Nuclear Physics B311 (1988/89) 140-170 North-Holland, Amsterdam

N O N - S U P E R S Y M M E T R I C FOUR-DIMENSIONAL STRINGS A.H. CHAMSEDDINEt

CERN, Geneva, Switzerland J.-P. DERENDINGER

Institut J~r theoretische Physik, ETH-HiJnggerberg, Ziirich, Switzerland M. QUIROS 2

CERN, Geneva, Switzerland Received 23 June 1988

The general formalism of non-supersymmetric four-dimensional strings is worked out in the Green-Schwarz formalism. The conditions on twisting parameters from world-sheet supersymmetry and Lorentz invariance in the light-cone gauge are shown to be identical. Tachyonic states can be projected out in many models by the modular invariance conditions. Tachyon-free models based on twisting parameters with Z2, Z3, Z 4 and Z 6 discrete symmetries are discussed. In particular, the four-dimensional analogue of the tachyon-free ten-dimensional O(16)× O(16) model is obtained. Two specific models with Z 4 and Z 6 symmetry are presented. The Z 6 model has E 6 X U(1) 2 as gauge group, 18 generations of chiral fermions and 17 multiplets of complex scalars in the 27 of__E6. The Z a model has a gauge group SO(10) × SU(2) X U(1) 4, the matter in 4. (16) + 2- (16 + 16) of SO(10) chiral fermions and complex scalars, and a Higgs scalar in the adjoint representation of the gauge group.

1. Introduction I n m o s t a t t e m p t s at the c o n s t r u c t i o n of realistic string t h e o r i e s [1], it is g e n e r a l l y a d m i t t e d t h a t s p a c e - t i m e s u p e r s y m m e t r y s h o u l d p l a y a n a c t i v e role. A l r e a d y at the l e v e l o f t h e s t r i n g theory, s p a c e - t i m e s u p e r s y m m e t r y has s o m e i m p o r t a n t a d v a n tages: t h e a b s e n c e o f t a c h y o n s is g u a r a n t e e d a n d t h e v a c u u m a m p l i t u d e p r o b a b l y v a n i s h e s to all orders, h e n c e e n s u r i n g the a b s e n c e o f a n i n d u c e d c o s m o l o g i c a l c o n s t a n t . It is t h e n c o n s i s t e n t to c o n s t r u c t string t h e o r i e s in a flat b a c k g r o u n d a n d t o e x t r a c t a n e f f e c t i v e l o w - e n e r g y a c t i o n in M i n k o w s k i s p a c e - t i m e . 1 On leave of absence from the American University of Beirut, Beirut, Lebanon. 2 On leave of absence from Instituto de Estructura de la Materia, CSIC, Madrid, Spain. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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The most important contribution of space-time supersymmetry is, however, when trying to understand the hierarchy of scales in particle physics. Scalar fields are chirally protected against quadratic mass renormalizations in supersymmetric theories, allowing for naturally light or massless Higgs bosons, and making plausible a (technical) solution to the hierarchy problem. The strategy which has then been used in supergravity unified theories is to identify the weak scale G~ 1/2 with the supersymmetry breaking scale A and to invent a super-Higgs mechanism with A << Mp. At first sight, it is then natural to extend this strategy to the case of string unified theories, and then to consider only superstrings. There are, however, some reasons to consider this approach critically. Firstly, supersymmetry is not necessary in order to have a tachyon-free theory: an example of a ten-dimensional, non-supersymmetric string theory without tachyons has been constructed [2-4]. We will construct some other examples in four dimensions in the present article. The problem of the induced cosmological constants is more difficult. Arguments have been presented [5] to suggest that a mechanism (based on the so-called Atkin-Lehner symmetry) could possibly cancel the one-loop vacuum amplitude in some class of non-supersymmetric theories. However, an example belonging to this class in physically relevant dimensions has not been constructed yet, and nothing is known beyond one loop. The problem of the cosmological constant in string theories must, however, be considered with care. Strings have only been constructed in a flat background and the plausible understanding of the cosmological constant requires understanding strings in a curved background. Also, using supersymmetry at low energies to protect the hierarchy seems rather troublesome in superstrings. It requires a mechanism for spontaneous supersymmetry breaking, with a breaking scale independent of, and hierarchically smaller than, the string scale ( - Mp). Studies of this problem [6], although not fully general, have reached the preliminary conclusion that such a supersymmetry breaking is impossible to achieve. It seems then that the supersymmetry solution to the hierarchy problem is in difficulty. We will now argue that non-supersymmetric strings could provide an alternative. To solve the hierarchy problem, one needs at least a mechanism to obtain naturally light scalars. We will construct below non-supersymmetric string theories with massless scalar fields. These massless states are essential for the modular invariance of the theory. It seems then reasonable to argue that these states, which are kept massless by quantum string loop corrections, should also stay naturally massless in the effective field theory, even without supersymmetry. The mechanism at work in the effective theory has not been elucidated yet, but it is natural to conjecture its existence. Notice that this discussion does not indicate a way to understand the weak interaction scale. Such an understanding is intrinsically related to the structure of the effective theory*. * Arguments along these lines have been developed in ref. [7].

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Our aim in this article is to study some aspects of non-supersymmetric string theories with fermions and in four dimensions. Various tachyon-free models are presented. We use the Green-Schwarz formalism, adapted to the construction of four-dimensional theories [8]. The main reason for choosing this formalism is the simplicity of the treatment of space-time supersymmetry. In sect. 2, the boundary conditions on string fields are discussed. The conditions arising from world-sheet supersymmetry in the covariant formalism are derived. The Green-Schwarz formalism for four-dimensional strings [8] is then reviewed in sect. 3. Several examples of non-supersymmetric string theories, with twists of order 2 and 3 are presented in sect. 4. In sect. 5 we present an example based on Z 2 ):< Z 3 = Z 6 twists which gives a gauge group E 6 )< U(I) 2, 18 generations of chiral fermions in the 27 and 17 generations of complex scalars in the 27 of E 6. In sect. 6 we consider a Z 4 based model which gives a gauge group SO(10) x SU(2) × W(1) 4, the matter in 4. (16,1) + 2. (16 + 16,1) chiral fermions and complex scalars and a Higgs scalar in the adjoint representation of the gauge group (which could potentially break the gauge group in the field theory limit). Finally, sect. 7 contains some remarks and the conclusion.

2. World-sheet supersymmetry and Lorentz invariance String theories with four-dimensional space-time symmetries can be constructed in various formalisms [9-11]. We will use the Green-Schwarz formalism, in which world-sheet fermions are also space-time spinors. This choice of two-dimensional fermions gives the simplest description of space-time supersymmetry in ten dimensions as well as in smaller dimensions. This simplicity is due to the fact that the string states in each sector and at each level appear in complete supersymmetry multiplets, contrary to the situation which prevails in the Neveu-Schwarz-Ramond formalism. The ten-dimensional Green-Schwarz formalism has been extensively studied, both in the light-cone gauge [12], and in covariant formalism [13]. The corresponding four-dimensional theory contains more freedom in the boundary conditions of the different fields, allowing us to construct many new theories (or vacua) with different gauge groups, massless spectra and space-time supersymmetries. These boundary conditions are subject to strong conditions due to world-sheet supersymmetry (or, correspondingly, Lorentz invariance in the light-cone gauge), and from modular invariance. Conditions from modular invariance as well as from super-Poincar~ invariance for superstrings have been studied in ref. [8]. We study in this section the general conditions from world-sheet supersymmetry. The four-dimensional covariant Green-Schwarz action is easily obtained by dimensional reduction of the ten-dimensional theory. It will possess an internal SO(6) - SU(4) symmetry resulting from the breaking of the ten-dimensional Lorentz group into SO(l, 3) × SO(6). This global symmetry will be broken by the boundary

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conditions imposed on the various four-dimensional string fields. Constraints on boundary conditions will, however, arise from world-sheet supersymmetry, which allows for the existence of a light-cone gauge with full Lorentz invariance. In order to discuss the boundary conditions, we start by considering the tendimensional theory and reduce it. The covariant Green-Schwarz action for rightmovers only (functions of • - o) reads [13] 1

S=

,7

2¢r do f d~" [A--t; h'~/~HMH~M ---f0

+ 2ie'q~( O,,xM)(OFMO~O)],

(2.1)

where F M, M = 0 , 1 , . . . , 9 are 32 × 32 Dirac matrices, and r l y = G x M - i~I"Mo.o.

(2.2)

Besides the usual bosonic coordinates X M, this action contains a 32-component Weyl-Majorana space-time spinor which is also a world-sheet scalar. We have then 16 fermionic real degrees of freedom. The action (2.1) is invariant under world-sheet reparametrizations and local world-sheet supersymmetry transformations (Siegel symmetry)

3XM= iOFM30 , 60 = 2i17 y I " M e " ~ K B ,

6(fZ-flh '~a) = - 16fZh-P~P¢8/~8 ( 0v0 ) ,

(2.3)

where K~(~" - o) is a space-time spinor, world-sheet vector parameter. Its chirality is clearly opposite to 0. The projector is given by

~o~/¢~-).

eo~ = ~(h°~ -

(2.4)

The action has also global supersymmetry:

3 X ~ = i~FMO , 68=

e,

(2.5)

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and a local bosonic symmetry

8XM= iOFMSO, 80 = ¢7

p-%0xo,

8(~Zh-h~a) = 0,

(2.6)

in terms of a world-sheet vector, space-time scalar parameter )%. U p o n dimensional reduction, the bosonic coordinates split in a space-time vector X ~', /* = 0,1, 2, 3, and six internal real bosons X I. It will be convenient to replace the SO(6) index I = 1. . . . . 6 by SU(4) indices by defining new complex fields 3

z k'= •

6

( a t X ' ) k ' + i £ ( f l ' X ' ) k'

1=1

The 4 × 4 matrices a I and satisfy

fll

(2.7)

I=4

appear in the Dirac matrices F M [14]. Since they

(~')mnemnpq=2(~l)Pq

'

(j~l)mnEmnpq=

pq '

-- 2(~1)

(2.8)

we have . _-- ~EmnklZ 1 kl , Zmn

(2.9)

corresponding to the fact that the antisymmetric tensor 6 of SU(4) is real. In the same way, the space-time spinor 0 will give rise to four spinors in four dimensions, with left-handed (right-handed) components 0k(0_k) in the 4(4) of SU(4). We now introduce the following boundary conditions for the four-dimensional fields. Each spinor will have a twist Ok: Ok+(r - o + 7r) = e2i'~ak0k+(z -- o ) , 0 k(r-o+rr)=e-2i'a~0

k(r-o).

(2.10)

Analogously, each boson z kt will have a twist d?kt, and a shift v kl which will be irrelevant in this section: all invariances of the action (2.1) involve only space-time derivatives of X M. Clearly, the space-time vector X ~ must be periodic in order to get the right space-time Lorentz transformations. These boundary conditions must be chosen so that the dimensionally reduced world-sheet supersymmetry transformations (2.3) have the correct twist behavior. It turns out that all constraints can be derived from the transformation of the spinors, which reads, after dimensional

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reduction k 80L=2i(0,, X ~,) ~'.eR,~k+ 27~-i( O,,zkt)e'~_, -

~

m

-m

ak

~

+2[OLmY aaOL +OR'}' OaORm]Y~e R

+ 4[O~O,fl~- O~O,,O~ - e""ktOt,,,O,flR,,] e~,.

(2.11)

From the first term, we see that ek must have the same twist Oh as OL k. The second term leads to eOk'= Ok + 8'

(mod 1).

(2.12)

Finally, the last term gives 4

E 0h = 0

(modl).

(2.13)

k=l

Notice that because of the reality condition (2.9), eq. (2.13) follows from eq. (2.12). It is then straightforward to verify that all world-sheet supersymmetry transformations (2.3), as well as the bosonic local symmetries (2.6) are not spoiled by twists satisfying (2.12) and (2.13). The same conditions could have been derived by considering the world-sheet supercurrent. Conditions (2.12) and (2.13) have an interesting group theoretical interpretation. Writing the fermion twists as a (unitary) matrix T = exp[2i~r diag(0x, 02, 03, 04)], eq. (2.13) simply tells us that det T = 1. T is then an element of SU(4) or SO(6), the fermions transforming like the representation 4 of SU(4). Then, eq. (2.12) means that the six bosons ~k, transform under T like the antisymmetric tensor 6 of SU(4) [which is also an SO(6) vector]. World-sheet supersymmetry thus implies that the group of our Abelian twists is a discrete subgroup of SO(6). This is a direct consequence of the equivalence between the ten-dimensional superstring theory and its dimensionally reduced, four-dimensional version. Notice also that orbifold compactifications lead to the same condition of the point group [11]. The light-cone formalism is obtained by going first to the conformal gauge, ~Zh-h~B = diag( - 1,1), and then imposing X+=

1

F+8 = O,

++

o), (2.14)

removing two bosonic and eight fermionic degrees of freedom. The equation of motion of the spinor S = 2V/p+ O is a two-dimensional Dirac equation: S is both a world-sheet and a space-time spinor, giving rise to four spinors S t and SRk after dimensional reduction. The constraints satisfied by twists reappear in the (non-lin-

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ear) Lorentz transformations: an explicit construction of Lorentz generators would also lead to (2.12) and (2.13), since it is world-sheet supersymmetry which permits the light-cone gauge choice (2.14) for fermions. We now turn to the relation of the boundary conditions with space-time supersymmetry, which arises in the fight-cone gauge from a combination of a global supersymmetry (2.5) and a compensating world-sheet supersymmetry transformation to preserve the gauge choice (2.14). After dimensional reduction to four dimensions, one obtains N = 4 supersymmetry, broken in general by boundary conditions. The relation between twists and space-time supersymmetry has already been discussed in ref. [8], by studying the generators of the non-linear super-Poincar4 algebra. One finds that if the twist of one fermion, say S 4, vanishes, then we have N = 1 supersymmetry. In this case, defining new scalar fields Z k = z k4, k = 1, 2, 3, eq. (2.12) indicates that each Z k has the same twist as its supersymmetric fermion partner S k, while the periodic S 4 is the partner of the periodic space-time coordinates X ". Eq. (2.13) implies that the group of twists is a discrete subgroup of SU(3) with fermions S k and bosons Z k (k = 1,2,3) in representation 3. Again, orbifold compactifications lead to the same condition [11]. In the same way, when two or four fermion twists vanish, the theory will have N = 2 or N = 4 supersymmetry respectively, the case N = 3 being forbidden, as it should, by condition (2.13). It is important to realize that having four non-zero fermion twists does not always lead to a non-supersymmetric string theory. Choosing q~kt= 0 ,

0 h = 21'

k, l = 1, " ' ' , 4,

(2.15)

which is a solution to (2.13) and (2.14) leads to a situation analogous to the Neveu-Schwarz and Ramond sectors of the old fermionic strings. In the twisted sector, fermions have massless first-level states which are space-time spinors. It is then possible to choose a GSO projection which retains these states, and also keeps in the untwisted sector their supersymmetric boson partners, restoring N = 4 space-time supersymmetry. This particular situation will be studied in more detail in sect. 4.

3. Construction of four-dimensional strings We have seen in sect. 2 that the condition to break space-time supersymmetry but preserve world-sheet supersymmetry and Lorentz invariance is a constraint on the boundary conditions of the fight-movers as given in eq. (2.13). The case of N = 1 space-time supersymmetry is a special case when the twist for one of the fermionic right-movers is zero. Thus, both supersymmetric and non-supersymmetric theories can be treated simultaneously, the only difference being the above-mentioned constraint.

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T o construct a consistently interacting string theory, we must ensure invariance under the modular group. To achieve this, it is sufficient to have modular invariance at the one- and two-loop levels [15]. This analysis, with the Green-Schwarz formulation and allowing for twisted boundary conditions for fermions and bosons, has been done in full detail in ref. [8]. To make this paper self-contained and for the reader to have a quick reference, we shall summarize here the main results that will be needed from that paper. For proofs, we must refer the reader to ref. [8]. The world-sheet of a one-loop graph is a torus with its points labelled by the complex variable z = o 1 + '/'02 where o 1 and 0 2 are real periodic variables and ~- is the modular parameter restricted to the fundamental region. The one-loop boundary conditions for a space-time spinor S t are 3 1 _ ( o 1 - t - ~ , o 2 ) = e-2irr°lsI (Ol, O2) , $1_(Ol, 02 q.- ~ ) = e 2i=°'SL(oa, 02).

(3.1)

The one-loop partition function for such a space-time spinor is given by ~ ( S a) = . . ~ =_ e2i~(o-1/a)(q,-1/2)

,

(01,),

(3.2)

being the Dedekind eta function and 0 the theta function [16]. The complex bosonic coordinates Z k satisfy the boundary conditions z k ( ol q- tiT" 0"2) = e - 2i'rOkzk( 01, 02) + crV k , g k ( o1, 02 q- ~ ) = e- 2i~fl~kzk( Ol,

0"2) -'F q'rU k .

(3.3)

The c o m m u t a t o r of the above relations gives uk(l

-- e -2i•0k) = v k ( l -- e-2i~ @ ).

(3.4)

In the case when Ok or ¢pk is non-zero, the string field is in a twisted sector but, when both are zero, it is in an untwisted sector and then there is no constraint on U k and V k. The partition function for a complex twisted boson is

~o (zk)twisted = (~,~ -1)@k, ok

(3.5)

while for the untwisted boson (i.e., when Ok = ~k = 0) it is ( z k )untwisted =

v~o~e ~u~u~

,

where Vx and Vy are the real and imaginary parts of V and similarly for U.

(3.6)

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We write the general partition function for a complex boson as

(1 -

eo,)(~-l)°,

"j-

~V.o~V.

D

* 0, *~U."~ o7, ,

(3.7)

where

Po, = 8008,0. Under the modular transformations, the function ~ f

1 T .,...> _ _ - - . ,.g

- ~ + 1:

,~f

_.+ e 2 i , r ( $ - 1 / 2 ) ( , -

transforms as 1/2)~,,~ 0 ,

~-,e,~(°'-e+l/6~L~.

(3.8)

We denote the set of boundary conditions in the o 1 direction by

w = (0r, ( ~ , w;, v{)l (,,, v;, vf)),

(3.9)

where the first four entries Or ( l = 1 , . . . , 4 ) are reserved for the right-moving space-time fermions and the triplets (q~, Vk~, Vky) are the boundary conditions for the right-moving bosons Z k (k = 1, 2, 3) while (q~,, V/~, Vtv) ( I = 1 . . . . . 11) are for the left-moving bosons. We decompose the vectors W into two orthogonal pieces, one corresponding to rotations and the other to translations

W= R ~ V,

R. V = O ,

(3.10)

where

R = (0r, ~ 1 ~ 1 ) , (3.11)

V = ( V~I U,).

Under the modular transformations the boundary conditions W and W' (in the o 1 and % directions) change, and modular invariance requires us to include a full set that closes. We denote the general set of vectors W by

AW= E a m R , ~ £ceVe, m

p

where each of the a m and cp take the possible values a,, = 0,1 . . . . . N m - 1 ,

Cp= 0,1 . . . . . Np-1

(3.12)

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and N m, N e are defined by

UmR m = 0 (mod 1),

NpVp = 0 (mod 1).

By taking all possible values for a,, and dp we generate the full set. However, since a m and ce are in general independent and we would like to accommodate the case where the action of a twist is accompanied with a shift (i.e., identifying some of the a ' s with some c's) we write the full set as AW

= ZamWm--]m

ECpWp p

- ~.~a,,,R,,, + Ecp,Vp,. m

(3.13)

p'

If we choose the entries of IV to belong to the interval [0,1[ then under modular transformations, they transform remaining in this interval. This is not true for the vectors A IV, and we thus define the new vectors A W, all of their entries belonging to the interval [0,1[, by

AIV=AIV

(modl).

(3.14)

We write the general partition function in the form ~=

~_, t~AW °leAW ,,-,BW,~-. BW , A,B

and require invariance under modular transformations. At one loop this amounts to the constraints

CS_~= exp{ ZiTr(A W - Wo) . ~ah. ( B W - Wo) } C~--~, C .4~ ~ - - exp{ i~r(A I v - a + Wo) • fi"b.(AW-- a_Wo)}C~w,

(3.15)

where /3ab is a diagonal matrix given by ~ a b _~_ diag(14, ( p f f b _ 1,

pkab'~P,~b)l(P['t'-- l,p]'b,P~b)),

(3.16)

p~b = 8~k, ° 8_~k,o while a += 1 + ~ / 3 and Iv0 is a vector whose components are all 1 Invariance under the two-loop modular transformations is more difficult to analyze because of the presence of space-time fermions and twisted bosons. We assume that this could

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be equivalently replaced by a generalized GSO projection [17] of the form ~ t--AW^ - 2 ~ i ( ~ -

Vo).['"h.N~-w_

e- 2~i~(,07/(~),

(3.17)

B

where e(a) takes the values zero or ½, and ~ is a function that takes the values zero or one. The components of the vector N ~ are the number operators with integer eigenvalues

N ~ = (N~(S'),(-N~k, Nv~)[(-Nh~,,Nv,)).

(3.18)

The number operators are given by

n=l

Na~(Zk) = ~ ( A ~ + ~ - I

-

A ,'k_ ~ ) ,

n=l

Nv(Zk)=(Kkx,Kky),

(3.19)

where the Sn's and An's are the number operators corresponding to oscillators appearing in the expansion of the string fields S and Z, while the K ' s are the momenta of the field Z belonging to a lattice. As a consequence of eqs. (3.15) and (3.17) we can solve for C~w Aw to obtain

)-1

C~-

I--[NmFINp exp(-Zi~raW. \

m

p

~aO. Wo )

J

_

(3.20) [q

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where the matrices k and T are defined by (modulo 1) (3.21a) ~ Wm •

• Wm -k W O"

ab _I_ kq,p ab = Vp. pab • Vq, kpq, 1 ~.°~- k ;?o= ~v~

~ab

Wm - k o m ,

(3.21b) (3.21c)

q' = q or m,

(3.2ad)

. v~,

and provided the vectors W,,, satisfy the constraints 2Ro. fi ab. Wm=O

(mod 1),

(3.22a)

k~__lP/~b)= 0

(mod 1).

(3.22b)

1/ 11

The total partition function can be written in the form &~= E C~T~['~H'(AVV~anr(AW)'~ 2i~(KW-V°)P°~'N~]

(3.23)

A,B

where H t and H r are the hamiltonians for the left- and right-moving fields respectively• By combining eqs. (3.20) and (3•23) we deduce that all states will be projected out except those satisfying Wm • p,,b. NA~= ~ Zn~nan ab __ Wrn. ~ab "aW n q- Z k m q,~bC q q Vp. e~a b .

N~=

Vm. ~ab . (aV + c V _ al2) + k ~

(mod 1), (3.24a)

E k p qab c q 7!- E k p maba m q m

- v~. b o~. ( a ~ )

+ ko~

(rood ~).

(3.24b)

From the above equations and the definition of N,, and Np, we can deduce the

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following constraints on the vectors Wm and Vp N m-rim T ab = 0

(modl),

(3.25a)

N,.kq~ = 0

(mod 1),

(3.25b)

Np,k qapb,= 0

(mod 1),

(3.25c)

1 Nm(~Wm, P~ab • Wm'-[- Wo. P ab. Win)=0

(modl),

(3.25d)

Np( I Vp. pab. Vp--}- Vo. P ab. Vp) = 0

(modl).

(3.25e)

By substituting the expression for the hamiltonian in the partition function, we can read the masses of the spectra. For the right-movers we have 2

m2(right,AW) = Z

~ nAa

a=l n=l

4

o0

+ z z /=1 n=l

3 tk + E (1--eka) ~ [(n+-d-~k--1)Akn+h~-l+(rt--adP--k )An-h~] k=l

n=l

..j_ k~=lek a 1 K~ + aVx + cVxk_ ½

+ n=lnA'x+x'~Y

(3.26) where Pka = ~ ~ ~,0"

Similarly for the left-movers mE(left,A-W)

11[ ()]

=

1 + ½ )-". a--~'- ~__~, 2 + E

-

I=1

E "~a

a=l n=l

11

+

y" ( I - P / a ) ~ [(n +a--~'-1)A/+~_1 + ( n - a q-J- 1 ) A~t, _1 ~ ] I=1

11[(

n=l

1 K ~I + a V ~ + - J ~ - 1 + I~=1 v,o -~ I

n.ff I x + x ,~ y

+

tl~l

1

(3.27)

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and must satisfy the level matching condition m 2 = m2(left) = ma(right).

(3.28)

We adopt the convention for the shifted vectors that instead of writing the x and y components, we shall double the range of the right-movers to six (denoted by k*) and the left-movers to 22 (denoted by I*). One further constraint that we must take into account comes from the non-abelian nature of the boundary conditions in eq. (3.3) and requires that the twists of the momentum lattice (pk*, p~*) have a crystallographic action on the lattice [11]. Those constraints are almost trivially satisfied for Z 2, Z 3, Z 4 and Z 6 twists but rule out anything interesting for the Z 5 twist. Finally, we mention that if we wish to impose the action of a discrete symmetry in coupling the right-moving states to the left-moving ones, then this discrete symmetry must be a true symmetry of the partition function otherwise one will encounter gauge anomalies a n d / o r missing opposite helicity states.

4. Some examples In this section we will construct simple examples of four-dimensional strings, without space-time supersymmetry and tachyon-free, following the rules of the preceding sections. It will be done mainly for illustration of the method. The reader interested only in realistic models can jump directly to sects. 5 and 6. 4.1.

Z2

MODELS

The simplest possibility is to allow only untwisted bosons both in the left- and in the right-moving sectors. In that case, world-sheet supersymmetry a n d / o r Lorentz invariance, eq. (2.13), imply for the twist vector R the unique choice R = ((1)4,03[011).

(4.1)

Concerning shift-vectors, we will make the choice V?, p = O, 1, 2 given by Vo=((1

~) ° I(~) 1 22 ) ,

V1 = (061014,

1 8

Vz = (06106, (½)8,08), leading to rank-22 gauge groups containing E 8 × E s or SO(16) × SO(16).

(4.2)

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A word of warning has to be said here. Z 2 models are exceptional in that the twist (4.1) does not guarantee supersymmetry breaking. World-sheet fermions in the right-moving sector with boundary conditions (4.1) are real (Ramond-NeveuSchwarz fermions) and supersymmetry can be restored in the twisted sector. This happens in models with one single twist accompanied by an arbitrary shift-vector. These models are generated by the basis vectors 2

AW=aW+

Y'~ CqVq, q=0

2 W = R -~- E bqVq, q=0

(4.3)

where the parameters a, Cq a n d bq can take the values zero or one. The set bq is fixed and determines the model. The conditions of one-loop modular invariance and GSO projection translate into the following set of (mod 1) equations, eq. (3.21) kpp = k p o , k pq = k qp, T = ½q- E b q k q o , q

where all

kpq

(p, q= 0,1,2)

(4.4)

are either zero or one-half, and

Vp" NA~

=

Ekpq(Cq -t- abq) + kop

(p =0,1,2),

q

R" NA~ = l a ,

(4.5)

where N ~ is the number operator vector in the notation of sect. 3. The simplest way of checking that models defined by (4.3) are supersymmetric is to evaluate the cosmological constant. It is given by e

d2r

aco, = J 7£771

(4.6)

where integration is extended over a fundamental domain of the complex plane, ~ is the Dedekind eta function and &~'T the total partition function corresponding to the

A.H. Chamseddine et al. / Strings

155

four right-moving Weyl fermions and three right-moving and 11 left-moving complex bosons,

~'r = ~ ~t~w~aw~,

(4.7)

A,A'

where the coefficients C~,~ are constrained by modular invariance. For models defined by (4.3) we can write ' A W _ ( o'fa/2]4[ O,l~Bo/Z]12[ Oja(Bo+B2)/218[ Olt,(Bo+B1)/2]8

~w--t~/2

] t~B6/2 : t~(B6+BD/2 : t~(B6+B{)/2 : ,

(4.8)

where

Bp -~- Cp "~ abp,

8; = c'p + a'bp

( p = 0,1,2)

(4.9)

and Bp, B~ = 0,1 (mod2). We can use (a, 177) as independent variables. Then, the coefficients Ca,~, Aw constrained by modular invariance, can be written as

C A~ W - _ (½)4exp{27riI½aa' + k ( Bp,

B;)]}

(4.10)

where A=

ZB;k,qBq+ P,q

+

+ koo

(4.11)

P

does not depend on a and a'. Using (4.8)-(4.11), we can factorize out in ~ T fermions as

the contribution from Weyl

1

~ T O~

Z e~'iaa'( °'fa/2~4 a'/2 ] " a,a'=O

(4.12)

The relevant partition functions here are given by

, ~ 1 / 2 ~---'1~- 1 ( T ) 04 ( 'r ) ,

6•1/2 1/2

-~ T ] - I ( T ) 03(q') '

(4.13)

and satisfy the identity 0 4 ( r ) + 04($) - 0 4 ( r ) = 0.

(4.14)

We obtain from (4,12)-(4.14) that =,~eT = 0 and so the cosmological constant vanishes. The reason is that supersymmetry is unbroken; actually, it is restored by

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the twisted sector. The massless spectrum can be worked out and corresponds to N = 4 supergravity coupled to an SO(12) × SO(16) × SO(16) [SO(12) x E 8 × Es] Yang-MiUs supermultiplet if k12 = 1 [k12 = 0]. So far we have studied the most general Z 2 models generated by a basis with one single twist and found that supersymmetry is restored by the twisted sector in all cases. Supersymmetry breaking is triggered in models with two independent twists and two independent shift vectors. One possible choice is AW=

a o W o + a l W 1 + coV0 + Cl(V1 + V2)

Wo = R + V o,

WI = R +

V1

(4.15)

where a 0, a 1, c o, c I = 0,1, and will be analyzed next. Any other choice with two independent twists would lead to equivalent results. The modular invariance conditions, eqs. (3.21) and (3.24) translate into eq. (4.4) for kpq (which are either zero or one-half), as well as

Too= ½ + k oo, T11 = 1 ..[_ kol,

Txo = To1,

(4.16)

which are constants that can take values zero or one-half, and Vo • N ~

= k o o ( a o + c o + 1) + k o l ( a 1 + Cl) -b ko2c1,

V1 • NW~= (a o + al)(Tol + ½) + kol(C o + c 1 + 1) + k 1 2 Q , 112"N~W= (a0 + al)(T01 + k01 + ½) + k o 2 ( a o + Co + ¢1 "[- 1) + k , 2 ( a 1 + ca), R. N~=

~(a 0 + al) + al(Tol + k m + ~).

(4.17)

In the untwisted sector [ a 0 + a I = 0 (rood 2)] only subsectors Vo, V0+ V1 and Vo + V2 contain massless states. For right-movers the massless state is the ground state la)R + IZk>R + IS~)R,

(4.18)

where Z k are three complex scalars corresponding to the six compactified dimensions and 1 is an index transforming as the 4 of SU(4) and corresponds to the four Weyl spinors in the right-moving (supersymmetric) sector. The introduction of the twist (4.1) makes possible the introduction of a Z 2 parity P under which la)R and IZK)R are even ( P = 1) while IS~)R is odd ( P = - 1 ) . Because the twist R is

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157

accompanied by V0 in (4.15) we can also define a Z 2 parity for states with quantized m o m e n t a in winding sectors as

P = e x p [ - 2i~rVo . NAw ]

(4.19)

The parity of a state is P --- PRPL and only states with P = + 1 will be retained in the physical spectrum. For left-movers the subsectors V0, Vo + V1 and Vo + V2 have to be analyzed separately. The massless state in the V0 subsector, consistent with (4.17), is Ib)L + (66,1,1) L + (1,120,1) L + (1,1,120) L where the last three terms are the adjoint representation of SO(12) × SO(16) x SO(16). These states have PR = + 1. The massless states in subsectors Vo + V1 and V0 + V2 are projected out by eq. (4.17) unless TOl q- kol = ~ and k12 = 0. In that case the retained massless states are (1,1,128(X,)) L + (1,128(x2),1) L, where Xe = e x p ( - 2 i ~ k o p ) is the chirality and coincides with the Z 2 p a n t y PL of the state by virtue of eq. (4.17). In the twisted sector [ a o + a 1 = 1 (rood 2)] massless states are in subsectors R + V o, R + Vo + V 1, and R + V 0+V2, and the tachyonic state in R + V o. The massless states consistent with modular invariance conditions, eq. (4.17), in the subsector R + Vo are* S l_ 1/21~2) R ~ [1b) L + (66, 1, 1) L + (1,120, 1) L + (1_,1, 12_O0)L] if condition T01 + kol = ~ is satisfied. If To1 + h'01 = 0, the massless state is St1/2[J2)g ® (16, 16)L, where 16 is the vector representation of SO(16). The state S~ 1/2[f2)g is assigned a Z 2 parity PR = + 1, as in ref. [2], and we denote by [COp the corresponding parameters in the twisted sector. The tachyonic state l l2)g ® 44 L, where 44 is the vector representation of SO(44), is projected out from the physical spectrum by conditions (4.17). Massless states in subsectors R + V0 + V1 and R + V0 + V2 consistent with (4.17) are S~1/21~2)R ® [(1,1,128(Xl))L + (1,128(X2),l)L ] if To1 + ] £ O l--5 1 and k12 = 0, and 12g ® [(1,1,128(_xl)) L + (1,128(_x2),l)elif To~ + kol = 0 and k12 _- ~-1 The chiralities of spinors __128 are again given by )tp = exp(-2i~r~:0p ) and coincide with the corresponding Z 2 parity PL of the state. The state 12R, in the vector representation of $O(12), has PR = - 1 ; this sign is compensated by the change of chirality of spinors 128 in the left-moving sector. It is worth noticing at this point that we can choose the chiralities of spinors 128 in the twisted (~p) and untwisted (Xp) sectors as equal [i.e., loop = kop ] or opposite [i.e., k0p = kop + ½ (mod 1)]. In general different values of the constants To1, k12, kop and kop will give rise to different models, supersymmetric or not. The best way of classifying the different models is by computing the one-loop cosmological constant as a function of the total partition function, eqs. (4.6) and (4.7). The partition function for the models (4.15) can be written as AW__ ( 7 A o / 2 ]

4

(7A,/2]

6

( 7 A2/2 ]12 { y A3/2 ]12

* [~2)R is the scalar ground state and S~a/2 are fermionic oscillator creation operators.

(4.20)

A.H. Chamseddine et al. / Strings

158 where

A0 = a o +

a I ,

A 2 = A 1 + c1,

and Ai, A ,' - 0 , 1 invariance to be

A1 = a o + co , (4.21)

A3 = A 2 + a 1

(mod2). The coefficients C ~Aw are constrained by modular

AW__ C~y w - (½)4exp{2rri[1AoA, ° + (To1 + koI q_ 1)

X[Ao(A ; - A I ) + A ; ( A

3-A2) ] +a]},

(4.22)

where A = koo(Al+

1 ) ( A [ - 1) + k m[(A 3 + 1 ) ( A ; - 1) - (A 1 + 1 ) ( A ~ - 1)]

q-ko2 [(A2 + 1 ) ( a ; - 1) - (A 1 + 1 ) ( A [ - 1)] +klz[(A3-A2)(A ~-A[) + (A;-AI)(A 2-A1) ]

(4.23)

does not depend on A 0 and A6. Notice that compared with eq. (3.21.c) we use here lo _- kop n = kop. ~ Using that ~ o _ 0, we can cast the notations k ~ = kop and ko°~= kop the total partition function as

"~)J'T~ E e~iA°A°['~)~'A°/2~4 k A;/2 ] Ao, A~

xexp{2~ri(Tol+/cOl + ½ ) [ A o ( A ; - A I ) + A ~ ) ( A 3 - A 2 ) ] } "

(4.24)

If To1 +/c01 = 1, we can see from (4.24), (4.13) and (4.14) that ~ r = 0 and the theory is supersymmetric. Its massless sector corresponds to N = 4 supergravity coupled to SO(12) x SO(16) x SO(16) [SO(12) x E 8 × E8] Yang-Mills if k12 = 0 [k12 =

If Tm +/cm = 0, ~'T =~0 from (4.24) and the theory is non-supersymmetric. The massless spectrum will depend on kpq and /COpand contains, in each case, a subset of the set of states (V, F, if, S) where V= [[a)R + [zk)R] ® [[b)L + (66,1,1)L + (1_,120,1)L + (1,1,120)L ] , F = ISIS)R® [(1,1_,128)L + (1,128,1)e ] ,

f = scl/2Lsa). ® (!,16,16),, S = 12 R ® [(1,1_, 128)L + (1,128, 1)C ] ,

(4.25)

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A.H. Chamseddine et al. / Strings

which corresponds to a non-supersymmetric theory with gauge group SO(12)× SO(16) x SO(16) and different matter contents. The states F, ff and S correspond to four fermions from the untwisted sector, four fermions from the twisted sector and 12 scalars in the corresponding representation of the gauge group, respectively. If k12 = 1 the massless spectrum is (V, if, S). If k12 = 0, the massless spectrum will depend on the relationship b e t w e e n kop and kop..If ~:op= kop the massless states are (V, if) and if /Cop= kop + 1 they are (V, F, F). This is the four-dimensional version of the (tachyon-free) ten-dimensional non-supersymmetric model of refs. [2,3]. The gauge group can be reduced to U 6 × SO(16)× SO(16) by using five compactification-like shift vectors as in ref. [8] and sects. 5 and 6 without introducing any new massless state. The extra U 6 gauge group would then correspond to the toroidal compactification from ten to four dimensions of the model of refs. [2, 3].

4.2. Z 3 MODELS

In models with Z 3 discrete symmetry, the conditions of world-sheet supersymmetry, Lorentz invariance, eq. (2.13), and no space-time supersymmetry fix (up to trivial world-sheet field redefinitions) the twists for fermions and bosons in the right-moving sector. The twists for left-moving bosons are constrained by the modular invariance conditions, eqs. (3.22) and (3.25d). These conditions imply that a shift-vector should necessarily accompany the twist. The only possibilities consistent with modular invariance and leading to gauge groups containing SO(10) are W 1 = R 1 -t- V 1, where R1 = ( ( 72) 2 ( 71) 2 ;17, 0 2117,010)' VIi = (0 6 021 1

(4.26)

or W2 = R 2 + V2' where R2=((2)2(1

7) 2 2 2

2 1 ~ 2 l ! (}2 (l 3) ; ~ , v 1 3 , v ,

V2t = (06[019 7,3, 2 1 0)

4

) (4.27)

both of them being symmetric orbifold-like constructions. Unfortunately the assignment of Z 3 for right-moving fermions in (4.26) and (4.27) does not give a chiral theory. Actually I S l ) R ( l = 1,2) transforms under Z 3 as ) t = e x p ( - 3 H r ) while ] S ] ) R ( ] = 3,4) transforms as X-1. So any left-moving state coupled to [S~)R to

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f o r m a fermion, will also be coupled to [S~+)R to form its mirror fermion. The way out would be to introduce a second twist under which at least one of the fermions is neutral. A possibility would be to introduce, as a second twist, W3 = R 3 + V3, where

R3 =

V3 = (06[019(1)3),

(4.28)

which is the twist, leaving N = 1 supersymmetry, used in the supersymmetric Z-orbifold construction. However, the n u m b e r of twisted sectors is eight and the massless spectrum is far from being realistic. A similar example with Z 3 × Z 3 discrete s y m m e t r y has been worked out in ref. [4] and we shall not dwell u p o n it any longer. A second possibility based on the discrete symmetry Z 2 x Z 3 = Z 6 will be discussed in sect. 5.

5. A rank 8, E 6 X U(1) 2 Z 6 model

H a v i n g examined models based on twists with the discrete symmetries of Z 2 and Z 3, the natural thing to examine next is based on Z 2 x Z 3 = Z 6 symmetry. W e consider the following set of b o u n d a r y conditions

(5.1)

a W + coV0 + q V x + 8~0c3V3,

where a = 0 . . . . . 5, c 0, q = 0,1 and c 3 = 0,1, 2. V0, V1 and V3 are given in eqs. (4.2) and (4.28) while W = R + V6

1 2 (~) , 2 ;111 R~_~((~) 176= (061019~½).

11 1 2 2 2 4

)

(5.2)

The entries in the vector W are chosen to satisfy all constraints such as eqs. (3.22) and (3.25d) as well as eq. (2.13). We stress that finding a vector satisfying all constraints is highly non-trivial and if one further requires a model which is not vector-like, avoids tachyons and breaks the rank-22 gauge group to rank 8, one will end up with very few choices. [Another consistent choice of W is possible with the last four non-zero entries in R replaced by (g) ~ 2(g) 5 2 but gives a two-family model with either E 6 x U ( I ) 2 or SO(10) X U(1) 3 gauge group.] Since Z 6 = Z 2 × Z 3, it is obvious that the twist vectors a W can be replaced by a l W 1 + a 2 W2 where

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W I = R I + V[, W z = R 2 + V2', al = 0 , 1 , a 2 = 0 , 1 , 2 and R1

[[1 "~2~2. 1"11 1 ~1 1~8~

2j ~ , ~ i 2 ~

001'

v,'= (o 6 10191

1,

'

'

),

1

V2'= (06[0192~0) .

(5.3)

Thus the model presented here can be thought of as a natural extension of those in sect. 4. It will be easier to deal with one twist and we will thus work with the set given in eqs. (5.1) and (5.2). When a = 0 we have the untwisted sector with the subsectors coVo + ClV1 + c3V3. By inspecting eq. (3.26) it is clear that massless spectra would occur if all the entries of the right-moving shift vectors are ½. Thus the only subsectors giving the massless states are V0, Vo + V1, Vo + V3, V0 + 2V3, V0 + V1 + V3 and Vo + VI + 2V3, and they share for the right-movers the massless ground states. These are given by the vectors [a)g with the eigenvalue 1 under the Z 6 symmetry, the complex scalar states IZk)R, k = 1,2,3 with eigenvalues (X -2, X-3, ~-3) where ~ k = e -2~i/6, and the fermionic states [St_)R, l = 1 . . . . . 4 with eigenvalues (X -1, X-1, X -2, X-2) as well as the opposite helicities for [Z k) and 1S/_) with the complex conjugate eigenvalues. For the left-movers starting with the subsector Vo we see from eq. (3.27) that the massless states correspond to the following possibilities. First we can either set A~ or .~2 to 1 (always taking all the other numbers to be zero), giving the vector ground state [a)L. Next we set one of the 22 AIx and Axle to 1 giving the states [A[)L = [Ax~x+ i.4[y) with eigenvalues ?~z= (~z, X3, ?d, ?~2,~2, ~4, ~4) for I = 1 . . . . . 7 with the complex conjugates taking the conjugate eigenvalues, while for I = 8 . . . . . 11 they are neutral and will provide the 8 Cartan elements for the surviving rank 8 gauge group. Finally a solution of eq. (3.27) is given by KI* = ( 0 . . . 0 + 1 0 . . . 0 +_ 1 0 . . . 0 ) ,

I * = 1 . . . . . 22

(5.4)

After imposing the constraints from eq. (3.24)

~.N~=~.N~=~.N~=W.N~=O. the solution in eq. (5.4) reduces to K I* = ( 0 . . . + 1 . . . + 1 . . . 0 ) ,

I* = 1 . . . . . 14,

(5.5)

A.H. Chamseddine et al. / Strings

162 or

K I* = ( 0 . . . _+1... _+1 . . . 0 ) ,

I* = 15 . . . . . 19.

(5.6)

It is easy to see that these correspond, together with the 8 Cartan generators given before, to the group SO(10)× U(1) 3 as well as all non-zero roots of the group SO(28). To avoid such SO(28) vector particles in the massless states, we introduce a set of vectors of a form that appears in the trivial compactification of string theories [12]. These vectors are of the form ( 1 0 5 g105 v , (116] v : + four permutations of the 1 in t h e first

five right and left entries, ( 1 0 5 [0 5130 16 ) + four permutations of the 1 i n t h e f i r s t

five fight and six to ten left entries, (g0 l0 g0 ) + three permutations of the ?1 in the first four right and 11 to 14 left entries. 1 5

101 15

Such vectors do not introduce any new massless states but imply the constraints K 1 = ... = K 14 = 0.

(5.7)

The above mechanism is equivalent to a rotation in the lorentzian lattice SO(6,14) from an SO(28) case to a U(1) 14. Here, however, the U(1) 14 is destroyed by the twists on the 14 bosonic elements (seven complex). From here on we shall therefore only have massless states coming from the last eight fields, I* = 15 . . . . . 22. Coupling left-movers to right-movers forming Z 6 invariant states, we obtain from the subsector V0 the following states: [a)R ® [b)L giving a graviton, an anti-symmetric field and a dilaton, [a)R ® ]SO(10) X U(1)3)L gauge vectors, ISt_)R ® IA[) L,

l=3,4,

I=1,4,5,

Is~-)R ® IAI*')I.,

•=3,4,

•=6,7,

Iz~)R®(I

I>L, IA4>L,I ~>n,

[Zk)R ® IA()L,

~ ]L,

k = 2,3,

1 }L],

•=6,7,

(5.8)

as well as the complex conjugates for the opposite helicity states to form ten fermionic and nine bosonic gauge singlets. From the subsector V0 + V1 by solving eq. (3.27) we get the state Kl*=(al

..... a8) ,

ale

{0,1}.

(5.9)

A.H. Chamseddineet al. / Strings

163

F r o m the constraints of eq. (3.24) on this solution we obtain Vo" Nv77pT, = V1 • Nv-Tv>7~= kin, V3 • Nv77pT, = 0,

V6 • Nv~W-~ = ~ + k61 ,

(5.10)

where kol = 0 or ~ and will always give equivalent states: we thus choose it to be 0. S i m i l a r l y k61 = 0 or ½ but give inequivalent possibilities since when k61 = 0 all the states in eq. (5.9) are projected out, while if k61 = ½ the states* [(05) + (03, 12) + (0,14)][0, 0, 0] + [(04,1) + (02,13) + (lS)][1,1,1] (which is 16 + ]-6) will remain. These couple to the vector la>R enlarging SO(10) x U(1) to E 6. Since the first possibility based on the SO(10) x U ( I ) 3 group gives two fermionic 16 + ]-6 families as can be seen later f r o m the analysis, we shall fix k61 to I" F r o m the subsector V0 + V3, solving eq. (3.27) we get the states

K'" = (0')(1,02 )

and

( --[-1 , 0 4 ) ( - 12,0) ,

1 * = 1 5 ..... 22.

(5.11)

T h e constraints of eq. (3.24) on this solution read

Vo" N ~ =

Va " N v ~ ¢ ~ = ½,

V3" N v o ; ~ = 1, V6 ' N v T v ~

=

k63

3x ,

(5.12)

where k 6 3 = 0 , ~ or 2. Only if k63 = 2 do the states K I * = ( 0 5 ) ( 0 , 1 , 0 ) + ( + 1, 04)( - 1, 0, 1)[1 + 10] survive. Since V6 • N determines the eigenvalue of the state u n d e r the Z 6 symmetry, we see that the [1 + 10] couples to the positive helicity fermions IS/)R, l = 3,4 and the complex b o s o n IZ1)R (with X 2 eigenvalue) to f o r m a neutral state. It is not difficult to check that the negative helicity states come f r o m V0 + 2173. The subsector V0 + V1 + V3 gives u p o n solving eq. (3.27) the states ai~ (0,1}.

(5.13)

2 V6"Nvo+VI+V3 = k63 + 3.

(5.14)

( a 1. . . . . a 5 ) ( 1 , 0 2 ) , T h e constraints of eq, (3.24) are ~.N~=~.N~+~+~=0,

V3"Nvo+vI+v3 = ±3'

Again, only if k63 = 2 the state [(05) + (03,12) + (0,14)][1,0,0] = 16 survives and since V6 • N = 3, the 16 will combine with the 1 + 10 to form a 27 of E 6 and couples '~ We use the notation (0m, 1") for all permutations of m zeros and n ones among the ruth entries.

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A.H. Chamseddine et al. / Strings

to the ISI)R, l = 3,4 and [Z1)R . As expected the opposite helicity states will come f r o m the Vo + V1 + 2V3 subsector. T o summarize the untwisted sector, we have the following massless states: a graviton, an antisymmetric tensor, a dilaton, E 6 X U(1) 2 gauge group (provided k61 = ½), 2- (27) chiral fermions, 1 • (27) complex bosons as well as ten fermions and nine complex boson gauge singlets. W e now analyze the twisted sectors. Those are a W , a W + Vo, a W + V 1, where a = 1, 2, 3, 4, 5. We shall write explicitly the cases a = 1, 2 and 3 only since a = 4, 5 give the opposite helicity states to a = 2,1 respectively. W h e n a = 1, the vacuum energy for the right-movers as evaluated from the last term in eq. (3.26) is zero. Since all the right-moving bosons and fermions are twisted except for the space-time coordinates, the ground state is a scalar bosonic doublet 7 (or equivalently a complex boson). The vacuum energy for the left-movers is 36 and b y looking into eq. (3.27) for the three possibilities W, W + V0, W + V1, we find that only the last two give one massless state each, K ' * = (05)(0,0, a ) ,

a~ (0,1),

I * = 1 5 . . . . . 22.

(5.15)

Writing the projection conditions Vo . N w ~ v o

= V 1 • N W ~ v o ° = O,

W. N~-~o

= 0

(5.16)

and similarly for W + V1, the remaining singlets are K t* = (05)(03), 1" = 15 . . . . . 22 f r o m the two sectors. Coupling them to the right-movers we have two complex bosons. The rotation vector in this case ( a = 1) has a large n u m b e r of fixed points, b u t fortunately in this case there are only singlets. The complex conjugate states c o m e from the sectors with a = 5. N e x t when a = 2 the vacuum energy for the right-movers is - 1. This could imply that tachyons could be present and we must show that they are projected out. The v a c u u m energy for the left-movers is - ~. 4 F r o m the subsectors 2W, 2 W + V1 and 2 W + V0 + V1, we get only massive states while from 2 W + V0 by writing eq. (3.27) we find that there is a tachyon for K t * = (05)(0,0,0), I * = 1 5 , . . . , 2 2 which can couple to the tachyonic ground state and massless states K ~* = ( 0 5 ) ( - 1 , 1 , 0) as well

as j~,l \ z~,4 \ tl,5 \ ~6 ~7 1/3/L, 1/3/L, 1/3/L, I 1/3)L, ] 1/3)L, which must couple to the massless right-moving states I

,3

,1 IAt/3)R •

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A.H. Chamseddine et al. / Strings

Writing the projection conditions from eq. (3.24) we obtain

W" N 2 ~ o

(5.17)

= ½

and it is easily seen that the tachyonic state is projected out since W. N gives 0 and not 1. The states satisfying the above constraints are

t4 ., 1 1/3) tl .) ® (05)(-1,1,0), (Is;33 ., Islj3 •=4,5,6,7

(IS1/3)R,[SZ/3)R)®([A~I/3)L,[A~/3)L),

(5.18)

with the opposite helicity states coming from the sector 4W + V0. We thus get from the a = 2 and a -- 4 twisted sectors 12 fermions and one complex boson singlet, to be multiplied by the number of fixed points (27 in this case). Finally, we have the a = 3 twisted sector. The vacuum energy for the right-movers is zero while for the left-movers it is - 3. From the subsectors 3W, 3W + Vx, we get only massive states. By solving eq. (3.27) for the vector 3W + V0 we obtain the massless left-moving states

K'* = (+1,04)(a1,0, a:) ( 0 5 ) ( a l , d-I, a2)

(I52

)( al,0, a2)

(5.19)

and must couple to the right-moving vacuum state. Since the a = 3 sector is self-conjugate it must provide both helicity states implying that the doublets of complex bosons (coming from X a and Z 1) and fermions (coming from S 3 and S 4) will split into helicity up and helicity down. Thus the ground state is one complex boson and one fermion. The projection condition constraints are ~'N3~oo=~N~-~, W . N 3 ~ + Vo = O .

_

1

(5.20)

The surviving states are ( _ _ _ 1 , 0 4 ) ( 0 , 0 , 1 ) + ( 0 5 ) ( 1 , 1 , 0 ) = ( 1 0 + 1 ) as well as ([52/2), 1.43/2))(0, 0, 0) and will couple to the right-moving state to give a 2 - ( 1 + 10) and four gauge singlets of complex bosons and chiral fermions. We note that in this case we obtain all helicity states from the a = 3 sector. Similarly, from the subsector

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3W + Vo + V1 we get the masless state KZ*=(al

.....

as)(O, a 6 , 0 ) ,

a i ~ (0,1),

I * = 1 5 . . . . . 22.

(5.21)

The projection conditions of eq. (3.24) give Vo " N 3 w + Vo+ vl -

VI " N 3 w + Vo+ ~

W " N 3 w + v0+ v, =

x3

_ 1

2,

(5.22)

and when applied to eq. (5.21) give the state [(05 ) + (03,12 ) + (0,14)](0,1,0) = 16, and will join the 1 + 10 from 3 W + V0 to form a 27 of E 6. Thus apart from the gauge singlets we obtain 1 • (27) chiral fermions and 1 • (27) complex bosons in the twisted sector. This has to be multiplied by the number of fixed points (tori). The sector 3R has 16 fixed tori which are also left invariant by R and 2R, implying that the multiplicity of the sector is 16. Combining this with the untwisted sector we have a non-supersymmetric four-dimensional string model with E 6 )< U ( I ) 2 gauge group [no extra U(1)'s or hidden sector] and 18 chiral (27) fermions and 17 complex (27) bosons as well as many fermionic and bosonic gauge singlets.

6. A four-generation Z 4 model

We will present a very simple model with four generations of chiral fermions and exhibiting two very interesting features. The first is that it contains, in the massless sector, a scalar in the adjoint representation of the gauge group. The second is that the massless spectrum in the sector of matter generations is supersymmetric. We consider the following set of boundary conditions

aW--}- coV0 -t- ClV1 -]- ~aoC3V3,

(6.1)

where a = 0 . . . . . 3, Co, ¢1 = 0 , 1 and c 3 = 0,1, 2. V0, V1 and V3 are given in eqs. (4.2) and (4.28). The twist W = R + V4 with Z 4 discrete symmetry is constrained by the requirement of world-sheet supersymmetry, eq. (2.13), and modular invariance, eqs. (3.22) and (3.25d). The unique choice leading to a theory with chiral fermions, tachyon-free and gauge group containing SO(10) is

([1"]2!3 3 1 ,,.o, ~,~lO,¼, 1

[/4= (06[ 019, 1, 1,0).

(1)4,o4 ) (6.2)

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Let us first examine the untwisted sector (a = 0). From the expression for mZ(right, A W), eq. (3.26), we see that only subsectors with all entries of the right-moving shift-vectors equal to ½ can have massless states. These are Vo + clV1 + c3V3 and the massless states correspond to the ground state (4.18). The vector la)R and the scalar IZI)R are neutral under the Z 4 symmetry, the scalar states JZk)R, k = 2, 3, transform under Z 4 with eigenvalues (~, ?d), where X = e x p ( - 2 ~ r i / 4 ) , and the fermionic states ISt_)R, l = 1 . . . . . 4 with eigenvalues (~2, ?~2,h3, ~) while the opposite helicity states transform with the complex conjugate eigenvalues. For left-movers we have to examine separately the different subsectors with m2(left, A W) given by (3.27). In the V0 subsector the massless states correspond to different possibilities. First, .~b = 1, for b equal to 1 or 2, gives the vector state [b)L, neutral under Z 4. When tensored with la)~ + IZ1)R it gives the graviton, an antisymmetric tensor field (axion), a dilaton and two graviphotons (remnants of N = 4 supergravity). Second, A [ * = 1 for some I* from 1 to 22. For I* = 3 , . . . , 1 4 we can form six complex states and their complex conjugates transforming under Z 4 with eigenvalues (~3, ~, ~2, X2, ~2, ~2). They can be coupled to the ground state (4.18) making gauge singlets: 12 Weyt fermions and four complex scalars. Finally, a solution to m2(left, V0) = 0 is given by K I* = ( . . . + 1 . . . + 1 . . . ) , I* = 1. . . . . 22. Since the Cartan generators of SO(24) have been broken by the twist (6.2) we need to introduce a set of compactification-like shift-vectors to get rid of the non-Cartan generators. This can be easily done without introducing new massless states as described in sect. 5. In this way K I*= O, I * = 1 . . . . . 14, getting rid of all non-Cartan generators of SO(28) which contains the SO(24) broken by (6.2) and an SO(4) which appears from compactification of the two untwisted extra dimensions. In the same way we can introduce an additional compactification-like vector as (1,051021, 1) that can be added to the set (6.1) without any conflict with modular invariance conditions (3.25) and making K 22 = 0 without introducing any new massless state. Consistency with the modular invariance conditions (3.24b) requires K z*= + ( . . . + 1 . . . - 1 . . . ) , I* =15,...,19, (K2°,K21)=(0,0) and KX*=0, I * = 1 5 . . . . . 19, ( K 2 ° , K 2 1 ) = _(1, - 1 ) as the only allowed solutions. It is easy to see that IK~*)L together with the 10 Cartan generators II*)L, I* = 1 , 2 , 1 5 . . . . . 22, correspond to the adjoint representation of SO(10)× S U ( 2 ) × U 4. These states can be tensored with the right-moving state la)R + IZ1)R making a gauge vector boson and a complex scalar in the adjoint representation of SO(10) × SU(2) × U 4. In the V0 + V1 subsector the massless states are in the 128 + 128 representation of SO(16) D SO(10) × SU(2) × U 2. However, they are projected out by the modular invariance condition V 4 . N ~ = ¼, where we have taken k41 = k40 = ~ in eqs. (3.25). The other choice, k41 = k40 = 0, would lead to tachyons as we will see next. In the I10+ V3 subsector the massless states are (05; 1 , 0 ) + ( 0 5 ; 0 , 1 ) + (±1,04;--1,--1). They are projected out by the modular invariance condition

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V4" Nv-U-~ = I- The complex conjugate states appear in the V0 + - ~ subsector and so they are projected out in the same way by the modular invariance conditions. In the Vo+ V t + V3 subsector the massless states are (16 + 1 6 ; 1 , 0 ) + (16 + 1--6;0,1). The choice k0a = 0 (the other choice k01 = ½ would lead to an equivalent theory) and the modular invariance conditions Vo.N~o~-v~v ~~ I/1" Nv~+ v~+ v~= x2 and V3. Nv00+-V~l+~= ~, will project out the 1~ from the physical spectrum. The modular invariance condition V4.NGTv~ + v~- ~ will finally retain the physical state (16, 2) of SO(10) x SU(2) transforming under Z 4 with eigenvalue X. This state will be coupled to [S3)R, IS4+)R, IZZ)x, [ ~ Z ) g providing two generations of bosons and fermions in the 16 + 16 of SO(10). The Vo + Va + 2 ~ subsector will provide the complex conjugate states with opposite helicities. In the twisted (a = 1) sector the vacuum energy is - ¼ for right-movers and - ~~6 for left-movers. For right-movers the ground state is a tachyon at m~ = - ¼. It does exist for subsectors W + Vo and W + Vo + V~. Massless states, m~ = 0, do exist for subsectors W + Vo, W + V0 + V1, W and W + Vv The massless states for sectors t2 __ VG and W + Vo + 1/1 have S(/4 = 1 or Sx')4 =- 1 and A~/4 - 1 or A~/a = 1. The massless states for sectors W and W + V~ are [K j) with K J = 0 or 1 for j = 1,2, forming the 4~ vector representation of SO(4). Massless and tachyonic states for left-movers have to be analyzed separately. In the W + Vo subsector the tachyonic state, miz. = - ¼, is the state K t" = (05; 02), I* 15 . . . . . 21. The massless states are K t* = (05; - 1 , 0 ) + (05; 0, - 1 ) and A~ltl2 4 _- _ 1 or A~/4 = 1, K ~* = 0, I* = 15 . . . . ,21. The modular invariance conditions are =

Vo. Nwv-~o = V~. Nw~-~o = k4o, W- N ~

= k40

(6.3)

and therefore, as was anticipated, only in the case k4o = ~ is the tachyon projected 3 out from the physical spectrum. In that case, the retained states* are [S_~/4J~2)R + s'~/4{~)~. + ,~3_~/41~}R + a_l/~l$2)R] ,, ®(1,2)L, tWO fermions and two scalars in the (1,2_) of SO(10)× SU(2), all multiplied by 64 fixed points. The complex conjugate states with opposite helicities are obtained from the 3W + V0 subsector. In the subsectors W(3--~ and W + V0 + V1 ( 3 W + V0 + V1), m 2 > 0 and so there are neither massless nor tachyonic states. Finally, in the subsectors "W + VI (3W + 1/1) the massless states are projected out by the modular invariance conditions. In the twisted sector (a = 2) the vacuum energy for right-movers is zero, and there are two fermions and two complex scalars which are untwisted. But since they must also provide the negative helicity states, the ground state for subsectors having rn~ --- 0 is one complex scalar and one fermion. The vacuum energy for left-movers is - ~ and the massless spectrum has to be analyzed independently for subsectors 2-W~-Uoo and 2 W + Vo + V1. * S/_1/4 and a~_~/4 are fermionic and bosonic creation operators acting on the ground state 19)R,

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In the 2W + V0 subsector the massless states a r e K I* = ( ± 1, 04; ala2) and A1/2 = 1 for some I* = 2, 3, K 1. = (0s; ala2). The modular invariance conditions V0 • N ~ - p , , = V1 • N2~-;~o= 0, W- N2~-;~ = 43-project out all gauge non-singlet states. The only retained states are A2/2 = 1, K t• = (0s; 0 2) and A~/2 = 1, K I* = (0s; 12). When tensored with the right-moving ground state we obtain two gauge singlet fermions and bosons. In the 2 W + V0+ V1 subsector the massless states are the SU(2) singlets (16 + 1--6;02). The modular invariance conditions V0 • N 2 ~ o + vl - 1'11" N 2 w + vo+ v1 = k0u W . Nzw + Vo+v, = 0 will pick the 16(1-6) for k0a = 0 (k01 = ½). When tensored with the right ground-state we obtain one chiral fermion and one complex scalar in the (16,1) of SO(10) x SU(2). This has to be multiplied by the number of fixed points (tori). The sector 2R has 16 fixed tori, but only four of them are left invariant by R implying that the multiplicity of this sector is 4. In summary, the model defined by (6.2) has SO(10) x SU(2) × U 4 as gauge group and the massless states form four generations of chiral fermions and complex scalars in the (16, 1) of SO(10)x SU(2), 2. (16 + 16, 1) of fermions and complex scalars, and one complex scalar in the adjoint representation of the gauge group. The possibility of the existence of massless scalars in the adjoint representation of the gauge group is a feature of non-supersymmetric strings which is not shared by string theories with N = 1 supersymmetry. Notice that Higgses in the 16 + 45 with appropriate vacuum values allow us to break SO(10) into SU(3) x SU(2) x U(1).

7. Conclusion

In this paper we have presented a general construction of consistent non-supersymmetric tachyon-free four-dimensional strings and emphasized that in this setting it is not difficult to avoid tachyons without invoking supersymmetry. We have illustrated the general formalism by studying some simple cases reproducing the known examples existing in the literature. We have found two new models exhibiting the general features of non-supersymmetric string theories. One is a Z 4 model, with gauge group SO(10)x S U ( 2 ) × U 4, a complex scalar Higgs in the adjoint representation of the gauge group and the matter in 4- 16 + 2. (16 + 16) fermions and complex scalars. The second one is a Z 6 model with gauge group E 6 X U12, 18 generations of chiral fermions in the 27 and 17 generations of complex scalars in the 27 of E 6. Our final goal is to find non-supersymmetric four-dimensional string theories having the particle content of the standard model in the massless sector. The models presented in this paper are just possible intermediate steps towards that end. Work along those lines is in progress by the authors. Also, the hierarchy problem and the effective action in non-supersymmetric theories with massless scalars deserve further study.

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We gratefully acknowledge useful discussions with K.S. Narain, H.P. Nilles, F. Quevedo, A.N. Schellekens and N. Warner. One of us (M.Q.) would like to thank the Institut ftir Theoretische Physik (ETH), where this work was partly done, for warm hospitality. References [1] M. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge University Press, Cambridge, 1987) 2 vols [2] L.J. Dixon and J.A. Harvey, Nucl. Phys. B274 (1986) 93 [3] L. Alvarez-Gaum~, P. Ginsparg, G. Moore and C. Vafa, Phys. Lett. B171 (1986) 155 [4] T.R. Taylor, Nucl. Phys. B303 (1988) 543 [5] G. Moore, Nucl. Phys. B293 (1987) 139 [6] M. Dine and N. Seiberg, Nucl. Phys. B301 (1988) 357; T. Banks and L. Dixon, Nucl. Phys. B307 (1988) 93; I. Antoniadis, C. Bachas, D. Lewellen and T. Tomaras, Phys. Lett. B207 (1988) 441 [7] R.R. Metsaev and A.A. Tseytlin, Nucl. Phys. B298 (1988) 109 [8] A.H. Chamseddine and J.-P. Derendinger, Nucl. Phys. B301 (1988) 381 [9] K.S. Narain, Phys. Lett. 169B (1986) 41; K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B279 (1987) 369; W. Lerche, D. Liist and A.N. Schellekens, Nucl. Phys. B287 (1987) 447 [10] H. Kawai, D.C. Lewellen and S.-H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. B288 (1987) 1; I. Antoniadis, C.P. Bachas and C. Kounnas, Nucl. Phys. B289 (1987) 87 [11] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 651; B274 (1986) 285; K.S. Narain, M.H. Sarmadi and C. Vafa, Nucl. Phys. B288 (1987) 551; L.E. IbaYaez, H.P. Nilles and F. Quevedo, Phys. Lett. B187 (1987) 25; B192 (1987) 332; L.E. Iba~ez, J.E. Kim, H.P. Nilles and F. Quevedo, Phys. Lett. B191 (1987) 282; L.E. Ibafiez, J. Mas, H.P. Nilles and F. Quevedo, Nucl. Phys. B301 (1988) 157 [12] M.B. Green and J.H. Schwarz, Nucl. Phys. B181 (1981) 502; B198 (1982) 252; B198 (1982) 441 [13] M.B. Green and J.H. Schwarz, Phys. Lett. 136B (1984) 367 [14] L. Brink, J. Scherk and J.H. Schwarz, Nucl. Phys. B121 (1977) 77; A.H. Chamseddine, Nucl. Phys. B185 (1981) 403 [15] N. Seiberg and E. Witten, Nucl. Phys. B276 (1986) 272 [16] L. Alvarez-Gaumfi, G. Moore and C. Vafa, Comm. Math. Phys. 106 (1986) 1 [17] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253