A comment on duality transformations and (discrete) gauge symmetries in four-dimensional strings

Physics Letters B 302 (1993) 38-46 North-Holland

PHYSICS LETTERS B

A comment on duality transformations and (discrete) gauge symmetries in four-dimensional strings Luis Ib~tfiez Departamento de Fisica Teorica, Universidad Aut6noma de Madrid, E-28049 Madrid, Spain

and Dieter Liist CERN, CH-1211 Geneva 23, Switzerland

Received 16 December 1992

We discuss the relationship between target space modular invariance and discrete gauge symmetries in four-dimensional orbifold-like strings. First we derive the modular transformation properties of various string vertex operators of the massless string fields. Then we find that for supersymmetric compactifications the action of the duality elements, leaving invariant the multicritical points, corresponds to a combination of finite K~hler and gauge transformations. However, those finite gauge transformations are not elements of a remnant discrete gauge symmetry. We suggest that, at least in the case of Gepner models corresponding to tensor products of identical minimal models, the duality element leaving invariant the multicritical point corresponds to the Zk+2 symmetry of any of the minimal N= 2 models appearing in the tensor product.

In four-dimensional strings, m o d u l i fields in general have non-vanishing charges under local gauge symmetries. This implies that parts o f the local gauge symmetries are spontaneously broken at generic points in m o d u l i space. H o w e v e r usually, a discrete gauge s y m m e t r y [ 1 ] survives this spontaneous s y m m e t r y breakdown. On the other hand, target space duality transformations [ 2 ], in particular target space m o d u l a r transformations [ 3 ], act non-trivially on m a n y light four-dimensional string fields. Moreover, particular elements o f the duality group act with simple, constant phases on the fields. Therefore duality symmetries act similarly to discrete symmetries. (We will show that they act like R-symmetries. ) The similarities between discrete gauge symmetries a n d some duality symmetries naively suggest considering these duality symmetries as just another example o f discrete gauge symmetries. However, this naive suggestion has to be qualified. F o r example, it is well known that the massless spectrum o f orbifold models typically has duality anomalies [ 4 - 7 ] whereas the enhanced gauge (discrete or continuous) symmetries are anomaly-free (at least for (2, 2) m o d e l s ) . Thus things are not so simple and, although indeed there is a connection between duality a n d discrete gauge transformations, the identification is not straightforward. The intention o f this letter is to clarify the relation between target space m o d u l a r transformations, broken gauge symmetries a n d discrete gauge groups in four-dimensional string models. As a specific example we consider the Z3 orbifold [ 8 ] #1. However, the discussion can be easily extended to other models. Let us d e t e r m i n e the transformation properties o f the vertex operators o f various fields under target space m o d u l a r transformations T-~ (a T - ib ) / ( i c T + d) for the Z3 orbifold. As explained in ref. [ 10 ], these transform a t i o n rules can be derived from the action o f the m o d u l a r transformations on the m o m e n t u m and winding a~ For previous discussions on the relation between duality symmetries and broken gauge symmetries in the Z3 orbifold see refs. [ 9,10 ]. 38

0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

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numbers and from the subsequent action on the Narain lattice vectors. Specifically, consider the left (right) moving complex coordinate XL(R) +i (i= 1, 2, 3) which is associated with the ith two-dimensional torus of the Z3 orbifold. For a general PSL(2, Z) transformation one finds --

+i (-iciTi+di\ XL (Z)--,2i~ ic--~-~-~ )

--1/2

+i [-iciTi+di "~1/2 XR (z)~X,~~c~T?~-d~) S~'(z),

X:'(~),

(1)

where 2iis a T-independent phase that depends on the parameters cz, d~, e.g. 2 =p = exp (2hi/3) for c= d= 1 (ST transformation ). Next consider the corresponding right-moving world sheet fermions ~+i with conformal dimension hR= ½. Their transformation behaviour can be deduced from the requirement [ 9 ] that the right-moving world sheet supersymmetry commutes with the target space modular transformations. This requirement follows from the fact that the action of the right-moving supercurrent connects equivalent (picture-changed) physical string states. The right-moving world sheet supercurrent has the form SR(Z):

3 Z (~I/RiOXRi'~-WR i O X ~ i ) ( Z ) "

(2)

i=l

Demanding SR to be invariant under modular transformations, one obtains -

¢/R+i (z)~2i~f - i~c i+T-i-' l~-/d)i ' ~

1/2

~i(z).

(3)

For many purposes it is very convenient to bosonize the fermions q/~ : ~u~i(z) = e x p [iH~t (z) ]. Then modular transformations act on the two-dimensional bosons Hi as --

['-iciTi+di\ Hk(z)~Hk(z)-ilog).i[ ~dii )

1/2

"

(4)

Next consider the bosonic twist field vertex operators a,,~ (z,- z) of conformal dimension hE = hR = {. Each twist field vertex is associated with one of the three fixed points ott (oft= l, ..., 3 ) of each complex i. The twist fields e,~, of different fixed points transform into linear combinations under target space modular transformations [ 11 ]. In addition there is a common field-dependent phase factor, which was determined in ref. [ 12 ]. In total one gets --

( - ici Ti + di \ ! \ ictTi+di ]

a~,(~, z ) - ~ [ . . . . .

1/6

A,~a+a~,(g, z)

(5)

Specifically for the ST transformation the matrix A has the form A--

¢

.

(6)

1 Now we are ready to examine the modular transformation properties of some specific massless string states. First the vertex operator associated with the modulus T~ has the form (we will only show the internal parts of the vertex operators)

~U~r(g, z) ~ OXc'( g)OX~'( z) .

(7)

Thus ~ - transforms under PSL(2, Z) as i

)+++(,+,z).

l'-iciTi"Fdi'~

(8) 39

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Next consider the scalar fields ~,2 whose mass depends on the background parameters T;. They become massless at the critical point Tc = - i exp (2hi/3 ). The corresponding vertex operators look like O'~.2(L z) ~ Vl.a ( ~, z, Ti) OX+i( z) ,

(9)

Vt.a (g, z, Ti) is a conformal field that depends on the background Ti (see refs. [9,10] for details). Thus they transform as

i - ~ [ - i c / F i + d i \1/2 (~,.2(z,z) 2 i ~ ~ ) q)il.2(g,z).

(10)

In the untwisted sector of the Z3 orbifold, each complex plane is associated with three fields, which transform under the 27 representation of E6. Their vertex operators are obtained by the action with the left-moving supercurrent on the vertex operators of the moduli T:

¢~Tu(g, z )

~ ~+J(g)Ox+'(z)

.

( 11 )

(We left out some part of the vertex operator associated with the gauge group E 6. ) Thus we obtain that the fields q~TU transform in the same way as the fields ¢[,2 (see eq. (10) ). Next let us consider the fields in the twisted sector of the Z3 orbifold. First we have 27 fields, associated with the 27 fixed points of the Z3 orbifold, which transform like the 27 representation of E 6. Their vertex operators are built by the product of the three bosonic twist fields a~,. In addition the twist also acts on the left- and rightmoving world sheet fermions. In total one obtains ~2°t~2°t3(Z% Z) --

3 H

i=l

exp[ ] i H [ ( g ) ]ei,(;L z) exp[-~iH[(z) ] .

(12)

Thus these fields transform as 1/3

-

OtlOt2Ot3 -~27T (Z,Z)--*

f i '~'i ~.-2/,3 [{ -. ici . . Ti. -b . di'~ l i=l ~ iciTi+di ]

fllfl2fl3 ( z,-- z) . Aotl~lAot2f12Aot3~3~)27T

(13)

Finally there are 81 E6 singlets in the twisted sector. They contain left-moving twisted oscillators. (Some linear combinations of them, the twisted moduli, are obtained by acting with SL on ~27x. ) Their vertex operators look like 3 it~l ot2t~ 3 i :j k (brr ( z , z ) ~ exp[ - ]i2H~.(;0 ] exp [ ~iH:L(;0 ] exp[~ink(g)]Z,~i(Z, z)tr~j(z, z)a,k(z, z) I-I exp [-~in[~(z) ] . l=1

(14) Here i~jv~ k and z i is an excited twist field obtained by the operator product ofOXL-~ and a i. Then ¢~a- transforms as -

5/6

-

rs.3f -iGTi+d~'X ~.3[-icjTj+dj'~ Iciri+di )

~

ic,Tj+ 4 )

1/3

.

-

y2.3{--iCkTk+dk~ "~k ~, iCkTk+dk ]

1/3

,~

.4

~ d~i#I#2fl3( ~ Z'lOtlfllX'lOt2fl2l"lOOfl3t/tIT \'

z)

(15) '

So far we considered the transformation rules of the scalar components of the chiral superfields. To obtain the action of the modular group on the corresponding space-time fermions, one has to examine the vertex operator of the space-time supercharge. Its internal part has the form 3

a ( z ) ~ I-I exp[ - ½iH~(z) ] . i=1

40

(16)

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Therefore Q transforms as

a(z)~

~=l i , z [ - i c , L +d/~-1/4 X~" ~ ~ + - d i i ) Q(z).

(17)

Thus we recognize that the fermions get an additional phase under modular transformations. Therefore target space modular transformations act like an R symmetry. From the field theory point of view, we will identify this additional phase as a K~ihler transformation. To summarize this analysis, the modular transformation behaviour of the massless string fields has the following form (up to the field-independent phase): • -nU2 -

i=l \ iciTi+di .]

Os.

(18)

n ~ is called the modular weight vector of 0s. Specifically, compared with our previous results, we have

(b~ : n= -- 2ei, ¢l,2,~i~Tu:n=--ei, i

¢27a-:n=(---~,--3z,--]),

O~w:n=(--],--],--~)--ei,

Q:n=(½, ½, ½).

(19)

(The e; are the three-dimensional unit-vectors• ) Notice that the above numbers for n correspond to the modular weights ofmassless states discussed in ref. [7]. Now let us consider the particular element, ?=ST, of the target space modular group acting as T,--) 1 / ( Ti i ) , i.e. a/--0, bi=- 1, ci=di=1. This transformation leaves the critical point T c = - i exp(2ni/3) invariant. Moreover, one can find a basis in which the STtransformation acts diagonally on particular linear combinations of the twist fields of the form Z,~c,~a~. Specifically, the S T charges of these particular linear combinations of twist fields are given by the eigenvalues of the matrix in eq. ( 6 ): A' = diag (/7,~6,p). Thus, at the critical point, all fields transform under the S T modular transformation as ~s ~exp(2zriQ~r)0~.

(20)

We call Q ~r the duality charge of each field. With ( - i 2Pc+ 1 ) / ( i Tc + 1 ) = exp (4zti/3 ) one obtains the following duality charges:

Oir: Qisr = 2, ~1,2,~7U: i "" Q~;T=2, ,,.2. Q~T=0, IT •

~l~2T: a s r = 0

0,13: Q ~ r = ]

~237x:Qsri = ~2, (21)

On the other hand, all fields are characterized by certain U ( 1 ) gauge charges. First each complex plane is associated with an enhanced U( 1 )~ × U ( 1 )~ gauge group which is left unbroken at the critical point T~= To. As discussed in refs. [ 13,14 ], at the critical point it is possible to "rebosonize" those parts of the vertex operators that involve the torus coordinates XL,±~R. In this new, so-called covariant lattice basis the left-moving (and also the right-moving) part of the vertex operators of all fields we have considered so far can be written as 3

VL(g) ~ l--[ exp(i[Q~ Y~ (~) +Q~ Y~(z) ]}, i=

(22)

1

where Y~,2 are the covariant lattice coordinates. The conformal dimensions are just given by hL= ½Z3 [(Q~)2+ (Q~)2]. For example, the left-moving torus coordinates can be written as i0X~ (Z)= 1/,4/3 Y.,,exp [ +iat. Y(g) ], where + at are the six root vectors of SU (3) with at 2 = 2. The gauge bosons of the enhanced U ( 1 )6 gauge group, which are massless for T = T¢, correspond just to ~ Y~.2(2). In the twisted sectors, the fields with definite U ( 1 ) charges just correspond to those linear combinations of twist fields, on which S T acts in a diagonal way. Let us consider the particular U ( 1 )~ subgroup of U ( 1 )~ × U ( 1 )~, with charge Qb(i), defined by the following linear combination: 41

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Q~j~,)=~-~-Q]+~6Qiz.

(23)

In fact, these charges Q'I and Q~ are just given by the charges that appear in the vertex operators eq. (22) in the covariant lattice basis. The various fields have the following U ( 1 ) gauge charges [ 13 ]: 0~, ' "

0,,~.Q~

=x/~,

q~7o:Q~=0,

Q~=0, 1

Q~,)=0, 1

Q~=-7~,

~217T: a~ = 3x--~'

0~¢: Q~-

1 3 i 2 Q~,,,=3, a~=-~Ti' O~'=+~Tg' a~'"=~'

Q~=0,

,/2

a ~ =0,

3 ' 1

Q~ro)=-~,

O ~ > = - ~2,

1

Q2= vTg'

~I~: Q~ = -

2

Qu(i) = 4 ,

2

.~_,

~x/Z

q~T: Q ~ - 2~/~

Q~ = ~ 7 '

Q~,l, = 4,

i

=4

0

3 ' Qz= ' Q~Jo) 2

O3T:Q~=- 3 ~ '

Q~=-~'

2

,

Q~J~l)=~.

(24)

Second each complex plane is associated with a U ( 1 ) holonomy charge QHot.This charge is a linear combination of the superconformal U ( 1 ) inside E6 and of the two Cartan subalgebra U ( 1 )'s of SU (3): 3

.

1

Qsup..... f= i~1Q~iot, QSU,3),=~7~(Q~ol_Q~o,),

QSO~3)2_ 1

_~7~(Q~o~+Q~o~_2Q3ol),

Q~iol is determined by the left-moving world sheet fermions ~u[. Specifically, the left-moving fermionic part of the vertex operators has the form exp (i s QhotH[). Thus we obtain T~: Qnoa i = 0 , q~l.Z-Qnol i . i = 0 , q~TO:Qhol=3,

q~27T:Q[ao,=~,

0~r: Q~iol= 4 .

(25)

Now comparing eqs. (21 ), (24) and (25 ), we recognized that the charges obey the following relation:

Qisr=Q{jo ) + Q[ao, .

(26)

For the fermions, an additional phase is involved. Thus an ST duality transformation acts like a linear combination of an enhanced U ( 1 ) gauge transformation and a U ( 1 ) gauge holonomy transformation. Moreover, the ST duality transformation acts like an R-symmetry. Eq. (26) becomes clear when the left-moving supercurrent SL is investigated. It has charges Q u o ) = 3, QHol-------3. Specifically, the vertex operator of SL in the covariant lattice basis has the form

Then, the requirement that Qsr = 0 for SL implies eq. (26). It is also interesting to consider the overall target space modular transformations, i.e. simultaneous transfor42

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mations on all Ti with al =az=a3, etc. With eq. (26) it is easy to see that an overall S T transformation acts exactly like a linear combination of all U ( 1 )9 gauge transformations. Note that the Z3 orbifold can be equivalently constructed by tensoring together nine c = 1, N = 2 superconformal models. Using this method [ 15 ], one always constructs the theory at the multicritical point Ti= T¢ with enhanced U ( 1 ) 9 gauge symmetry. In this case, the massless bosonic states of the model may be labelled by giving the nine q-charges of the nine chiral states with (l, q, s) = ( 1, 1, 0) appearing in the tensor product. For example (see e.g. ref. [ 16] ), the untwisted 27's correspond to charges (ql, q2..... q9) = ( 1, 1, 1; 0, 0, 0; 0, 0, 0), (0, 0, 0; 1, 1, 1; 0, 0, 0), (0, 0, 0; 0, 0, 0; 1, 1, 1 ), .... Twisted 27's correspond to states labelled (1, 0, 0; 1, 0, 0; 1, 0, 0) where the underlining indicates permutations. We have grouped the charges in sets of three to explicitly show the correspondence of each set of three factors with one complex dimension. The symmetry assignments of each massless state with respect to the Zk+ 2 symmetries of the model are obtained by computing the scalar product of vectors of the form y = (yt, ..., Yg) (Y~= integers) with the (q~, ..., q9 ) vectors of each state using a diagonal metric with g~j= - 8 u / ( k + 2), i, j, = 1.... ,9. In particular, we find that the action in the first complex plane of the S T generator on the massless bosonic fields of the Z3 orbifold (eq. (21 ) ) is identical to the symmetry generated by y= ( 1, 0, 0; 0, 0, 0; 0, 0, 0), as the reader may easily check. The above fact suggests that, at least for Gepner models [ 15] of the type ( k = n)m (m identical copies of a k = n minimal model), the duality generator that leaves invariant the multicritical point corresponds to the Zk+ 2 symmetry generated by symmetries of the type y = ( 1, 0, ..., 0). Now let us discuss the target space modular transformations and their relation to the U( 1 ) gauge transformations within the four-dimensional effective field theory of the orbifold compactified heterotic string. The kinetic energies of the moduli T~and the chiral "matter" fields As are determined by the K~ihler potential of the following (tree level) form: 3

K=-

3

~, l o g ( T , + L ) + 1--[ (Ti+L)n~lAsl 2 i=1

(27)

i=1

Invariance of the matter kinetic energies under target space modular transformations requires that the chiral superfields as well as their bosonic components (we denote them by the same symbol As) transform like (up to constant matrices and phases): 3

as--. I-[ (iGT,+a,)"IAs •

(28)

i=l

Therefore we call the numbers n~ the modular weights of the matter fields. The modular weights n~, i.e. the kinetic energies of the matter fields, were previously computed [ 17,7,18 ] by ccomparing string calculations with the effective lagrangian, and for the matter fields of the Z3 orbifold the result is given in eq. (19). The transformation behaviour of the fermionic components of the chiral fields in the supergravity lagrangian follows from the action of the target space modular group on the K~ihler potential• Specifically K transforms with a Kiihler transformation like 3

K-~K+A÷,~,

A = ~, (iGT~+di).

(29)

i=l

Then it follows [ 19 ] that the fermions ~A, transforms with an additional K~ihler phase as

g x A , ~ e x p t l ( A - . ' t ) ] ( l c i•T i - d i )

"'' V A , = [~-~i c i T ~ + d i \ -) ' / 4

(lciTi+di) " ,,,"g/As .

(30)

Likewise, the gauginos ~uztransform as --

( - ici Ti + di ~ ~,~--,exp[--~ ( A - d ) ] g a = \ ~lCi,i.ai .]

1/4

~;,.

(31) 43

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It is now easy to show that the transformation rules of the fields A, in the supergravity lagrangian agree with the rules we have obtained in the string theory by examining the corresponding vertex operators. In the string basis, the fields Cs are represented by vertex operators, which create normalized states with canonical kinetic energies. Therefore to relate the string fields with the supergravity fields, one has to perform the following nonholomorphic field redefinition (except for the moduli fields Ti, see ref. [ 10] ): 3

0, = 1-[ (Ti+L)"~nAs •

(32)

i=1

Then, using eq. (28), we immediately obtain the correct field-dependent phase eq. (18) of the string vertex operators. Moreover, the string theory also provides the information about the field-independent phases and matrices, which cannot be obtained by considering the effective supergravity lagrangian. Analogously, the I~hler phase exp [ ] ( A - , ~ ) ] just corresponds to the non-trivial modular transformation behaviour, eq. (17), of the space-time supercharge in the string basis. Let us now show that also within the effective supergravity lagrangian, the ST-modular transformations act like a linear combination of field-independent U ( 1 ) gauge transformations plus, for the fermions, a constant K~ihler phase on the chiral fields. To achieve this, one has to perform a field redefinition to a new supergravity field basis, which allows to couple the charged chiral fields to the U ( 1 ) vector gauge fields. Specifically, for the moduli fields Ti one has to perform the following holornorphic field redefinition [ 10 ]:

T~-'r¢+T'T¢-T T~ = - i exp(2rti/3) .

(33)

Then ~r~transforms under the ST transformation Ti --, 1/ ( T~- i) as L--*exp (47ri/3) L .

(34)

Thus i~; transforms under ST exactly like the vertex operator CTi at the critical point T~, i.e. STacts on Ti like a U ( 1 ) gauge transformation. In fact in the literature about modular functions (see e.g. ref. [ 20 ] ) the variable ~ is nothing else than the uniformizing variable around the elliptic fixed point T~ = - i exp(2izc/3) of the element ?= ST. In general, the uniformizing variables are conveniently used to expand meromorphic functions F around the fixed points of modular transformations. A function F is single valued if it can be expressed in terms of integer powers of T~. This is the so-called uniformization. Similarly, the matter fields have to be redefined as follows: i

' ~ - \ ~c+T /

(35)

As.

Then the matter fields transform under ST with a constant phase like displayed in eqs. (20) and (21 ). For the fermions similar field redefinitions can be performed and one obtains that the fermions transform with an additional constant phase, which shows that ST acts like an R-symmetry in the supergravity lagrangian. The K~ihler potential in the new field basis has the following form: 3

/~=-

3

~ log(1-1T~[2) + 1-I (1-1T~I2)"'~IA~[ 2. i=1

(36)

i=1

Note that in this field basis the K~ihler gauge function is a purely imaginary number, .4= -4i7r/3. Therefore ST does not act on/('. It is obvious that the lagrangian can now be made locally gauge invariant under the U( 1 )9 gauge symmetry by the gauge covariant replacement ~s ~-~s exp ( ~ 9=~ Q~ Va) (here As also includes ~, ), where Va are the U ( 1 ) 9 vector fields and the Qa are the corresponding charges. Since the moduli fields T~ are charged under U (1)~ × U ( 1 ) ~ (see eq. (24) ), non-vanishing vacuum expectation values of these fields spontaneously break these U ( 1 ) 44

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symmetries, a n d the c o r r e s p o n d i n g gauge b o s o n s b e c o m e massive. (See refs. [ 9,10 ] for details). ( N o n - v a n i s h ing v a c u u m e x p e c t a t i o n values o f twisted m o d u l i also break the two further U ( 1 )'s, which are l i n e a r c o m b i n a t i o n s o f U ( 1 )fio~. M o r e o v e r , in general there are also m o d u l i that are charged u n d e r E 6. ) H o w e v e r a n i n s p e c t i o n o f the U ( 1 ) charges o f the v a r i o u s fields shows that U ( 1 ) ] X U ( 1 )~ is n o t b r o k e n completely, b u t the l a g r a n g i a n is still i n v a r i a n t u n d e r a discrete gauge s y m m e t r y Z3 X Z3. C o n s i d e r for e x a m p l e the group U ( 1 )i d e f i n e d by eq. ( 2 3 ) , with charges as d i s p l a y e d in eq. ( 2 4 ) . Since all u n t w i s t e d fields, i n c l u d i n g the s y m m e t r y - b r e a k i n g field Ti, have charge 3, 2 whereas all twisted fields have charges o f u n i t s 3, a discrete Z3 s y m m e t r y r e m a i n s u n b r o k e n . U n t w i s t e d fields are n e u t r a l u n d e r this discrete gauge s y m m e t r y , whereas twisted fields have Z3 charges -~, 2. T h i s result has to be c o m p a r e d with the discrete Z3 group generated by the m o d u l a r e l e m e n t ST. L o o k i n g at the S T charges o f all fields, eq. (21 ), we see that the S T discrete group cannot be identified with the discrete gauge group discussed above, since the s y m m e t r y - b r e a k i n g field lri is n o t inert u n d e r ST. F i n a l l y let us m e n t i o n the i n t e r e s t i n g possibility [21 ] that the discrete gauge groups are a n o m a l o u s . I n this case the c o r r e s p o n d i n g a n o m a l y o f the u n d e r l y i n g U ( 1 ) m u s t be cancelled b y the G r e e n - S c h w a r z m e c h a n i s m , i.e. by a n o n - t r i v i a l gauge t r a n s f o r m a t i o n o f the u n i v e r s a l a x i o n field. Similarly target space d u a l i t y transform a t i o n s , which i n v o l v e a n a d d i t i o n a l K~hler phase for the fermions, m a y be a n o m a l o u s [ 4 - 7 ]. T h e n the a x i o n t r a n s f o r m s n o n - t r i 2 v i a l l y u n d e r m o d u l a r t r a n s f o r m a t i o n s . H o w e v e r note that there is no direct relationship b e t w e e n a n o m a l o u s discrete gauge s y m m e t r i e s [ 21 ] a n d a n o m a l o u s target space m o d u l a r t r a n s f o r m a t i o n s [ 4 7 ]. In fact, for the Z3 orbifold, l o o k i n g at the massless s p e c t r u m target space m o d u l a r t r a n s f o r m a t i o n s are a n o m alous, whereas the e n h a n c e d U ( 1 ) s y m m e t r i e s a n d t h u s the discrete gauge s y m m e t r i e s are anomaly-free. We t h a n k S. F e r r a r a a n d J. Louis for useful discussions. T h e work o f D.L. was p e r f o r m e d as a H e i s e n b e r g fellow.

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[ 17 ] L. Dixon, V. Kaptunovsky and J. Louis, Nucl. Phys. B 329 (1990) 27; V. Kaplunovsky and J. Louis, paper to appear. [ 18 ] D. Bailin, S. Gandhi and A. Love, Phys. Lett. B 269 ( 1991 ) 293; B 275 (1992) 55; D. Bailin andA. Love, Phys. Lett. B 288 (1992) 263. [ 19 ] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Nucl. Phys. B 212 ( 1983) 413. [20 ] J. Lehner, Discontinuous groups and automorphic functions (American Mathematical Society, Providence, RI, 1964). [ 21 ] L.E. Ibitfiez and G.G. Ross, Phys. Lett. B 260 ( 1991 ) 291; Nucl. Phys. B 368 (1992) 3; L.E. Ib~ifiez, preprint CERN-TH.6662 (1992).

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