trial we concluded that the correct form of v'~(z) is

v"(z) = ~z ~'2/" :exp(i ot.~(z) ~. X [:exp(i a'~(z)'~.

-exp

:exp

(lOb

since it is periodic for all M'e tAG. Furthermore, if wc define the modes o f h ' ( z ) and v~(z) as _ _

h,,,-

d~ni Z m h t ( z ) , -

a,,,(a)=

~ 2~iz dz z,,,v~, ( z ) '

rneE, then a straightforward but somewhat lengthy computation (using both the well known OPE for the exponentials of O'(z) and the less well known one [10,11]

371

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:exp [ i a ' ~ ( z 2) ]: :exp [ifl.~(w ~ ) ]:

( z - w y ~B = \7~w] :exp{il°t'~(z")+fl'~(t°2)]}: for the exponentials of ~ ' ) gives that

[ h',, h'. 1 = 2mJ"JJ., +. [hk,, a ~ ( a ) ] =a'a.+,.(a) .

[am(a),a~(fl)]=O,

a.fl>~O, a.fl=-l,

] = 2rn&., +. +a.h . . . .

i.e. h',., and a.(a) satisfy "the almost commutation relations" of a level-two (~ currcnt. To convert them into the needed commutators we must construct the appropriate cocycle factors, c,~'s. Since the p' momentum eigenvalues corresponding to M ' live on the weight latticc of G scaled by I / v / 2 we can readily adapt the construction o f t . dcscribed in refi [ 8 ]. Indeed, if our ground state m o m e n t u m is/~ (which is eitherzero or is l / v 2 times a minimal weight o f G ) thcn the hcrmitian cocyles, satisfying a , , , ( a ) ~ =c_,,a,,,(a) are given by r-

c,=

~. c(a,H)lfllx/5_+p)(p+fl/v721,

tic

where the e(a, ,8) is the quantity constructed for all simply laced lattices in rcf. [8]. Defining (,~ by absorbing the exp(iot.q/x/-}) factor in 6~, ('~ = c x p ( i a . q/x/2)c,, we see that g'a?B = ( - I)"'~t~B~,~= e(a,,8)(,~+ ~; therefore finally we define the components of our level-two (3 current as e , , , ( a ) =

a,,,(a)c,,. Once this is completed we can form the Sugawara e n e r g y - m o m e n t u m tensor associated to this current; its Laurent coefficients, L~ -'~, form a VA with central charge c 2. However. in ~,rw we can also define Virasoro generators built in the usual way from the a',, and ,8',, [101 oscillators: L,, = L,,(a) + L,,(fl) + ~lo,,.o , ~-

~ E .O~ . . m O. l . a _

m .

,

n,1el

re/+

1/2

(where a~ =p') who form a VA with central chargc 372

r.5

•

--

PDAO

ro. "0 "

A~

z..(a)

c' =21. Since the conformal weight of the lcvcl-two current (h',,,. a,,(o~) ) with respect to L,, is one (provided ot 2 = 2) - i.c. their commutator is of the canonical form - we can again split L,, into the sum o f two commuting VAs: L,, = L ~ 2~ + K,, with K,, again corresponding to the same conformal system that appeared in the untwisted case. Therefore in those cases when c ' - c 2 < 1 we can decompose .,'/,,, .rw into linear combinations of L(A )® v'~"~ like in the case of the untwisted sector. The main difference between the two cases lies in the fact that the eigcnvalues of Lo (N,~+Np+M2+~I) are rather different from those o f ~ 6 . Indeed in all conjugacy classes of ziG the ground state values are scaled by ' and shifted by j~l rclative to the untwisted sector and also within a conjugacy class the eigenvalues are spaced in half-integers instead of integers. This implies that in the decomposition o f ~,,r~, new combinations between L ( A ) ® v ~''° must appear. Since the single ground state o f .,,f/,-r., w (i.e. 10) with ,.14"2=0 in thc conjugacy class of the root lattice) is not annihilated by Lo but LoJ0) = i},ll0), in all cases with c' - c Z < I listed earlicr. there must be a Virasoro representation v~"~ with highest weight ~l; and to account for this state the decomposition of.,~rw must contain L ( 2 A o ) ® V c,,,~ This combination - as can be checked directly "

(am(a),a.(fl)}=a,,,+~(a+fl). Jam(a), a . ( - a )

19 October 1989

- never appears in rcf. [ 51. To obtain the dccomposilions in detail, we follow the procedure outlined in the casc of the untwisted sector: first we find those L(A ~~ 1 " ~ combinations that give such values tbrL ~o2~ + Ko that appear among the eigenvalucs of Lo. Then we analyse the multiplicitics of/-o'S cigenvalues in the following way: at a given eigenvaluc (2) we first count the actual number of states. Then we determine how many of them can be accounted for by going to higher "grades" (eigcnvalues o f Lo) in the various L(A , ~ -,.~v~"Jcombinations introduced at lower eigcnvalues. If not all the states are accounted for then the number o f the remaining ones and the actual value of 2 must select some of the possible combinations L ( A ) ® V , [ ,"~. These. ofcourse, then would contribute to the counting of states at even higher 2"s. Since in general thcrc are only a few possibilities this procedure terminates reasonably fast. We carried out this analysis in detail for the twisted sectors of the E8 and A, examples. The E8 is cxcep; ~ -

r,$

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tional in two rcspects; on the one h a n d it has only o n e c o n j u g a c y class, on the other the eigcnvalucs o f Lo are just ~n, n = 1,2 ..... F r o m the procedure in this case we f o u n d . ~ ,T,~ = L ( 2 A o ~"9- - V ~., ~ '

+L(AT)•

Vltl, ) + L(AI ) ® I:cI~ " 2.2 •

We e m p h a s i z e that the first two c o m b i n a t i o n s are new, they have not a p p e a r e d in ~ , t . T h e A, e x a m p l e is m o r e c o m p l i c a t e d since its weight lattice consists o f three conjugacy classes denoted as ( 0 ) , ( l ) a n d ( 2 ) ( b u t as far as the n u m b e r o f states is c o n c e r n e d ( I ) a n d ( 2 ) are e q u i v a l e n t ) . M is in the ~ ( 0 ) class if the W~, 14"2 vectors in eqs. ( I ) . ( 3 ) are from the ( 0 , 0 ) . ( 2 , 1 ) , (I ,2) classes. In this case Lo's eigenvalues have the form ~ + ~n, n=O, 1,2,..., and the procedure described above yields TW

•~,.t ( ~ ( O ) ) = L ( 2 A o ) ® V ~ 3 4 +L(AI+A,)®_

I/~.23~

+ L ( A , + A 2 ) ® I,'~.~+ L ( 2 A o ) ® V,~.~'. If 14"~, I,I:2 belong to the (I,0), (2,2) classes then M~ ~ ( I ); in this case Lo's eigenvalues have the form I + ~, + ~n, n = 0 , 1 . 2 ..... a n d we f o u n d f / T,W, , , t~ t( I ) ) = L ( A o + A ~ ) ® V t 3 2

~

+ L(Ao +A, ) ® V~s2 + L ( 2 A , ) ® V,~2 + L ( 2 A , ) ® V,~.s2. C o m p a r i n g with eq. ( 6 ) [ a n d with what we o b t a i n from ( 6 ) by the s u b s t i t u t i o n ] we see that all the comb i n a t i o n s a p p e a r i n g here are new. In c o n c l u s i o n , in this paper we c o n s t r u c t e d the twisted sector, ..~rw,,,,, o f a closed string m o v i n g on G X G using an orbifold c o n s t r u c t i o n based on the t r a n s f o r m a t i o n e x c h a n g i n g the two G's. We d e r i v e d the explicit form the level-two ¢3 c u r r e n t built from the " ' u n t w i s t e d " (ot~,) a n d " t w i s t e d " (,6':,) oscillators together with the "half-lattice'" vector ML U s i n g the a n a l o g u e o f the "coset space c o n s t r u c t i o n " we dec o m p o s e d the v a r i o u s conjugacy classes o f xt,T,w into p r o d u c t s between a p p r o p r i a t e c o n f o r m a l a n d leveltwo Q r e p r e s e n t a t i o n s if 2 r a n k G - 2 d i m G /

19 October 1989

( 2 + g ) < I a n d exhibited that most o f these c o m b i n a t i o n s n e v e r a p p e a r in the "'untwisted sector" [5]. Finally, we w a n t to e m p h a s i z e that there are alternative ways to establish a c o n n e c t i o n between these p a r t i c u l a r c < 1 systems a n d the c o r r e s p o n d i n g (3 algebra: in rcf. [ 1 1 ] it was achicved using " t w i s t e d " fields only, while in ref. [ 12] "'untwisted" fields were used b u t thc roots o f G were scaled to n o n s t a n d a r d values. O n e o f us ( L . P . ) wishcs to thank the U K Science a n d E n g i n e e r i n g Rescarch C o u n c i l for a Visiting Fcllowship.

References [ 11 M.B. Green, J.H. Schv,arz and E. Witten, Superstring theory, Vols. !, I1 (Cambridge U.P., Cambridge. 1987); D.J. Gross. J. Harvey, E. Maninec and R. Rohm, Nucl. Phys. B 256 (1985) 253, B 267 (1986) 75: J.H. SchwarT, Intern. J. Mod. Phys. A 2 (1987) 593. [ 2 ] W. Lerche. D. Liist and A.N. Schellekcns. Nucl. Phys. B 287 (1987) 477; W. Lerche. B.E.W. Nilsson and A.N. Schellekens, Nucl. Phys. B294(1987) 136. [ 3 ] L. Dixon and J. Harvey, Nucl. Phys. B 274 ( 19861 93. [4] P. Forg:tcs, Z. Horvtith, L. Palla and P. Veesernyes, Nucl. Phys. B 308 (1988) 477. [ 5 ] V.G. KaY:and M. Wakamoto, Adv. Math. 70 ( 1988 ) 156. [6] L. Dixon, J. Harvey, C. Vafa and E. Witten. Nucl. Phys. B 261 (1985)651;B274(1986) 285; C, Vafa, Nucl. Phys. B 273 (1986) 592; K.S. Narain, M.H. Sarmadi and C. Vafa, Nucl. Phys. B 288 (1987) 551. [71 P. Goddard and D. Olive, Nucl.Phys. B 257 [FSI4] (1985) 226. [81 Fora review see P. Goddard and D. Olive, Intern. J. Mod. Phys. A 1 (1986) 303. [9] E. Corrigan and T.J. Hollowood, Nucl. Phys. B 303 ( 1988 ) 135; E. Comgan, in: Nonpenurbative methods in QFT, eds. Z. Horv~ith et al. (World Sctent|fic, Singapore, 1987 ) p. 24 I. 110] E. Corngan and D. Fairlie, Nucl. Phys. B 91 (1975) 527. I I 1 ] E. Corngan. Phys. Lett. B 169 (1986) 259. 1121G.V. Dunne, I.G. Halliday and P. Suranyi, Phys. Lett. B 213 (1988) 139; A. Neveu and P. West. preprint CERN-TH.5196/88.

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