Orbifold models and modular transformation

Nuclear Physics B302 (1988) 291-329 North-Holland, Amsterdam

ORBIFOLD MODELS AND MODULAR TRANSFORMATION Ikuo SENDA* and Akio SUGAMOTO**

National Laboratoryfor High Energy Physics (KEK), Oho-machi, Tsukuba, Ibaraki 305, Japan Received 23 June 1987 (Revised 30 December 1987)

The orbifold models of the heterotic string are constructed on the quotient spaces of generalized tori by translational and rotational discrete symmetries. In order to obtain the consistent orbifold models, the conditions of the modular invariance are derived from a one-loop vacuum amplitude. Z 3 orbifold models satisfying such conditions are searched systematically.It is shown that there are infinite possible models with N = 2 supersymmetry. Among these models, two examples having E6 and E 7 gauge groups are discussed. The orbifold models with N = 1 supersymmetry are also discussed in detail. It is shown that there are only five consistent models in the class of these models based on E 8 ® E~ heterotic string in which the extra six-dimensional torus and the E8 ® E~ maximal torus are modded out by the rotational and the translational Z 3 symmetries respectively.

1. Introduction T h e superstring theories can n a t u r a l l y incorporate gravity into their u n i f i c a t i o n scheme w i t h o u t difficulties of divergences a n d anomalies. The unification occurs, however, at the Planck mass scale (10 a9 GeV) which is far from our energy scale of 10 TeV, a t t a i n a b l e i n the near future. If the superstring theories can be the ultimate theory of n a t u r e , it must describe the p h e n o m e n a at low energies. O n e of the i m p o r t a n t p r o b l e m s left unsolved is the structure of generations such as: (i) H o w m a n y g e n e r a t i o n s we have. (ii) W h y do these generations mix with each other, giving the flavor m i x i n g matrix of C a b b i b o - K o b a y a s h i - M a s k a w a . I n order to approach such p r o b l e m s , we have to k n o w superstring models in which six transverse d i m e n s i o n s are compactified o n a compact space, leaving f o u r - d i m e n s i o n a l M i n k o w s k i space-time. Several attempts have been m a d e o n this p r o b l e m a n d some good results have been given [12-14]. O n e of the most p r o m i s i n g approaches is the o r b i f o l d compactification [1], in which various amplitudes are calculable. The orbifold is o b t a i n e d b y dividing a torus b y its discrete symmetries. There are two ways to describe the orbifold, n a m e l y the bosonic f o r m u l a t i o n [2,10,11] a n d the * On leave of absence form Tokyo Institute of Technology (TIT), Meguro, Tokyo, Japan. ** Present address, Ochanomizu University, Otsuka, Tokyo, Japan. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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fermionic one [6, 8, 9]. The orbifold compactification modifies the string variables slightly from the corresponding ones in the torus compactification and research of mass spectra and of the amplitudes is not so difficult. As the orbifold is obtained by dividing a toms by its discrete symmetry, the toms compactifications should be seriously studied. On a toms, string variables satisfy certain periodic boundary conditions induced by a lattice. We know that the consistent lattice, on which the heterotic string [4] is compactified, is given by the SO(d, d + 16) transformation of (Pz)d® F16, where P2 denotes a two-dimensional lorentzian lattice with the signature [ ( + ) , ( - ) ] and /~16 is a root lattice of the E s ® E s or Spin(32)/Z z [3]. The degrees of freedom of the SO(d, d + 16) transformation can be interpreted as constant background fields [5]. Therefore we must investigate the symmetry of the toms having various constant background fields to see what kinds of orbifolds are available. As is well-known, modular invariance is one of the most important consistencies in closed string theories. Thus modular invariance must be imposed on the loop amplitudes of the orbifold models. Some of our results including the constraints of modular invariance and several models are reported briefly in refs. [10,11]. This paper gives detailed derivation of the constraints coming from modular invariance and of models from a somewhat different viewpoint, which can supplement our previous papers. In the next section, after introducing various notations we discuss the discrete symmetry of the toms. We will consider two kinds of discrete symmetries of the toms, namely the translational symmetry and the rotational one. When Z u translational symmetry of the lattice is obtained, the constant background fields gij, Bij and A~ are not restricted and they take arbitrary values. As for the rotational Z N symmetry of the lattice, there are two ways in imposing twisted boundary conditions on the string variables, the left-right independent way and the non-independent one. In the left-right non-independent formulation, the rotational twisted boundary conditions are imposed on the sum of left and right movers and only the constant background fields gij are restricted while the others are arbitrary [20]. On the other hand, all the constant background fields gij, B~ and A~ are restricted in the left-right independent manner. Throughout this paper, our discussion is based on the heterotic superstring in the light-cone gauge, in which the right-moving sector is given by the Neveu-Schwarz-Ramond formalism and the twisted boundary conditions are imposed on the left- and right-moving string variables separately. To represent fermionic variables, we will use a bosonization technique and fermionic variables are replaced by four real bosonic ones. Such bosonic formulation of the superstrings is adequate for the unified description of the orbifolds and also for the counting of the number of supercharges. In sect. 3, we derive the constraints from modular invariance of the vacuum amplitude. Our orbifold models are made up of various subspaces, which we will call suborbifolds. Each suborbifold is obtained by modding out a certain torus by its

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discrete symmetry. First, we investigate the modular transformation properties of a vacuum amplitude in each suborbifold. Then the consistency conditions are derived by imposing modular invariance on the vacuum amplitude of the total orbifold. In the sect. 4, we give some examples of Z 3 orbifolds starting from the heterotic string with E 8 ® E 8 or Spin(32)/Z 2 gauge symmetry. Our Z 3 orbifold models are composed of two suborbifolds, o d , d+16 =

odl,dl ® o d 2 , d 2

+16

= (Tdl,a,/Z3R) ® (Td2'd2+16/zT), where T al,al and T a2'a2+16 are d 1 and (d 2 + 16)-dimensional tori respectively. Z R and Z3T represent Z 3 rotational and translational discrete symmetries. In the construction of 0 d''a', we use a lattice discussed in sect. 2 so that T al,al has a rotational Z 3 symmetry left-right independently. The models with N = 2 supersymm e t r y are obtained by putting d 1 = 4 and d 2 = 2. These models have infinite possibilities depending on the constant background fields as parameters. Among these models, we give two examples having E 6 and E 7 gauge groups, which are left-right symmetric. The E 6 model is left-right symmetric having three right-handed and three left-handed families in the 27 representation of E 6. The E v model has two families in the fundamental representation 56 in each chirality. In these models, there is one peculiar feature, namely the states in the twisted sectors are all massive. The models with N = 1 supersymmetry are given by putting d I = 6 and d 2 = 0. The sixteen-dimensional torus is either the maximal torus of E 8 ® E s or Spin(32)/Z 2. We study b o t h cases and detailed investigations are done in the case of E 8 ® E 8 gauge symmetry. We survey all possible embeddings of the Z 3 translational symmetry into the entire group of E 8 ® E 8. We find that there are only five distinct models of which the gauge symmetries a r e E 6 ® E8, E 7 ® SO(14), SO(14) ® SU(9), E 6 ® E 6 and E 8 ® E 8, where the model with E 6 ® E 8 gauge symmetry is just the Z-orbifold model [1]. The representations of the chiral fermions are also given. The last section is devoted to the discussion.

2. Orbifold compactification in the presence of constant background fields In this section, we discuss the discrete symmetries of the torus and introduce several notations. The most general torus compactification of the heterotic string is given by Narain, Sarmadi and Witten [3,5, 19]. According to their results, the consistent torus is given by Rl°-a'1°-a+16/Flo_a, lo_a+16 if d of ten dimensions are uncompactified. Here, Flo_a, lo_a+16 is a S O ( 1 0 - d, 1 0 - d + 16) transform of the lattice (P2)I°-d® F16 , where P2 denotes a two-dimensional lorentzian lattice with the signature [( + ), ( - ) ] and/"16 is a root lattice of the E 8 ® E 8 or Spin (32)/Z 2. The parameters of the S O ( 1 0 - d, 1 0 - d + 16) transformation are represented by the

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constant background fields, &j, Bsj. and A~, (i, j = 1 - 10 The vectors in Flo_a, lo_d+16, (U, L', La), are as follows,

-

d,

a,b=l

- 16).

!2g ijrAj(m a a 1 ~ g a b n :ka kb) ,

Li= --ni+lgiJmjq-giJBjknk+

1 ijrAja ( ma _ 5gab kAk Li = ni -[- 1-°iJm2l~,,,j + gOBjknk + 5g L a = gabm b

-

-

g,j = gji,

niA a '

B,j = - B j i ,

(2.1)

where n', rn i and m a are arbitrary integers. Explicit calculation shows that the lattice is even lorentzian self-dual lattice which is necessary to obtain modular invariant loop amplitudes [5,11]. Introducing the basis vectors a(k)= (Ot(k)i; ~(k)i, ~(k)a), fl(k) =_ (B(k)i; ~(k)i [~(k)a), and 7 (b) = (~,(b)i; q(b)i, ~(b)a), a [ 2 ( 1 0 - d ) + 16J-component vector L = (L i,/~i, £a) on the lattice in eq. (2.1) is expressed as: 10- d

L = E

10- d

Or(k)n k +

k=l

E

16

[~(k)mk+ E "Y(a)ma"

k=l

(2.2)

a=l

The inner product is defined with respect to the lorentzian metric,

L%bL '~.

L . L ' = L i g i j L 'j - £ * g i j L 'j -

(2.3)

The inner products of the basis vectors defined above are as follows, a ( k ) . a ( l ) __. a ( k ) . y ( a ) =

Ol(k) . ~(l) = ~l~ ,

fl(k), flU) = fl(k),

y(a) = 0,

.y(a) . ,y(b) .~_ __g(a)(b).

(2.4)

The inner product between vectors L and L' becomes, 10-d

16

(nkm'k + n'kmk)

L.L'=

k~l

-

E

rnag(a)(b)m'b,

(2.5)

a,b=l

where the dependence on the constant background fields disappear. If g(a)(b) is unimodular, the lattice F becomes even lorentzian self-dual lattice without regard to the constant background fields. Next we consider discrete symmetries of a torus, which is important in considering the orbifold compactifications. The compactification on the torus T p'q =

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R e' q/Fp, q is realized by the identifications of the string variables, as X i -- X i - - q r L i ,

.~a _ )~a + ~rLa,

i= 1~

p,

a = 1 - q,

(2.6)

where (L'; L a) is a lattice vector o f Fp, q. In order to define a discrete symmetry, G, consistently on the toms, there is a constraint which G must satisfy. If X and X' are identified on the toms, X - X ' m o d ~rL,

L ~ Vp, q,

then an arbitrary element of G must satisfy the following equation,

Vg~O

g(X)-g(X')mod~rL,

L ~ F e , q.

(2.7)

We can consider two kinds of discrete symmetries, one of which is a translational discrete symmetry and the other is a rotational one. As for the translational discrete symmetry, we can get those of order N(ZN) by using a vector ( L ; / ~ ) ~ F as follows,

g( X ) = X - rrL/N, g ( 2 ) = )( + ~r[./N,

(2.8)

where vector indices are suppressed for brevity. In this case eq. (2.7) is satisfied and we can choose arbitrary N as an order of G. Thus we can get various kinds of translational discrete symmetries out of the lorentzian self-dual lattice F for any choice of the constant background fields. For rotational discrete symmetries, a little consideration is required. First, consider the case in which left and right movers are not treated independently. In this case, the twisted boundary condition is imposed on the string variable of* --- l(Xi + )~i), where X ~ and )(~ are right and left movers, respectively. The winding vector of y-i is w i= ½ ( - L i + L i) = n i, where L i and £i are those in eq. (2.1). The boundary condition of R "~ on the toms is represented by, Lrs_ f ~ m o d ~F', where F ' is an integer lattice (n ~} and the inner product in F ' is defined in terms of the constant background fields &j, W ' •,

w¢i

~ ~

¢,

w i • W ¢i = g i j W i W ' j .

We can obtain lattice F ' having various discrete symmetries according to the choices of &j. Because there are no constant background fields A~ and B~j in F', A~ and Bij can be introduced without changing the discrete symmetry of F'. It is possible to discuss the symmetry breaking of the model using such a lattice [20]. For example, if we choose gij as a Cartan matrix of SU(3), we obtain a lattice with Z 3 rotational symmetry. Whereas in the left-right independent formulation, the boundary condi-

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tions of the left and right movers are imposed separately. In this case, we must investigate the discrete symmetry of the lattice in the left-right independent manner, taking the existence of the constant background fields B,j and A~ into account. If we consider the more general orbifold in which the Z N rotation parameters of left and right movers are different, so called asymmetric orbifolds [15], it is natural to impose the Z N rotational symmetry on the left and the right movers independently. In the following of this paper, we will employ the left-right independent formulation and the twisted rotational boundary conditions are imposed on string variables left-right separately. However, our discussion of modular invariance is also applicable to the left-right non-independent formulation and the condition of modular invariance is obtained as a special case of that derived in this paper. Let us examine the rotational discrete symmetry of a two-dimensional lattice having left and right movers. The lattice vectors on the two-dimensional lattice F2, 2 are given by,

L(~,i ") _-

_ n i + ½giJmj + g i J B j k n k '

-i -- n " + ½ g " J m j + L(n,,n)

gOBjknk

(2.9)

where the index i runs 1, 2 and we have put A7 = 0. Let us introduce the zweibein e/ satisfying el% , i _- gij and define

Li= eiiL i '

£i= e,~i.

(2.10)

It is convenient to represent a two-component real vector by a one-component complex vector,

L(,,,,,)

=

L i(n,m) +iL~,,,~),

t +iL~,,m ) . L ( n , m ) = ~L(n,,,)

(2.11)

If this lattice is invariant under rotation by an angle 2~r~/ in the two dimensions left-right separately, the following is expected; for arbitrary n, m ~ Z there exist n', m' ~ Z satisfying, L(n', m') = e2~rinL(n, m),

Z(n,, m') ~- e2'~irlL(n, m) "

(2.12)

We can show that the particular solution of eq. (2.12) is,

eli=C,

el2 =0,

e2x = ccos (2~r~),

e22 = c sin (2~r~l),

Bt2 = arbitrary real constant,

(2.13)

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1 1 1 1 where c is a real constant. The consistent values of 71 are ~, ~, ~, g and multiples of them. In these cases, the lattice has Z N, ( N = 2, 3, 4, 6), rotational discrete symmetry for left and right movers independently. In the presence of the constant background fields A~, we can not find the solution to eq. (2.12). It is not clear whether there are any other solutions or not, but we will use those in eq. (2.13) to ensure Z N rotational symmetry of /'2, 2 in the following of this paper. Next we consider the mode expansions of string variables on the orbifold which is a quotient space of a torus by a rotational discrete symmetry. We consider rotational discrete symmetries in the space which is a product of 2-dimensional tori, ®D=IT(t) , T(O=R2'2/F(,/). To divide each 2-dimensional torus by a rotational discrete symmetry, we must ensure that the torus has the symmetry. Thus we use the constant background fields in eq. (2.13) so that the 2-dimensional torus has Z N ( N = 2,3,4,6) rotational symmetries. On each 2-dimensional torus, we use the complex notations,

z'(~-o)

= ~ ( x ~'-~ + i : ' ) ( ~ - o ) ,

~'(~-o) = ~( x ~'-~- ix~')(~-o), for the right movers and similar expressions are given for the left movers, 2~I, ~:. M o d e expansions in the untwisted sector are as usual, namely we have: a ~ [0, ~r]

/V

z,(~ - o) = z" + p'(~ - ~) + ~i E - - e -2""-°~ , v:g0 P

2 ' ( ~ - o) = e ' + y ( ~ -

o) + ½i E --a: e_2.(._o); v=/:0 P

and

ff_I(T..[_O) = ~ I ~.fiI(,r +o).+. ~l E l . __e-2iv(r+a), j~: vva0 P "21( T-~- ~ ) -1- ~l-{- ~l('r-~ (Y) nt- li E

~: e-2iv('r+°), v:¢o 1,'

(2.14)

where we treat left and right movers separately. The commutation relations are given by,

[z t, fiJ] = [~,, py] = [yI, ~j] = [~:, fi:] = ½i6t:, [fl~,fl~l = [ f i ] , / ~ ] = v6'a6~+,',0 •

(2.15)

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The zero-mode momenta are quantized on the lattice FzD.2 o. As we will see later, the twisted sectors are necessary to maintain the modular invariance. In the k times twisted sector (k-twisted sector), the boundary conditions of the string variables are,

z l ( o = 7?) ~- e 2 " " ' Z t ( o = 0)mod ~rF2D,2 o , ZZ(o = 7r) = e2"~'2~I(o = 0)mod ~rF2o.2 o , ~/g = n ' / N ,

(~,; ~z) = k ( , ~ ; ~to),

~lZo= ~ ' / N ,

(2.16)

where n z, ~ i = 1 - N - 1 and I = 1 - D. Untwisted sector (k = 0) has integer-mode oscillators and the 0-mode momenta are on the lattice F2o,2 D. Whereas in the twisted sector (k 4= 0), oscillation modes are non-integer and the coordinates of the center of mass are located on the fixed points of the Z N transformation. The contribution of the string variables on the orbifold, 0 2°'2°, to the Virasoro operators L o and L 0 reads, --I Lo= E I=1 n=l

°( E

Lo= E

I=1 n=l

I --I + J~--n+~lj~n-B I) -- ~22 -{- 1711(1 -- ~1)

I

t

(/~-n+~'/~n-~ -:' "' '-~ ~/n+l-~tl~n-l+~ t) -- 2~22"~ 1~/( 1 _ ~ I ) , (2.17)

where (~1I; ~ t ) = k01~; ~1~) and in the untwisted sector fld(/~d)= v[~ 2 p I ( ~pI . )- The operator which realizes Z~¢ rotational transformation in k-twisted sector is given by, P(k) = exp

--4rr

(~g(zI~ol--2tp')+flo(er~'-~1fiz)}

D ¢,o{ i(

-2~ri I=1 ~ n~= 1 71

q-~

1

_i

I

In+glt~l_~l--

I

1

n - 1 + nlB--.4-1--~I~.--lq-BI--

n

-

-

--I

)

'0Ifl-n+nlfln-~ll

1 _ ,~i~_n+l_glll~n_l+~t =' )1) ,

n-l+~

(2.18)

where p t and fit must be set 0 for k ~: 0. The above operator acts on string variables as follows, p(k)zlp(k)-i

= e2~ri~toZ 1 '

p(k)2Ie(k)-I

= e2wi~lZoZt"

Here we have used the same group, Z N, for both left and right oscillation modes, but in principle we can use different group for each mode since both oscillation modes are mutually independent.

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When we discuss the translational discrete symmetry, we adopt the real notations of string variables. Let us consider the compactification on Ta'd/Z(ua3 having d right and a~ left movers, where Z ~ is a Z u translational discrete symmetry and T d, d= R a, g/Fd ' d. The mode expansions are, i

X i ( ' r - o ) = x i + p i ( ' r - o ) + l i E a-'-~ne - 2 i n ( r - ° ) , n4:O Fl

i=1 -d,

~i

Xi(,'f+o)-----)~i

+fii('/'"~ O) + ~i E --a~e_2i.(:+o) '

i = 1 - d'.

(2.19)

nva 0 /'/

The commutation relations of the oscillators are

[+o,,,'m] = [<,,vm] : . 3 % ÷ m , o . The boundary condition of the string variables in the k-twisted sector is, (k = 0 N - 1)

( X i ; ) ( ' ) o=~ -- (X'; X-i )o=o + ~r(-3'; 6 i ) m o d ~rFa , d,

(2.20)

where (3i; g ' ) = k . ( 3 ~ ; 36) and N.(3~; 3~) belongs to the lattice Fa, d. These boundary conditions modify the values of the zero mode momenta, p ' = L ' + 8',

ffi= £ i + ~i,

(2.21)

where (Li; L i) ~ I'd, d. The contribution to the Virasoro operators from these string variables are given by,

Lo

y"

L i + 3i) 2 +

i=l

Oti_ nOlin -n=l

The operator which realizes the transformation in eq. (2.20), p
_ e-2~ri(36p'-g~/3').

X i

and .~i induced by the operator p
e+(x,; .V)e+-l= (x'; X') +

(2.23)

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We can consider the Z u discrete transformation which is a combination of rotational and translational transformations. Let us employ the complex notation of the string variables and consider the boundary condition of the k-twisted sector (k = 0 - N - 1):

( zl, 21)o=. =

(e2~in~zl,e2"i'°tZ1)o=0-

"lr(31,~l)mod~Fzd,zd,

(2.24)

where 7 I, ~t, 31 and 8r are given by

71= k*lI,

31=

~t= k~,

8'=

l=k-1 Z

)

e2~rilnt° 31,

,=0E ~

]~o,

(2.25)

where N . (31; 31) ~ F2a,2d.Untwisted sector has integral oscillation modes and its zero-mode momenta are quantized on F2d,2d. The twisted sector gives non-integral modes and vanishing zero-mode momenta, which is the same as in the case of rotational discrete symmetry. The center of mass in each twisted sector is associated with a fixed point which is obtained as Z I = e2~i~tZl _ ~ I ,

2~I = eZ"i~'2~t - ~r3t, rood F2d,2d.

(2.26)

The operator which realizes the Z u transformation is given by the product of those in eq. (2.18) and (2.23). The right-moving sector of the heterotic string has a supersymmetry. We use the bosonic formulation of the Neveu-Schwarz-Ramond model in the light-cone gauge to represent this supersymmetric sector, since it is convenient in analyzing the breaking pattern of the supersymmetry and also investigating the spin structure. The representations of the NSR model are those of SO(8) affine Lie algebra, which can be constructed by bosonic fields. Thus, we introduce extra right moving bosonic variables q¢(~ - o), t = 1 - 4, instead of fermionic variables. Mode expansion of q¢ is given as,

,ff('r-o)=q'+pt(r-o)+½i E

at~e-2i"('-°),

n4:0 /l

o ~ [0,1r].

(2.27)

It is well-known that if the zero mode momenta pt belong to the vector representation of SO(8), the bosonic system, q~t, is equivalent to the GSO projected Neveu-

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Schwarz (NS) model and if they belong to the spinor representation of SO(8), it is equivalent to the Ramond (R) model [21, 22]. The weight lattice of SO(8) vector and spinor representations are defined by using the orthonormal basis e~ (i = 1 - 4), e i • ej

=

~ij: Fso(8)~ =

nie ~

ni~Z),

n~ = odd,

i=1

4 Fso(8).. =

)

( n i + ½)ei E n / = e v e n , i=1

ni~Z

(2.28)

,

i=1

where SO(8)v and SO(8)sp indicate vector and spinor representations of SO(8) respectively. The Virasoro operator of the bosonic system, L0(40 , is given by, L0(q~) = 1~p 2 + ~. t=l

a_,a,t,

6~.

(2.29)

n=l

When we treat the system of fermions with twisted boundary conditions in each two dimensions among eight transverse dimensions, we must deal with complex fermions, ,pa (a = 1 - 4). We impose twisted boundary conditions on these complex fermions, ,Pa(o = ~r) = + e2"~n"~/,a(o = 0),

(2.30)

0 <~/a< ½,

where + ( - ) sign shows that ~a,s are Ramond (Neveu-Schwarz) type fermions. In the discussion of the orbifold compactifications, the parameters ~ are put equal to ~1 in eq. (2.16). For R-type fermions, the mode expansions of ~a and their canonical momenta H a are,

~,a(~ __ O) = E O.%oe -ia°+"°~"-~, nEZ

n a ( ~ -- °) = Z ~g-ee -~a"-e~'-°~, nEZ

{ 0ff+~, L.-¢ b

) = ~a~.+,.,0"

(2.31)

The mass operator L(om is, L(oR)=

~_, ( m + -aa~a a a "q ]~_m_rt~ 0rn+rl rn >~O

+ Z m>0

(m--rfl)O~-m+n°~,~-n°+~+lY'~((Tla)2--~la) a

•

(2.32)

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For NS type fermions, we obtain the similar expression of mode expansions and L~oNs) by putting 71" ~ ~a .q_ ½. The equivalence of the bosonic system and the fermionic system is easily seen when the partition functions are calculated. In calculating the partition function of the fermionic system, we must use a projection operator, which is a generalization of the GSO projection. The projection operator is defined as G = ½(1 + (_)F), where the operator F is a fermion number operator of NS or R type fermions. The partition function of the NS type fermion system, A(NS)(q 2, ~), and that of R type fermion system, A(R)(q 2, *]), are given by A(NS) (q2, ~/) = tr(Gq 2L~s' )

4

(2.33) A(R)(q 2, 4) = tr(Gq 2L~') = ½q2/3 [

a=lfi '(

(q(n°?-n°(1 +q2n ~) m>01-1(1 + q2m-2n")(1 + q2m+2n°))

))]

+ 1-'I q(~O)2_~° 1--I (1 - q2m-2n°)(1 _ q2m÷2~o a~l

m>0

. (2.34)

As for the bosonic system, let us consider the bosonic variables in eq. (2.27) with their zero-mode momenta belonging to the shifted SO(8) weight lattice F: F(v) = { w t + n t l t = l - 4 ,

wt ~ Fso(8)v },

/-(sp) = { wt + ~tJt = 1 - 4,

wt~- Fso(8)sp }.

(2.35)

The partition functions of the bosonic system with p ~ F (v) and F (sp) become, ½(1- ( - ) E n ' ) q EAn'+n')2 ,

~(V)(q2,~) = 1-I (1-q2n)-4q -1/3 n>0

~(sp)(q2,~)

=

H n>0

(n')cZ 4

( 1 - - q 2 n ) - 4 q -1/3

E (nt)EZ 4

½(1 +(-)Xn')q x'~n'+'/2÷n'~2, (2.36)

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The equivalence between A and ~ can be seen by the use of the product formula of theta functions [23], o

,Ao, , ) = e 2"'°b

e2 i"be n~Z

= e2eriab+rria2r H (1 -- e2"in~') n~N X H (1 4- e2~ri((m-1/2-a)r-b))(l Av e 2 ~ i ( ( m - 1 / 2 + a ) ' + b ) ) . hEN

(2.37)

Thus the system of fermions with twisted boundary condition in eq. (2.30) is equivalent to the bosonic system of which zero-mode momenta are quantized on the shifted SO(8) weight lattice. These facts are as expected because a fermionic field ~k can be expressed by a bosonic field ~ as x v - e 2~' and we can easily guess the translation on q~ induces the rotation on 'P. Notice that we concentrated on the cases of 1 > ~ t > O, but we can prove the same fact if some of ~t are zero. The operator which realizes these translations of the bosonic field are similar to those in eq. (2.23),

p,(k) = e - 2 ~ s ¢ / ,

(2.38)

where the eigenvalue of pt belongs to the k-shifted SO(8) weight lattice. In the construction of the orbifold, the action of the translational ZN symmetry on the SO(8) lattice must be defined consistently. The consistency condition is that the vector N - (~/t) belongs to the root lattice of the SO(8), which is equivalent to

E NT1t

even integer.

---

(2.39)

t

This condition is also necessary for the modular invariance of the vacuum amplitude. The space of the Z N orbifold, O d'd is obtained by the direct product of subspaces, Oa, d = @ Oa,'d,, Od~,d~= Td,,d,/Z(~), Y

gd,=d, Y

EL

(2.40)

Y

where Z(ff) is a translational and rotational Z N discrete symmetries. The total Hilbert space, ,~otot, is a direct product of the Hilbert space of k-twisted sector, 9/0 (k), (k = 0 - N - 1): v~tot =

k=N-1 ~ .~(k) . k=O

We must project the Hilbert space in each sector onto a subspace which is invariant under Z N transformation. In k-twisted sector, we define the vacuum such that it is annihilated by the positive modes of the string variables and without excitation of

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the zero-mode momenta:

ae.10)k =/3/_~+n, lO)k =pq0)k = p'10), . . . . .

O,

where we have denoted the vacuum in the k-twisted sector by 10)k. An arbitrary state [~P)k in o~ (k) is written as ~10)k, where ~ is an operator. To define the Z , transformation property of [q0)k, we must define the transformation property of the vacuum [0)k. We define the Z u transformation of [0)k by introducing a constant phase factor f(~) [7],

[O)k --, ei/'*'iO)k

Zu:

Hereafter, we will call this phase factor as the vacuum phase of the k-twisted sector. The consistency condition with respect to these factors is the Z N property, namely (eif(*~) N= 1. Since the Z N transformation of the operator ~ is obtained using Z N transformation operators in eqs. (2.18), (2.23) and (2.38), the Z N transformation of the state Iq~)k becomes, loP) ~eif'*~P(k)lq~)k;

ZN:

the Z u transformation operator p(k) is given by (1-[rP~(k)) • P~g), where P~(k) is a transformation operator of the bosonic part in eqs. (2.18), (2.23) or their combination according to the choice of Z(ur), and P~tk) is that of the fermionic part in eq. (2.38). The states which can exist consistently on the orbifold must be invariant under this Z N transformation. In other words, we must project the states in ~,~(~) on those having eigenvalue 1 with respect to ei/(k)P (k). Therefore, we introduce a projection operator in the k-twisted sector as, 1 N-1 a

-

E

(2.41)

N h=0

Using this projection operator, we will obtain a vacuum amplitude of the heterotic string compactified on the torus in the next section. Here, we should notice that these vacuum phase factors become very important in the model building since the spectrum of the model would be completely different due to the different choices of the vacuum phase factors. This problem will be discussed again in sect. 4 when we perform the model search. 3. Constraints from modular invarianee

In this section, we study the constraints coming from the requirement of modular invariance in the one-loop vacuum amplitudes. The condition of modular invariance for the left-right symmetric orbifold has been already derived in ref. [6], in which the condition is obtained that for the cancellation of the global anomaly with respect to the modular transformation. The string variables with k(h) twisted boundary

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conditions in o ( z ) direction contribute to the vacuum amplitude in the form:

A (~'h) -Tr[G(k)z/£k~L~'], where p(k~ and L~0k) (L~0k)) are the Z x transformation and the mass operators in the k-twisted sector. From the view-point of the path-integral, the total amplitude is the sum of A (k' h) over k and h and the relative phases of A (k' h) are arbitrary. If the global anomaly is canceled, it is always possible to determine the relative phases in the summation of A (k'h~ consistently with the modular transformation. As we have discussed in the previous section, we have to project the states on those which are invariant under the Z u transformation. Therefore, we must check if it is possible to choose the relative phases of the amplitudes A (k' h) so that they are consistent with both the modular transformation and the projection onto Z u invariant states. In fact, it is always possible which we will see at the end of this section. Thus the condition for the cancellation of the global anomaly is a necessary and sufficient condition to obtain the modular invariant amplitudes. In this paper, we will use the bosonic formulation of the superstring and derive the condition of modular invariance for more general class of the orbifold models including left-right symmetric and asymmetric orbifolds. We will perform the derivation of the one-loop vacuum amplitude maintaining the projection onto Z u invariant states, namely we insert the projection operator in eq. (2.41) to calculate the amplitude in the k-twisted sector. After that, we will require the invariance of the amplitude under the modular transformation ~---, ~-+ 1 and ~- ~ - 1/~. If the amplitude is invariant under these transformation, it is apparent that there is no global anomaly of the modular transformation. In this way, we will obtain the consistency conditions on the Z N transformation parameters (~/, ~; 6, 6) as well as the vacuum phases f~k), which become important in the model search in the next section. As was discussed in the last section, the space of Z u orbifold, O a' ~ is the product of suborbifolds, O d~' d~, each of which is obtained as a quotient of the torus Td, ' dy modded out by the Z~uv) symmetry. We evaluate the modular transformation properties of the one-loop amplitude in each suborbifold, od,'d,, comparing with those of the original torus, T d~'d'. Using the results in each suborbifold, we obtain the consistency condition of the orbifold models by imposing the modular invariance on the vacuum amplitude of the orbifold as a whole. The vacuum amplitude of the heterotic string compactified on the orbifold is a sum of the amplitude in each twisted sector. Using the projection operator G (~) in eq. (2.41), the integrand of the amplitude in the k-twisted sector is obtained as Tr[G(k)z L~'~2L~], z = e 2"~'. Since these integrands are defined independently in each twisted sector, we define the total vacuum amplitude introducing the relative phase factors between different twisted sectors, which are denoted a s eivtk)(y (°) = 0 ) * . These relative phase factors are determined later consistently with the modular * We can see that the modular invariance of the amplitude allows only the relative phase factors as the relative weights of the different twisted sectors.

306

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invariance. Hence, the vacuum amplitude, Z, is obtained as,

d 2'r N-leiv(k>TrlG(k)e2~i,L~ok,_2~i~L~ok)] z=fw E k=0

-

( d2~" N~I 1 N~ 1

A(k'h)(,)

JI-

(3.1)

, k-0 N h=0

where G (k) is the projection operator of k-twisted sector given by eq. (2.41) and L(O*),/],(ok) are right and left Virasoro operators in the same sector. The term A(k,h)(,r) in eq. (3.1) is defined as,

h>(,)_-

H A(: h>(, x

A(~"hl('r)

-- tr((p(k)

"f

h>(,), /

)h e2,,,L~o*) ,,i,L(o~>)F(k,h),

A~'hl('r)=tr((p(ok))he2'~i'L~°k,>*),

(3.2)

where the index • labels the suborbifolds. In eq. (3.2), p(k) is a Z ~ ) transformation operator acting on the bosonic variables on the 7-suborbifold and p(k) is a similar operator for the bosonized variables 0 t. The Virasoro operator L(O,k~~0,t/'(klvj ~ is that of right- (left-) moving bosonic freedom in the k-twisted sector of the y-suborbifold and L 0,4, (k) is a Virasoro operator of 0 t in which zero-mode momenta are quantized on the k-shifted SO(8) weight lattice. The factor F (k, h) is related to the degeneracy of the zero modes (center of the mass coordinate) and it is introduced only if the suborbifold is obtained through dividing a torus by its rotational discrete symmetry. We analyze the modular transformation properties of the bosonic degrees of freedom in each suborbifold and the fermionic one separately. Let us investigate the modular transformation property of the bosonic degrees of freedom in the ~,-suborbifold, OG 'a', =Ta,'d,/Z(u'r~, in which the torus TG 'd, is divided by Z N "translational" discrete symmetry. Notice that the toms T d,'d, has arbitrary constant background fields discussed in the previous section. The contribution of bosonic variables on ,/-suborbifold to A(k'h)('r) is given by eq. (3.2) and L(k),o.r L(k)o,rand Pr(k) are defined as follows, O, y =

2

L+

+

i=

O,y =

2

L

+

nOln - -

~i ~i OtnOtn --

i=

Pr(k) =

Og n=l

e-2~(~/-~

n=l

,

where U and L ~ are quantized on the lattice

(3.3)

I'G,d,

and the parameter of the

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discrete transformation is given by, (~i; Si) -~'~k ( ~ ; 8~), N(8~; S g ) ~ Fa~,d~ . Then, A(vk' h)(r) becomes

2~'i'dv/24nI~I>0((1 -

A'vk' h)(¢) = e-e,,~d,/24+

X

e2"'"') -d" (1 - e-2~rin') -dr

)

E (e-2~ih['°(Li+ai)-g6(Li+g')]+~ir(L'+a')2-=i~(L'+gi)2). (L, k)~ra,,d "

(3.4)

~

The sum over the zero-mode momenta differs from that of the torus compactification. To treat the momentum sum in eq. (3.4) generally, we define the summation over the zero-mode momenta with parameters 01i; ~i) and (8i; gi),

S(~;g)(n;fl)a['rFd,

\ ' ~1 =

E e-2~i((U+~')n'-(L~+g')rl~) (L; L)Er~,a X e ~i(~(L+~)2-÷(L+g)b ,

(3.5)

where we assume N. (~; ~s) and N . (8i; 8~) belong to the lattice Fa, d to preserve Z u symmetry. The function S r is a generalization of the theta function in eq. (2.37). It is very helpful to understand the function S r as a partition function of the spin system taking values on the shifted lattice of F + (8; 8), which couples to the external field (~; ~) and has complex temperatures ~-1(~-1) for right (left) movers. Since the modular transformation is generated by • ~ • + 1 and ¢ ~ - l / r , we must examine the response of S r to these transformations. For ~---, r + 1, we have S(rS;S)(n;~)($+l,~r+l)= +e-~i(~2-gbS(rS;S)(, 8 ; ~ - g ) ( , , ? ) ,

(3.6)

where the sign + ( - ) on the right-hand side of eq. (3.6) denotes that the lattice F is even (odd) lorentzian lattice. If the lattice is an integer lattice, {Zd;Z g}

_~_

{(ni;~J)[i

.

1.

d,. j .

1

c [ , n ,i

~j ~ Z}

,

we obtain a similar expression as eq. (3.6), S(n;g)(n;~)/_ + 1, ~ + 1) Za; Z d

k "t

=

e-~ri(a(8~:l)-g(g~Zl)),r((6;g)(n-a+-l/2;fl-g+-l/2)('r, S-r), (3.7) _ _Zd; Zd

/

As for the inversion ~-~ - 1/% the following identity is derived by the help of the Fourier transformation or Poisson's summation formula, 1

Sba; S3(,; 0)( _ 1/~-, -- l / F ) = 1 - ~ ( - i t )

d/2(i~) &2e -2wi(~is'-~ligi)S(.g;-fl)(8; S)(,r, "r), (3.8)

where F* stands for the dual lattice of F and [g[ is a determinant of the metric on

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the torus. Therefore, the lattice F must be self-dual, F = F*, to obtain a modular invariant vacuum amplitude. The fact that the lattice F is replaced by F* under the transformation r ---> - 1/~" is familiar in the solid state physics, where high and low temperature expansions are inverted, T <-->l / T , by the Fourier transformation. We can prove the following identities which are important to show the modular invariance, S~r~; *)(n; ~) = S(r~'; g')(~; ~),

(3.9)

S(r~; ~)(~; ~) = e2~(~a- ~a)S(rn; ~){~'; ~'),

(3.10)

if (8; 8) - (8'; 8') ~ Fd, d, and

if 01; ~) - 01'; ~') = (A; A) ~ Fd, d. Let us investigate the modular transformation properties of S(r8;g)(~;~) comparing with that of S(r°'°)(°;°). We define the factors E1(s; g)(n; ~) and E2(8; 8)(71; ~) as: S (r,~; g)(,); .r))

~eo;~

S(rS; g)(,)- 8 + (a); .~- g+ (3_))

( r + I) = e ~ ~;*)~";~>

S(r~; g)(,); ~))

s~O;O)(O;O ~

(r),

S(ra-,); >:-- ~)(~; g)

s o;o> o;o>

(3.11)

where E 1 and E 2 read, E (8; g)(n; ~)) = e -~ri(~2-g2) X

1; e2~i(~A-SA) ;

without ( A ; f ~ ) , with (A; A ),

E 2(& g)(m ~) = e - 2 , r i ( n s - ~ ) g ) .

(3.12) (3.13)

Since the oscillation modes are not altered in modding out the torus by the translational discrete symmetry, we can easily express the modular transformation properties of the vacuum amplitude in eq. (3.4). Comparing with A(°'°)(r), these properties are obtained as, A(~,h)(r + i)

_ . A~k,~-~+>(~) _ ~,(kSo; kSo)(h(~o; hSo) 3'

+ 1) _ p~k~o; k~o>~hSo; h~o) A ~ u - h, k> ( ~ )

a~'°)(-I/r)

--2,3"

(3.14)

where the index 7 of El, 3" and E2, 3' labels the suborbifold. As the parameters k and

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h are defined in the region 0 > N - 1, N must be added in the first equation of (3.14) if h - k < 0, which is necessary to obtain the modular invariant amplitude. Next we investigate the modular transformation properties of the vacuum amplitude coming from the fermionic degrees of freedom. We represent the fermionic degrees of freedom by the bosonic variables q¢, of which zero-mode momenta are quantized on the weight lattice of SO(8). The contribution of q~t to the vacuum amplitude is A(~'h)(~ -) in eq. (3.2) and its explicit form is given by,

= ( nI->[0 (1

-- e 2~'in~" ) -4)e-~rir/3

w t ~ Fso(8)~

wt ~ Fsots)~p

where the parameter 7/~)is taken equal to that of discrete rotation for the fight-moving bosons, (~/~} = { .... (,/~) .... ), when we discuss particular orbifold models in the next section. Since the momentum sum is obtained as in eq. (2.36), it can be written as a sum over four dimensional integral lattice Z 4. Then the zero-mode momentum sum in eq. (3.15) becomes,

y" (l(l _ (_ )Z"')e-2~ihqo(~'+kn'o)+~i~(~'+k~'o)~ (nt)EZ 4 --1(1

-~(--)~"t)e-2~rih~t°(nt+l/2+k"t°)+~ri~'(nt+l/2+k*lt°)2). (3.16)

To treat this summation generally, we define the following summation over Z 4,

S(z~)(~)(T) -

E

e-2~i("'+~')"'+~i~("'+~'?,

(3.17)

ntEZ 4

which is a special case of eq. (3.5). We denote the zero-mode momentum sum in eq. (3.16) as a(~)(~) ~ so(8) and using S~z~)(n) it is rewritten as A(8)(~) SO(8) { ~ ~-]] =- ½( S (z{)(n)( ~.) _ e =iz'8~ (z~)(n+ 1/2)( q" ) -- S ( ~ + l/2)(rt) ( 'r ) - e~iZ,(W + 1/2)S Z(~+ 1/2)(7; + 1/2) [\ "r 1] ]] '

where 8 t = k~/~ and ~/t= h~/~). Since the properties under the modular transforma-

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310

tion are given in eq. (3.7), we obtain

A(8)(~)/r SO(8) ~, + 1) =

1,4,

~"SO(8)

~,'!

A(*)(~) ( - 1 / , ) = ( - it ~4/z E 2,(*)(n)A(W-*)(~) [ r), so(8) ~ ] 4, so(8)

(3.18)

where w' is a root vector of SO(8), w t ~ Fso(8),a; The extra factors EL4, and

E2,e#

are

E(8)(n ) _

1,4,

e_,.n2 × [ 1 ;

without w t with w t ,

~ e2~riS"w ;

--

E 2,4, (~)(~) = e -2'~inn

(3.19)

Using these results, the modular transformation property of A(~' h)(r) normalized by A(~'°)('r) becomes, A ~ ' h ) ( , + 1)

E(k%)(hno)

+ 1)

h>( _ A(~,o)( _ i / , )

a(}'h-k+(U))('r) v

,

a,+

= E(kno)(hno)

z,+

A(N-h, k>( '9 ¢

A(~.o)(z)

(3.20)

Notice that A(~'°)('r) is zero because of supersymmetry or the equivalence between the vector and the spinor representation of SO(8), but we treat it as a nonvanishing object in order to see the modular transformation properties. Let us examine the contribution to the vacuum amplitude coming from the bosonic freedom on the orbifold obtained by dividing the torus by its "rotational" discrete symmetry, O 2D'2b-~- T2D'2/5//Z(NR). Here we assume the existence of Z N rotational symmetry in each two dimensions of T 2D'26, namely we choose the constant background fields as in eq. (2.13). The Virasoro operators in the twisted sector is given by eq. (2.17) and the Z ~ ) transformation operator by eq. (2.18). The vacuum amplitude A(rk' h)(r) in eq. (3.2) becomes, (k, h) ~ (0, 0), D

A(~'h)(r) = F ( k ' h ) F I A'(k~Lhn'°)('r) " X'(k~L h~)(O(?), I=1

(3.21)

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where A' and A' are defined as, e 2 = i ~ ( - 1/12+ 8(1 - 8)/2)

A,(8,n)(r) -_ I-I (1 - e-2=in+2"i'("-l+a))(1 - e 2~i'7+2~i'("-a)) n=l e - 2~ri7(- 1/12 + ~(1 - ($)/2)

A'(&~)(r) --- i~I (1

-

e-2~rirT+2~rir'(n-6))(1

-- e 2~ifl-2rri~(n-l+g))

(3.22)

n=l

With a help of the product formula of the theta functions in eq. (2.37), we can express the infinite product in the denominator of eq. (3.22) by the sum over the integer lattice Z. Let us define the oscillator part of (k, h) = (0, 0) by

[(

A'(°'°)(r) --- tr exp 2~rir

(fl_,/3, +/~ ,ft,) 1

= e2~i,(-a/12)(f(r))

])]

-2,

where f ( r ) = FI n ~ N ( 1 - - e2='*"). Normalizing A '(8, 7) by A '(°'°), we obtain the contribution of each complex variable Z I for two different cases. For the case 8 t 4 =0 and ~/i= arbitrary:

A'(8', n') A,(O,O) ( r ) =

e#i,/4 ( f ( ,r ) ) 3 e2'~i(8'-1/2)('7'-1/2)S{z '~'-1/2)(0' 1/2)('r) ,

(3.23)

for 8 * = 0 and ~ff~0:

A,(O,. ')

e~i./4( f( r ) )3(1 -

e-2.,.')

A t ( ° ' ° ) ( T ) = e 2~ri(- 1/2)(7//- 1/2)8 ( - 1/2)(~7'

(3.24)

1/2)(T).

We already know the modular transformation properties of S z ( r ) in eqs. (3.7) and (3.8). Furthermore, there are identities,

[,-,,,4<:<,>>'],_,+,

= o-,,4

q ,

[e'~i'/'*(f('r))3]~_l/,=e-3'~i/4"r3/2[e""/4(f(r))3]

.

(3.25)

Under the transformation ~"---)r + 1, the functions in eqs. (3.23) and (3.24) are

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312 transformed

as,

A,(~~,,~') A,(n',,'-~'+m) A,(O,O) ( , + 1) = e "iS'O-n') NO,o ) (~'),

(3.26)

where m is an arbitrary integer. Under r ~ - 1/% the transformation properties are a little bit complicated; for 8 ~ 0 and ~ r ~ 0: A,(,

A , ( 1 - n ~ , #)

I , n ~)

A,(0,o) ( - l / r )

=

( - i ' r ) e -2~ri(nt-1/2)(U-1/2)

A,{O,O~

(r),

(3.27)

for 8 z = 0 and ~/i~ 0: A,(1-n~,o)

Ap(0 ' ~i)

A,(0,o) ( - I / r ) = (-i~)2sin(~rT] t)

A,(0,0) ( r ) ,

(3.28)

and for 81 :~ 0 and ~/l = 0: A,(~',o) A,(O,n') A,(O,O) ( - 1 / 1 " ) = (-i'r)[Zsin(~rSt)] -1 A,(O,o) (~').

(3.29)

In the above discussions, we do not include the summation over the zero-mode momenta. The zero-mode momentum sum appears only in A~'°~('r) because of the insertion of the transformation operator p(k) and the twisted boundary conditions of the twisted sectors. The amplitude A'-v °,°~ is well-known and we need not investigate the modular transformation properties. As for the degeneracy of the zero-mode, F (k'h), w e must consider the number of fixed points in each (k, h) sector. Since the insertion of the Z ~ ~ transformation operator corresponds to imposing the twisted boundary condition in the ~--direction, the fixed points in the (k, h) sector must satisfy, Z I = e2'~i*n~°Z' (mod ~rF),

(3.30)

Z I = e2'~ihn~°ZI (mod ~rF),

and similar conditions are imposed on the fixed points of the left-moving sector, 5 / . The factor F (k'h) is written as a product of the contributions from right and left movers, F (k'h) = F~ k,h). F~Lk'h). To obtain the modular invariant amplitude, F~ k,h) (F~ k'h)) must be a square root of the number of fixed points in the fight- (left-) moving sector. F~ k, h) and F~Lk" h) are modular invariant and satisfy,

The same identities hold for the left m o v e r s , F(Lk'h). An explicit form of F~ k'°) is

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313

given by F~*'°) = I-[ [2 sin(~rk~/)]. I

Notice that the factor 2sin(~rT/I) appearing in eqs. (3.28) and (3.29) are just the square root of the number of fixed points. It is surprising that the factor corresponding to the degeneracy of the zero mode appears automatically in (0, h) sector, which is a part of the vacuum amplitude in the untwisted sector with the insertion of (p(O))h. Thus we need not add the factor F (°,h) in the untwisted sector, but F (k'h) must be added in the twisted sectors. Finally we obtain the response of A(vk' h)(~.) in eq. (3.21) to the modular transformation as follows, A(~'h)(~ + 1) = E(kno; k~o)(hno;h~o)A(~k'h *+(~))(r) 1,7 A(O,O) ( , r ) A(v°'°)(~"+ 1)

A(~'h)(--1/'r) A~'°)( _ l / T )

= j~'(kTlo; k%)(h~0; h~0)--Y

+2,~

A(O,O)(+)

(3.31)

where the factors E 1 and E 2 are given by, D

b

E1,y (~;g)(n;~)= I-I e~'~'(1-~)FI e-~ig~(l g'), I=I

D

E(~; g)(n;~) = I-I 2, y

I=1

I=I

e-2~ri(n'-l/2)(St-1/2)

b

H

e2~i(nz-1/2)(g' 1/2)

(3.32)

I=1

Combining the results in eqs. (3.14), (3.20) and (3.31), we can derive the condition of modular invariance in the total orbifold. The responses of the amplitude in each suborbifold are the same except for the extra phase factors E 1 and E 2. Since the total amplitude is given by eqs. (3.1) and (3.2), these extra factors and the relative phase factors e ~v(~)÷ihf(*) must cancel with each other to obtain the modular invariant amplitude. Then we have the following conditions,

eir(*'+ihf*'-ir(u-h)-ikf (N h)(I~IvE~,k~h ) E ~ h)) = 1 ,

(3.33)

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314

for arbitrary k, h ( = 0 - N - 1). More explicitly, eq. (3.33) can be written as

A2 -

2 - E( 0 )2 + i

j

~j

2t

I

2

J

nEZ, .~(k) .~_ h f ( k ) __ ~I(N h) __ k f ( N - h )

n~Z,

F =

@ Fd~ ' d~ = lorentzian even self-dual lattice,

(3.34)

3'

where the conditions in eq. (3.34) are independent of the constant background fields, which will be shown in the next section. As we have mentioned in the beginning of this section, we can see from eq. (3.34) that there is a unique choice of the vacuum phase factor which is consistent with the modular transformation. Since we treat the parameters of the discrete transformation in left and right sectors differently, the compact space O a' d+ 16 is not the usual orbifold. They can be called asymmetric orbifolds [15]. The other possibility is that we can choose different N ' s to divide the torus. The condition of modular invariance in such models can be derived in the same manner as discussed in this section. We apply the results in eq. (3.34) to particular models in the next section. In this paper, we have reformulated the amplitude as a sum over certain lattices, but we can obtain the same results by using the theta functions and their modular transformation properties [10].

4. S e a r c h for the orbifold m o d e l s

In this section, we investigate some examples of the orbifold models satisfying the constraints of the modular invariance in eq. (3.34). As we have seen in the previous sections, the twisted sectors must be included to acquire a modular invariant amplitude. Thus the total Hilbert space is a direct sum of each twisted sector, N-1 tot _ ~orb-

.I~ (k) ~) J~orb • k=0

The Hilbert space of string variables with kth twisted boundary conditions must be projected onto a subspace which is invariant under the discrete transformation to acquire ~o~bg). This projection onto the invariant subspace is equivalent to choose states which have eigenvalue 1 with respect to eif~k~P~). String states are described

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as a direct product of left- and right-moving states: I )R ® I )L. The physical state condition is also imposed on these states, (M~t'))2=(M(Rk))2 ,

where M(Lk) (M(Rk}) is a mass operator of left- (right-) moving modes. Since we concentrate on the orbifold compactifications of the heterotic string, we must consider how the supersymmetries are realized on the orbifolds. As we have shown in sect. 2, the GSO-projected Neveu-Schwarz-Ramond model is described by the bosonic variables ~t (t = 1 - 4) whose zero-mode momenta are quantized on the SO(8) weight lattice. Supersymmetry charges which connect NS and R sectors are given by, V ( u ) = e 2i"'¢ ,

(4.1)

where fit is a bosonized string variable and u t belongs to the SO(8) conjugate spinor(cs) representation. Supercharges which remain after the orbifold compactifications must satisfy, =0,

[V(u),P,]

(4.2)

where P, is a Z u rotation operator in eq. (2.38). The condition in eq. (4.2) is equivalent to sin ( 2 ~ r u ' ~ ) = O,

u' E S0(8)o S.

(4.3)

If the number of u t satisfying eq. (4.3) is known, we can obtain the number of supercharges, 1

{ # of supercharges in D dim} = ~

{ # of u' satisfying eq. (4.3)},

(4.4)

where D is the dimension of the uncompactified space-time and M D is 1, 2, 4 for D = 2, 4, 6 respectively. Let us discuss particular models. We concentrate on the heterotic string compactified on the orbifold O d, a+ 16: od,

d+16 =

Ta, a+16/G = ( T a - a , / G 1 )

G 1 = Z N rotation, d

=

d I + d R,

6 2 d I =

=

®

(Td2"d2+16/G2) ,

Z N translation,

even,

d ~< 8,

(4.5)

where we choose the Z N rotation parameters of G 1 as ~/1 = ~1. In the construction of Td,, d~, we divide T d,' d~ into (T2'2) dl/2 and choose the constant background fields in each T 2'2 as those in eq. (2.13) so that TZ'2's have Z N ( N = 2,3,4,6) rotational

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symmetry left-right separately. The constant background fields o n T d2,d2+16 are arbitrary, which are gu, Bu, A~, (i, j = 1 - d2, a = 1 - 16). The string variables of our particular models are as follows,

xi('F,O),

i=l-8-d,

Z ' ( r - o),

2'(~+ o),

I=1-

Xt(r-o),

2t(~+ o),

2~(r+ o),

¢(¢-

a),

½d,, l=l-d2,

a=l-16,

t = 1 - 4,

(4.6)

where we use complex notations for the string variables in d~ dimensions. The parameters of the orbifold O u, d+ ~6 are, {{.~}={

.... 0 , ( , t o ) , 0 . . . . } , 7 1 t = ~ t , a n d N , t o , N , t o ~ Z ,

8 ~ { 80t;~, 8~ },

(4.7)

N8 = N{ 8or;~ol, 8~ } ~ Fa2,a ~+ t6,

where ~1t ( , ' ) ' s are parameters of Z u rotation and 8 is a parameter of Z N translation. The conditions of modular invariance in these models are as follows [101,

dr~2 N ~ ,to = even integer, I=1

hf(O) _ y(u-h) = 2~rn,

n E Z,

-vrk2A 2 + k f (k) = 2qrn ,

n ~ Z,

~[(k) -- 7(U-h) + hf(k) -- kf(U-h) -- 2~rkhA2 = 2vrn,

n ~ Z,

£ = Fa, ' d, ® Fu2,u2+ a6 lorentzian even self-dual lattice,

dl/2 /12-

d2

16

E (.to)Z+ Y', (((~ot)2- (~ot)2) I=1

l=l

E ($~)2,

(4.8)

a=l

for arbitrary k, h = 0 ~ N - 1. If we choose f ( k ) as f ( k ) = ~rkA2, we will find that eq. (4.8) becomes equal to the condition of the level-matching, which is equivalent to that of the cancellation of the global anomaly with respect to the modular transformation. The discrete translation vector 8 can be written as 8 = L / N with L = ( U ; L t, L ~) given by eq. (2.1). Then A2 in eq. (4.8) is rewritten as

A2-~- dl/2E( ~ 1 i ) 2

_{._ ~ 1

(2nlml_magabmb).

(4.9)

I=1

The constant background fields do not appear in eq. (4.9). This is because square

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terms of 8 in A2 appear with the lorentzian signature ((+)d2,(_)d2+16). Thus the condition of the modular invariance is independent of the constant background fields. We can see that the condition in eq. (3.34) are also independent of constant background fields. Unfortunately, we have no way to determine the dimension d in eq. (4.5). Therefore, we put d = 6, since the dimension of our Minkowski space-time is supposed to be four. For the compactification of the orbifold in eq. (4.5), the mass operators in the untwisted sector, L
i

i + 1.l::.,-

n>0

n>0

t t + ~(L') 2 + E d . d . + ½(w~/+ E ~-~° n>0

L'o°~= 2 ~' s o + ~'~' + 2 (~'-o~: + n>0

1 2,

n>0

n>0

~'o~'.)

+~(L') 2+ E a'_°a'o+ ~(£o)2+ E a~-.a.~- a, n>0

n>0

(DI. ~I) ~ rdl,dl,( tl; fL~',L a)

E JPd2,d2+16 ,

w t ~ Fso(8)~ or Yso(8),,,

(4.10)

where i = 1 - 2 , I = l - i d ~ ,1 l = l - d 2, a = 1 - 1 6 and t = l - 4 . In the usual notations, the Virasoro operators L o, Lo are defined including the zero-mode momenta in the uncompactified spaces, but hereafter we use the same notations Lo, L o as mass operators in which these zero-mode momenta are suppressed. The Z N transformation operator in the untwisted sector, p(0), is

P'°) = exp ( - 47r ~ ~1°(z'~' - £'pt ) - 2~ri ~i

I

-

I

2rri(3gp t -

For the state, [ )L ® following conditions,

,>0 ~

l ( fll-"fl: - fl1-"fl: )

n>0

ggfit_ 3op-~-~+ COP')).

(4.11)

>R, to be massless on the orbifold, they must satisfy the

Leo°)[ >R=£~o°) I ) L = 0 ,

P(°)eif'°'(]

>L~

I > R ) ~--

(4.12) ] ) L ® I )~.

The equation L(o°)] )R = 0 is satisfied if w e belongs to SO(8) vector or spinor

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representations. The weight vectors in these representations are given in terms of the orthogonal basis, e t (t = 1 - 4): vector

w = +_e t,

t = 1 ~ 4,

spinor

w = +_~e 1 +_ ½e2 ± ½e3 ± ½e4,

even ( - ) signs.

(4.13)

The coefficient of e 1 is observed as a helicity in four dimensions. In the k-twisted sector, the mass operators L
L'o~,--- E ~"-.< + E (~'-.-~.,~L~-e+ ~'~.-e~'~e) n~>O

n>O

+~(L'+~')~+ E ~'-°<+ ~("/+¢)~+ E <.<+~o, n>0

£(o~)= E a'~.< + n>0

n>0

~'+(~I

"I ~I ~I ~-.-~+~'&+,-e + B-.-.,B.+.,) ~o ~

1 ~a ~a ~a +~(L'+~') 2+ E ~-. ~' s'+~(r. +~°)2+ E ~_.~.+ao, n>0

n>0

dl/2

%= -½ + ½ E ~'(1-~'), I=1

4/2

4o=-1+{

Y'~ ~ z ( 1 - ~ ' ) ,

(4.14)

I=1

where ~i = k ~ , (81; gl, ~ ) = k(3ot; g~, g~). The Z u transformation operator, p(k), is

p(k)

= exp

\

-2~ri~ I

I

2 .~-'-~_lj~I~,"l--l'~I~°c~'l I -~>0kn-~

[ E !

n>0[ n+

1

~I

I --I n + l - ~ 1fl-"-l+n'fl"+l-n'

- 2~ri(3~p'- 3ol/5'- 3g/3" + ~p')).

1

t~I

1 -"l] ip-n-l+~;pn+l-ff'

--

-

"I

-ZI

]

rt+~lfl-n-ntfln+n' -

J

(4.15)

Notice that we assume 0 < 7/I< 1. If k ~ exceed 1 in the mode expansions of Z ~, we redefine *1* by subtracting certain integers from k ~ so that ~1t becomes smaller

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than 1. The massless states in the k-twisted sector satisfy the similar conditions as those in eq. (4.12). Let us comment briefly on the relations to the Calabi-Yau compactifications. In the theories in which six dimensions are compactified on the Calabi-Yau manifold, we obtain the E 6 grand unified theory (GUT). In E 6 GUT, quarks and leptons are contained in the 27 representation and the number of families, Nf, are given by, Nf= N ~ - N ~ . ,

(4.16)

where NL7 (NLT.) is the number of left-handed 27 (27*) representations. It is well-known that there is a relation between N~ and the Euler number of the Calabi-Yau manifold X [17,18], Nf = ~ x .

(4.17)

In the derivation of the above relation, the spin connection on the six-dimensional Calabi-Yau manifold is identified with the gauge connection of E 8 gauge symmetry. The relation in eq. (4.17) is obtained in the orbifold models in which the discrete rotational symmetry in the six-dimensional torus is identified with the translational one in the E 8 maximal torus. We call such embeddings of Z N rotational symmetry into Z s translational one as a standard embedding [1]. If we consider the compactifications on general class orbifolds such as those in eq. (4.5), the unbroken gauge symmetry is not always E 6 and the embedding of Z N rotational symmetry into Z N translational one is not necessarily a standard one. Thus there seems to be no such relations between the topological quantities and the number of families in general. Next we have to reconsider the way to count the number of families. In grand unified theories, the number of families is defined as eq. (4.16). One of the reasons to define eq. (4.16) is that if there are (27L,27L*) pairs they have naturally superheavy mass terms and can not survive at low-energy scale. In string theory, however, the circumstances are very different from those of particle theories. It becomes clear to consider in terms of the string field theory, in which the action is given by, S - l(,l, x l ( ~ z l g l R ) t 2 + ~g(~tl(~zl(~h31 1V)123,

(4.18)

where [R)I 2 is a reversion operator, ] V)123 is a three-string vertex operator and K is a kinetic term. We can not add mass terms of (27L, 27~) pairs by hand but such mass terms are generated only through a dynamical symmetry breaking. Thus there are possibilities that some particles in the (27L, 27~ ) pairs have light masses. In our model searches, we investigate the number of massless states and their representations with respect to the unbroken gauge symmetry by the explicit construction of the massless states.

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Let us concentrate on the Z 3 orbifold models, N = 3. To search Z 3 orbifold models, we p u t the following constraints: 1. T h e c o m p a c t space is O a+16 in eq. (4.5) with N = 3. 2. d = 6 (four-dimensional Minkowski space-time). 3. Supersymmetries remain in the resulting theory. 4. T h e conditions of modular invariance in eqs. (4.8) and (4.9) are satisfied. U n d e r the constraints of 1, 2, 3, and NET/I0 = even in eq. (4.8), the allowed dl, d2, ~0 and the n u m b e r of supercharges are severely restricted. They are: Case 1

d 1 = 4,

d z = 2,

"Ot = (0, 71 , 71, 0 ) ,

N = 2 SUSY,

Case 2

d 1 = 4,

d 2 =

2,

2 2 r( = (0, 5, 5,0),

N = 2 SUSY,

Case 3

dI = 6,

d 2 = O,

~t = (0, 5, 1 31 , 2 ) ,

N= 1 SUSY,

Case 4

d1= 6 ,

d 2 = O,

~t

N= 1 SUSY.

= (0, ~ 2 , 52, 2 ) ,

If we choose d 1 = 2, d 2 = 4, 7/0 = (0, 7, 2 0 , 0), we get non-supersymmetric models and if d~ = 0, d 2 = 6, we obtain models with N--- 4 SUSY. In the above list, cases 1 and 2 are equivalent and cases 3 and 4 are also equivalent. Thus, we have only to consider cases 1 and 3. Since the states which exist consistently on the orbifold are singlets with respect to eif~k~P(k), the spectrum of the model would be completely different according to the choice of the v a c u u m phase f~k). Due to the Z 3 p r o p e r t y 4 of the v a c u u m phase, f~k) takes value 0, -~rr or 7rr. F r o m the condition of the m o d u l a r invariance in eq. (3.34), the consistent values of the v a c u u m phases and the c o r r e s p o n d i n g choices of As are restricted to the following three cases: A

f(o) = f ( t ) = f ( z ) = 0,

A2 = 2n

n ~ Z,

B

f(o)=0,

f(1)=ur2,

f(2)=ura,

As = 2 n + 2

n~Z,

C

f(O)=o,

f(1)=43i r,

f(2)=_~r,

Az = 2 n + 4

n~Z.

(4.19)

In the following of this section, we will investigate the cases 1 and 3 for the above three choices of the v a c u u m phases. 1 Case 1. d 1 = 4, d 2 = 2, Bt= (0, ½, 7,0). As we have discussed above, we choose the b a c k g r o u n d fields in d l ( = 4) dimensions as in eq. (2.13) and A a = 0 (i = 1 - 4, a = 1 - 16) so that the dl-dimensional space has a Z 3 rotational s y m m e t r y left-right separately. The other constant background fields AT, gtk and B~k (l, k = 1 - 2, a = 1 - 16) are still remaining as free parameters of the models. Since the condition of m o d u l a r invariance in eq. (4.19) is independent of these b a c k g r o u n d fields, we can count the n u m b e r of integer sets (n t, ml, m a ) which satisfy eq. (4.19) in case 1.

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As an example, we have calculated the number of such integer sets for the choice of the vacuum phase A) in eq. (4.19), f ( 0 ) = f { 1 ) = f { 2 ) = 0. In this particular choice of the vacuum phase, the condition of modular invariance becomes:

(2

)

2 + ~ 2ntmt- m~gabrnb

= even.

1=1

The Cartan matrices, gob, of E 8 and Spin(32)/Z 2 are given by [4, 19], 2 -1 0 0 0 0 0 0

E 8•

-1 2 -1 0 0 0 0 0

0 -1 2 -1 0 0 0 -1

0 0 -1 2 -1 0 0 0

0 0 0 -1 2 -1 0 0

0 0 0 0 -1 2 -1 0

4 -1 1 -1

-1 2 -1 0

1 -1 2 -1

-1 0 -1 2

... ... ... ...

1 0

0 0

0 0

0 0

... ... ...

0 0 0 0 0 -1 2 0

0' 0 -1 0 0 0 0 2,

(4.20)

1 0 0 0

0 0 0

-1 2 0

-1 0 2

0 ~

Spin(32)/Z2 : 2 -1 -1

(4.21)

If we restrict the discrete translation vector 8 in F2,2+16 to those contained in a unit cell of F2,~+16, the integers n l, mr, m~ can take values 0, 1 and 2. After a little calculation using a computer, we find the number of integer sets, (a)

397728342

forg°b=CartanmatrixofEs®E~,

(b)

389281851

for gab = Cartan matrix of Spin(32)/Z 2.

Since there are as many different orbifold models as the choice of constant background fields in d 2 + 16 dimensions, we have - 108 x oo possibilities, where comes from the freedom of the constant background fields. The similar situation will be found for the other choices of the vacuum phases B) and C). We have, however, several examples among the case 1 models, which is helpful to understand

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L Senda, A. Sugamoto / OrbifoM models

the characteristics of these models [10]. Here, we will give two examples among case 1 models with the choice of the vacuum phase A). Examples 1. We choose the lattice as

r~,6+~ = r4,, ~ (P2) 2 * r E ~ . ~ , and the parameters of Z 3 transformation are given by,

1- o-))

3~3~

,

where 1 = 1 - d 2 ( = 2), a = 1 - 16. Since 01o) 2 + (8~) 2 - (gol) 2 - (gff)2 = 0, this model satisfies the condition of the modular invariance in the choice of the vacuum phase A). This model has N = 2 supersymmetry in the zero mass sector. We separate the right- and left-moving parts of p(k) in eq. (4.11) as p(k) _-- p(k)p(k) K --L and define its eigenvalues of right (left) moving states I )R(I )L) as ~R()tL). In the untwisted sector (k = 0), the right-moving states which contribute to the massless states come from N S R sector, Iwt)R, where w t belongs to SO(8) vector (8v(R)) or spinor (8~})) representations. The representations -8v(a) and 8--sp (a) are classified according to the eigenvalues ~ a,

{ {

)~v. = 1,1(1 ) • 1(_1) • 2(o) ,

8_(R):

)kR

eZ~r//3,2(0),

~R

e-2~i/3,2(o),

(4.22)

~ R = 1 , 2 ( 1 / 2 ) (~ 2 ( _ 1 / 2 ) ,

8 (R)"" --sp

~R

~'R

~2~ri/3 ~ ,1 ~(1/2) • 1 ( - 1 / 2 ) , - 2~ri/3 e ,1(1/2 ) • 1 ( _ 1 / 2 ) ,

(4.23)

where the n u m b e r m(h ) denotes the multiplicity m and the helicity h. The lattice vectors in P2 can be written in terms of integers m t, n l ( I = 1,2),

(L'; L') =

m,; . ' + m,).

(4.24)

T h e square (Lt) 2 can not be ½ and I L t ) a do not contribute to the massless states, b u t ILt)L can be massless since the (Ll) z = 2 is obtained if n t = m r = +1. The left moving sector has_also SO(8) vector representation 8(vL), which is realized by ~ / 1 I 0 ) L' /~I_ 1 ] 0 ) L' /~/- I I 0 ) L and ~t_ 110) L- The classification of left movers accord-

I. Senda, A. Sugamoto

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323

ing to X, is

A, = 1,1(l) @1(-l) @2,) 2 X,=e

t3$‘-’: i

Classification

2ni/3

2

’

(4.25)

(0) ’

XL = e-2ni/3, 2,, .

of left-moving massless states with internal momenta are,

X,=1,(8,1,1),(1,78,1),(1,1,248),l~’),,

IO, :

XL = e2ni/3, (3,27,1),

(4.26)

i h, = e-2ni/3, (3*, 27*, 1))

where the number in the parenthesis indicates the representations with respect to SU(3) 8 E, 8 Es. The massless states in eq. (4.26) have helicity 0 and Ii’)r realize SU(2) @ SU(2) gauge symmetry accompanied with Zi/_,(0) L. Gauge bosons are obtained by BzR’8 (t)L with X, = XL = 1. The gauge symmetry induced by these states are SU(2) @ SU(2) @ SU(3) 8 E, 8 Es. Their superpartners are given by the product Q,(R)Q IL)r with h, = XL = 1. The matter supermultiplets are obtained by the products 8:R)@ Ii), and S$)@J Ii)L with X, = e*2nr/3 and h, = er27ii/3. We have three 27 and three 27* with helicity i, together with the same number of their antiparticlestates, namelythree 27 and three 27* with helicity - 5. In the twisted sectors, the massoperators a;given by eq. (4.14) with CQ= - &, Zi, = - $. After a little research, we find that there are no massless state and no tachyonic mode. This fact can be easily seen by investigating La’, which is given by, + + i

(Wt+kqg2-

A,

(4.27)

t=1

where k = 1,2. The minimal value of the first two terms in the right-hand side of eq. (4.27) is & and we obtain L,(k) > 0. Since this model is nonchiral, we have theory with zero family (Nr = 0) in the ordinary way to count the number of families. As we have discussed previously, there may be some mechanism which give some of left-handed 27 light masses. In the nutrino mass problem, we encounter similar problems andthese problems are solved by the see-saw mechanism. Even though the direct application of the see-saw mechanism may not work, we hope to have a similar mechanism which makes right-handed 27 superheavy but left-handed 27 light. Then we can ignore three 27ci,2j and obtain three left-handed chiral families having reasonably small masses. Examples 2. We again choose the lattice as

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I. Senda, A. Sugamoto / Orbifold models

and the parameters of Z 3 transformation are given by,

,o=(0.b 1

1

1

1

1

2

14

Let us see the untwisted sector, first. In this model, E 8 ® E 8 is broken down to E 7 ® U(1) ® E 8. The classification of the right-moving sector is the same as those in eqs. (4.22) and (4.23) of the example 1. The massless states in the left-moving sector is classified with respect to E 7 representations,

IL)L :

XL= 1:

(13__33),

hE = eZ~i/3:

(!) ~ (56),

XL

(!) ~ (56),

e -2"i/3"

(4.28)

where 133, 56 are the adjoint and the fundamental representations of E 7. In the twisted sectors, there is no massless states, which is similar to the first example. Then we obtain massless states 2 ( 1 3 3 ) • 2 ( 5 6 ) • 2(1) with the helicity ~ and the representations with the helicity - }. Hence, this model is left-right symmetric and we obtain two 56 representations with ½ helicity and two 56s with - ½ helicity. We have discussed only two examples of case 1, but there are infinite numbers of the possible models among which there may be interesting models. Case 3. d 1 = 6 , d 2=0,*1 t = (0, !3 ' ± to 3 ' z~ 3 I " Since we must put A ~ = 0 ( i = l - 6 ) maintain the Z 3 rotational symmetry in d l ( = 6) dimensions, there is no background field in 16 dimensions. Then the lattice 1"16 is a root lattice of Es® E~ or Spin(32)/Z 2. In this case, we also restrict the shift vector in F16 to those in a unit 1 a ma, where ~ t a lattice of F16. A shift vector in F16 can be written as 8 = ~et (a = 1 - 16) is a simple root of E 8 ® E 8 or Spin(32)/Z 2 and m a = 0,1,2. Then zlz in the conditions of the modular invariance in eq. (4.19) becomes A2=~(6-m,~gabrnb).

Let us concentrate on the models in which F16 ~-~FEs®E~. We have investigated all possible embeddings of the Z 3 translations into E s ® E~ lattice for three cases A), B) and C) in eq. (4.19). We have found that there are only five distinct models, of which unbroken gauge symmetries are (1) (A 2 - E6) ® E~, (2) E 7 ® D7, (3) D 7 ® A s, (4) (A 2 - E6)® (A 2 - E6) and (5) Es® E~. Three choices of the vacuum phase factors in eq. (4.19) and the corresponding unbroken gauge symmetries of the models are: (1) - (4) for A), C) and (1) - (5) for B). The model with unbroken gauge symmetry (A 2 - E6) ® E~ is the Z-orbifold model and its detailed derivation will be found in ref. [11]. Similar to case 1, the massless gauge bosons appear only in the untwisted sector with ?~L = ?~a = 1. We have the following representations of

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325

vector supermultiplets with respect to the unbroken gauge symmetries, (1).

(A 2 - - E6)

(2).

E7® D7: (133,1) * (1,91),

(3).

D7 ®As: (91,1) • (1,80),

(4).

(A 2 - Z6) ® (A' 2 - Eg): ((8, 1), (1,1)) • ((1,78), (1,1))

~ E8:

((8, 1), 1) ~ ((1,78), 1) ~ ((1,1), 248),

((1, 1), (8,1)) * ((1,1), (1,78)). (5).

E 8 ® E~: (248,1) • (1,248).

(4.29)

The representations of chiral supermultiplets are, (1).

( A 2 - E6) ®E8: 3((3,27),1),

(2).

ET ® DT: 3((56,1) ~ (1,1) ~ (1, 64*) ~ (1,14) ),

(3).

D7 ® As: 3((64",1) • (14,1) @ (1,84)),

(4).

(A2 - Er) ® (A'2 - E~)" 3((3, 27), (1,1)) • 3((1,1), (3,27)),

(5).

E 8 ® E~: (1,1),

(4.30)

where the factor 3 is a multiplicity of _3* representation of SU(3) which is a subgroup of the transverse group SO(8). We also have the conjugate representations for the antichiral supermultiplets which correspond to antiparticles of those in eq. (4.30). We should notice that there is no chiral super-multiplet in the model (5), since E 8 ® E~ symmetry of the heterotic string is unbroken. Next, we study the massless states in the twisted sector. The mass operator of the right movers without the excitation of the oscillation modes is, 4

L g)- ½ E (p'+

_1 6~

t=l

where k = 1, 2. There is no tachyonic state and the massless states are given by Iw'),, where w ~ is w ' = (0,0,0, - 1 ) a n d ( -17,-7,1 _ _2,x _ ½) w ' = (0, - 1 , - 1 , - 1 )

and (½,

The eigenvalue with respect to P~Rk) is 1,

1

2' ~R =

1

2'

for k = 1, 3)

for k = 2.

1. Thus the left-moving part of the

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L Senda, A. Sugamoto / OrbifoMmodels

massless states must have eigenvalue h L = e -if(*~ so that the products of left and right movers become singlets under the Z N transformation ei/(k)P (k). The mass operator of the left movers is L(ok) in eq. (4.14) with d 0 = - 2. We have to search for the states with L(0k)= 0 and e~/C~)P(k) = 1 among the models ( 1 ) - (5) for three choices of the vacuum phase factors separately, where f(k) is non-zero for B) and C). Then for k = 1, we obtain the following representations of the gauge symmetries satisfying such conditions,

(1).

(A 2 - E6) ® Es: ((1,27),1) ~ 3 ( ( 3 , 1 ) , 1 ) ,

(2).

ET® DT: (1,14) ~ (1,1) • 3(1,1),

(3).

D 7 ® As: (1,9),

(4).

( A 2 - E6) ® ( A ' 2 - E~): ( ( 3 , 1 ) , ( 3 , 1 ) ) ,

(5).

E 8® E~: 9(1,1).

(4.31)

For k = 2, we obtain the conjugate of the above representations. It is interesting that the representations appearing in the twisted sectors are the same even if the value of the vacuum phases in three cases A), B), C) are different. Since there are 27 fixed points in the six-dimensional orbifold, 0 6'6, we have 27 copies of the states in eq. (4.31). Combining the results in eqs. (4.30) and (4.31), we can count the number of families. In model (1), we have 36 chiral families if we assume that quarks and leptons are in 27 of E 6. In models (2) and (3), we do not know in what representations quarks and leptons are contained. Thus the number of the chiral families in (2) and (3) is undetermined. In model (4), we have nine chiral families if we assume quarks and leptons are in 27 of E 6 or E~. Because model (5) has no chiral supermultiplet and E s ® E~ remains unbroken, the number of families can not be discussed in this model. F r o m the results in this section, we can see that the condition of modular invariance works very well in restricting the orbifold models. Our Z3-orbifold model searches of cases 3 agree with those of ref. [1], in which the classification of the models is performed under the condition of the level-matching which is equal to the cancellation of the global anomaly. In this paper, the classification of all the chiral super-multiplets belonging to the twisted and the untwisted sectors is performed by carefully considering the vacuum phase factors and the massless spectra of the chiral supermultiplets are listed. Thus we have completed the model search of E 8 ® E~ heterotic string compactified on the Z 3 orbifold in eq. (4.5) with d I = 6 ( N = 1 supersymmetry). In our search of the case 3 orbifold models, we have found that all possible embeddings into E s ® E~ results in only five distinct models. As pointed out in ref.

L Senda, A. Sugamoto / Orbifoldmodels

327

[1], this is due to the automorphism of the lattice, namely the Weyl transformation of the lattice vectors. The Weyl transformation associated with the root vector ~, W~, transforms the discrete translation vector 8 as,

(e, Since this transformation does not change the value of A2 in eq. (4.19), the condition of modular invariance is reserved. One can see that the embeddings 8 in each E 8 lattice are classified in three Weyl equivalences. This is a consequence of the Z 3 property of the discrete translation vector 8, 33 ~ FE~®E~. It might be possible to classify the embedding vectors in the case 1 models using the automorphism of the lattice.

5. Discussion The orbifold models, which we have discussed in this paper, are obtained as the direct product of the suborbifolds. Starting from the torus having arbitrary constant background fields gij, Bij, and A~, each suborbifold is constructed as a quotient space of the torus by the translational and rotational Z N symmetries, where the rotational Z N symmetries are considered in two dimensions. In the case of the translational Z N symmetry, the order N and the constant background fields are arbitrary. It is well-known that there are four kinds of Zz¢ rotational symmetries in two-dimensional lattice, namely N = 2,3,4,6. When we divide a torus by Z N rotational symmetries left-right separately, the constant background fields are restricted so that the torus has such a symmetry. The solution of the constant background fields in eq. (2.13) has no excitation of A a, A~= 0. As we have mentioned in sect. 2, we can introduce arbitrary non-zero A~ if the twisted boundary conditions are imposed on the sum of the left- and right-moving string variables [20]. Since we can discuss symmetry breakings of the model using A ai , the existence of A~ makes the variety of the models richer. It is true, however, that the condition of modular invariance in eq. (4.8) is the same even if we employ the left-right non-independent formulation in imposing twisted boundary conditions. But for left-fight asymmetric models in which the twisted boundary conditions are imposed left-right separately, ~14~ ~i, the condition of modular invariance in eq. (3.34) must be applied. In such models, the constant background fields A7 are restricted by the existence of Z N rotational invariance on the torus left-right separately. In this paper, the degrees of freedom including fermionic ones are represented in terms of bosonic variables. In the usual bosonization technique, the rotational transformations of the complex fermions are replaced by the translational ones of

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the real bosons. It is difficult, however, to represent the rotational transformations of the complex bosons in terms of some sorts of transformations of the fermionic variables. Thus the bosonic formulation of the orbifolds seems to make it possible to treat wider range of the models. Among the models satisfying the condition of the modular invariance, there exist infinitely many possible models having N---2 supersymmetry. This is because we can introduce arbitrary constant background fields keeping the condition of the modular invariance. As for the models with N = 1 supersymmetry, we survey all possible embedding of Z 3 translational symmetry into the entire E 8 ® E 8 maximal torus and found that there are only five distinct models, E 6 ® E8, E 7 ® SO(14), SO(14) ® SU(9), E 6 ® E 6 and E 8 ® E~. The spectrum of the massless chiral supermultiplets are also investigated. One of the most important problems left unsolved is how we can calculate variety of small masses for quarks, leptons and gauge bosons starting from the superstrings which have only two parameters of the string tension and the three string coupling constant. We do not know whether this can be achieved by the dynamical breaking of symmetry or just by the introduction of tiny breaking terms through the excitation of the background fields. If the latter case is realized, then we may obtain a fancy model to generate masses under fine tunings. If the former case of the dynamical breaking is realized within the framework of the string theory, we will have a different picture on the symmetry breakings. One of the origin of the finiteness of the superstring is the supersymmetry in ten dimensions. Another reason comes from the modular invariance, especially the invariance under the transformation "r ~ - 1 / r . This is something like the duality between s and t channels at finite temperature, but the interchange of s and t channels should be associated with the inversion of the temperature r, or that of infrared and ultraviolet regions. This similarity between infrared and ultraviolet behaviors contributes to the finiteness. Therefore the dynamical breaking in the string theory may have very different features from the usual ones in the particle theories. The decoupling of massive modes at low energy region should be reanalyzed keeping the similarity between the infrared and ultraviolet behaviors. In this situation, the necessity of supersymmetry in four dimensions from the view point of the naturalness should be also reconsidered. The authors would like to thank Dr. K. Hagiwara, Dr. M. Kato and Dr. M. Kobayashi for useful discussions.

References [1] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 651, B274 (1986) 285 [2] V.P. Nair, A. Shapere, A. Strominger, F. Wilcsek, NSF-ITP-58 [3] K.S. Narain, Phys. Lett. 167B (1986) 41

L Senda, A. Sugamoto / Orbifoldmodels [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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D. Gross, J. Harvey. E. Martinec and R. Rohrn, Nucl. Phys. B256 (1985) 253 K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B279 (1987) 369 C. Vafa, Nucl. Phys. B273 (1986) 592 Da-Xi. Li, Phys. Rev. D34 (1986) 3780 H. Kawai, D.C. Lewellen, and S.H. Tye, Phys. Rev. Lett. 57 (1986) 1832, CLNS 86/751 (1986) M. Mueller and E. Witten, Phys. Lett. 182B (1986) 28 I. Senda and A. Sugamoto, KEK-preprint TH-143 I. Senda and A. Sugamoto, KEK-preprint TH-155 K. Ito, Phys. Lett. 184B (1987) 331 A. Strominger and E. Witten, Comm. Math. Phys. 101 (1985) 341 A. Strominger, Phys. Rev. Lett. 55 (1985) 2547, NSF-itP-85-109 K.S. Narain, M.H. Sarmadi and C. Vafa, HUTP-86/A089 R. Gilmore, Lie groups, Lie algebras, and some of their applications, (Wiley-lnterscience) P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Nucl. Phys. B258 (1985) 46 E. Witten, Nucl. Phys. B258 (1985) 75 P. Ginsparg, HUTP-86/A053 L.E. Ibanez, H.P. Nilles and F. Quevedo, Phys. Lett 187B (1987) 25 P. Goddard and D. Olive, in Vertex operators in mathematics and physics, ed. by J. Lepowsky, S. Mandelstam and I.M. Singer (Springer, New York, 1985) [22] F. Gliozzi, J. Sherk and D. Olive, Nucl. Phys. B122 (1977) 253 [23] D. Manfold, Tara Lectures on Theta, Prog. in Math. vol. 28