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Allowed Yukawa couplings of ZN×ZM orbifold models
Volume 262, number 4
PHYSICS LETTERS B
Allowed Yukawa couplings
of7/NX
77M
27 June 1991
orbifold models
Tatsuo Kobayashi a and Noriyasu Ohtsubo b a Department of Physics, Kanazawa University, Kanazawa 920, Japan b Kanazawa Institute of Technology, lshikawa 921, Japan
Received 3 August 1990; revised manuscript received 8 April 1991
SO(10) momenta and fixed points are studied for obtaining non-vanishing Yukawa couplings of ZN× YMorbifold models. As an example, we investigate explicitly three-point couplings of 714M774 and Z3×Z3 orbifolds with the standard embedding and no discrete torsion. Comparing with the 26 and 19 model by Gepner's coset construction, respectively, we find discrepancies with respect to the numbers of (27) (27) (27) couplings, while the numbers of 27 are identical.
U p to now, m a n y four-dimensional string vacua have been found through orbifold compactification [ 1 ], fermionic string constructions, [ 2 ], self-dual lattice construction [ 3 ], coset construction [ 4 ] and so on. They gave us various kinds o f gauge groups and m a t t e r contents through (0, 2) world-sheet supersymmetric models. F o r example, 4, 12, 58, 61, 39, 246, 248, 3026 and 3013 i n d e p e n d e n t models have been o b t a i n e d for 2r3, 224, 2261, 226-II, 227, 718-I, 7/8-11, 212 -I and 22~2-II orbifolds without Wilson lines, respectively [5]. Even if one limits the consideration to the (2, 2 ) model, there exist m a n y possibilities. Their gauge groups and m a t t e r representations are restricted so as to include E 6 X E8 and 27 and 27, but the numbers of the representations and the allowed interactions d e p e n d on the topologies o f internal spaces. It has been found that some o f G e p n e r ' s models have the same numbers o f m a t t e r representations and the same types o f Yukawa couplings as those o f models on the Calabi-Yau manifold [ 6 - 8 ] . These facts indicate that the C a l a b i - Y a u manifolds have the same topological structures as the c = 9 internal sectors o f G e p n e r ' s models. It is supported by a more detailed analysis o f fourgeneration models o f Y(4; 5)/275×7/5 and 35/22/5×225 [9,10]. A coincidence between some orbifold models and G e p n e r ' s models has been reported with respect to their matter contents as well [11,12]. R e m a r k a b l e examples are given by 7IN×Z× orbifold models with standard embeddings and no discrete torsion [ 1 1 ]. All the models correspond to at least one G e p n e r model with respect to the numbers o f 27 and 27. However, the correspondence among the Yukawa interactions has never been investigated explicitly. Although the general features o f Yukawa interactions on the orbifolds have been discussed [ 13,1 4 ], concrete investigations have been restricted to the case o f the 223 orbifold. In ref. [ 1 5 ], structures o f physical twisted states have been clarified and then calculations o f the Yukawa couplings have been m a d e possible in the case o f the other 22zvorbifold models. In this letter we extend the previous work to the ZN×22M orbifold models, and compare them with the corresponding G e p n e r models. First o f all, we study a constraint for the Yukawa couplings due to SO ( I 0 ) invariance o f the N S R fermion. In table 1 we list all the three-point couplings allowed by the SO ( 10) invariance. Next, we study structures (e.g., fixed p o i n t s ) o f six-dimensional Lie lattices possessing 22N× 22at rotations as their automorphisms. A generalized G S O projection is given, with which one can obtain a degeneracy factor and derive physical twisted states belonging to a representation. Space group conditions are also investigated in the 224× 224 orbifold as an example. Finally we examine a correspondence between the 22~vX 22~1orbifold and a G e p n e r coset construction through concrete examples o f 774×22 4 and 2 6, and 223×223and 1 9. 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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Table 1 Twisted sectors in/Z,,,,X7/,,~r orbifolds. The fourth column lists one example of the six-dimensional Lie lattices possessing the point group ZNXZ~ as the aulomorphisrn. The twisted sector Tkt in the last column denotes that the massless matter superfields of the sector are allowed under the shift kvl + lv2 of SO ( 10 ). Point group v~
vz
6D Lielattice
Twisted sector
272X~2 Z3XYs Z2×Z4 Z2XZ; ~2)~Z6 ~-4X/74 ~TSXZ6 ~6)~6
(0, 1, - 1)/2 (0, 1 , - 1 ) / 3 (0, 1,--1)/4 (1,1,-2)/6 (0, 1,--1)/6 (0, 1,--1)/4 (0, 1,--1)/6 (0, 1 , - 1 ) / 6
SO(4) 3 SU(3) 3 SO(4) X S O ( 5 ) 2
Tob T~o, TH Tol, To2, Tio, Tll, T.2, ]20, 121 Tol, To2, To3, Tlo, Tj b TI2 lo~, Toz, To3, T~o,T~, T~s, T~4 Tot, To2, To3, 704, To5, Tlo, TII, TI2, TI3 Tol, To2, To3, T,o, TII, TI2, TI3, T20, T21, T22, 730, Tst Tol, To2, 703, 704, ]/05, Tlo, TII, TI2, TI3, Ti4, T20, T21, T22 ]rOb 702, To3, To4, To5, Tlo, TII, TI2, Tt3, Ti4, TIs, T20, T2b 7"22, T23, T24, T3o, T31, T32, T3s, T4o, T41, T,2, I20, Ts~
( 1, 0, - 1)/2 (1,0,-I)/3 (1,0,-1)/2 (1,0,-1)/2 (1,0,--1)/2 (1,0,--1)/4 (1,0,--1)/3 (1,0~-1)/6
G~ SO(4)×G 2 SO(5) 3 SU(3)xG~ G3
The NS and R fermions in the right-moving sector o f the heterotic string can be represented by vectors and spinors o f S O ( 1 0 ) , respectively, in the bosonized version. Their m o m e n t a are Pv = ( -+ 1, 0, 0, 0, 0), where the underline means permutations, and ps = ( _+ 1, _+ 1, _+ 1, _+ 1, _+ 1 ) / 2 with an even number o f + signs. The ~NX~M rotation becomes shifts of a S O ( 1 0 ) root lattice under the bosonized expression. They are written as v~ = (a~, b~, q , O, O ) / N and v2= (a2, b2, c2, 0, O ) / M , where a~, b~ and c~ (a2, b2 and c2) must be a c c o m m o d a t e d with exponents o f the ZN (ZM) rotation due to preserving the right-moving N = 2 world-sheet SUSY. We impose on them the conditions a ~+ b~ + c~ = 0 and a2 + b2 + c2 = 0, which guarantee conservation o f at least one spacetime SU SY. (The values o f the first three c o m p o n e n t s of v~ and v2 o f each 7/x × 7/M orbifold, which conserve only one ( N = 1 ) spacetime SUSY, are given in the second and the third column of table 1, respectively.) Since the N = 1 spacetime supercharge Q must satisfy p ~ vt = p ~ / ) 2 = 0 , it is represented by p ~ = ( 1, 1, 1, _+ 1, _+ 1 ) / 2 , where + ( - ) denotes the upper (lower) c o m p o n e n t s o f the four-dimensional spinor. We define four-dimensional chiral m a t t e r of the untwisted sectors by Pv = ( 1, 0, 0, 0, 0) and their superpartners with p + =Pv - P ~ , where the sign conventions of p~ are same as those of Po- The three untwisted sectors are denoted by their vector m o m e n t a as U,:pv=(1,0,0),
U 2 : P v = (0 , 1 , 0 ) ,
U 3 : P v = (0, 0, 1 ) .
Here and hereafter, the fourth and fifth components o f the S O ( 1 0 ) m o m e n t a are ignored, because they are preserved automatically. The m o m e n t a o f a (k, l)-twisted sector are represented by Pv + kvj + l/)2 and Ps + kv~ + l/)2 for a boson and a fermion, respectively, where p+ + k/)~ + Iv2 =Pv + k/)~ + [/)2 - P ~ . The massless conditions o f chiral matter superfields for the (k, /)-twisted sector are given by 3
3
(p~ + k/)~ + l/)~2)2= ~ t=
I
~=
1
(p's + kV~ + l/)tz)2 + ½= l -- 2Ckl ,
(1)
1
3
c k l = ~ ~.= ( I k / ) ] + l / ) ~ l - I n t I k / ) ~ + l v ~ [ ) ( 1 - 1 k / ) ] - l / ) ~ l + I n t
Ikv]+l/)~l).
(2)
The last column o f table 1 denotes the twisted sectors satisfying the massless condition ( 1 ) in each ZNX ZM orbifold. The SO (10) m o m e n t a must be conserved in order to give a non-vanishing scattering amplitude. This means that
p!,) ..Fp+(2)+ps+ (3)=pv~l) +p!2)+pv~3)--Pf2 --PQ = 0
(3)
for three-point interactions o f one boson and two fermions with S O ( 1 0 ) m o m e n t a p~v~), ps~2) and p}3), respec426
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tively. Conditions for more than three point interactions are obtained by means of a picture changing operator [16]. When rn bosons have m o m e n t a pv~), ..., pv~") and 2n fermions have momenta p~m+ ~) , -.., ps(m+2n) , the coupling condition for these states is that m+2n i=1
P ~vO+ E
j=m+l
P~)=
m+n--2 Z (+1,0,0) k=l
.
(4)
From (3) we can conclude that only U~UzU3 is allowed for the three-point couplings of the pure untwisted sectors of all the 7N × ZM orbifolds. The (k, l)-twisted sector requires that the momenta p~ (p + ) of (3) and (4) are replaced by p~ + kv~ + lv~ (p+ + kv~ + lv:). Thus we can get the allowed three-point couplings including the twisted sectors. They are summarized in table 2. Next we shall consider the condition due to the six-dimensional orbifold. Fixed pointsf~ of the (k, /)-twisted sector should satisfy T~=O%~
+A ,
(5)
where A is a lattice vector of the six-dimensional torus and 0~o ~is an automorphism of the lattice. The 7/U× 7/M rotations can be constructed from a product of three Coxeter elements of Lie lattices with rank 2. The Z2 rotation of a two-dimensional space is generated by Coxeter elements of SO (4) and SU (2) × SU (2), a squared Coxeter element of SO ( 5 ) and a cubed one of G2. There exist four fixed points respecting eq. ( 5 ). Their reversed vectors A are written as (0, 0), ( 1, 0), (0, 1 ) and ( 1, 1 ) under bases of simple roots of S O ( 4 ) . The 7/3 rotation is realized by a Coxeter element of SU ( 3 ) and squared ones of SU (3) [ 2 ] (including an outer automorphism of order two) and G2. Their three fixed points have the reversed vectors of (0, 0), (1, 0) and ( 1, 1 ) based on S U ( 3 ) . TheZ4 (?76) rotation is realizedby Coxeter e l e m e n t s o f S O ( 5 ) a n d Sp2 (G2 and S U ( 3 ) [ 2 ] ) . T w o f i x e d Table 2 Three-point couplings allowed by SO ( 10 ) momentum conservation. Orbifold Three-point coupling ~2X~2
UI ~[2U3
UiTolTol U2TjoTlo U3TIITII
ToiTioTjl
'~3X~3 U1~72~73
UITolT02 U2TIoT20 U3TI2T21 TolTIIT21 TolTI2T20 To2TloTzl TozT11T20 TloTliTl2
UIU2~;3
UITolT03 elTo2T02 UzTIoTIo U3TI2T12 TolTIITI2 To2TioTl2 To3TIoTjj TllTliTo2
~2X~4
712XZ~ UI U2~f3
UITI3TI3 U2TloTlo U3To3T03 TolTozT03 TolTiiTi4
~2Xff6
el ~72U3
el ~02To4
To4Tll TI,
TosTjoT,~
~4X~4
TllTllTll
To2TozT02 To2TIoTI4 To2TIITI3 To3TIoTI3
elTo3T03 UITolT05 UzTloTlo U3TI3TI3 Tol TI2TI3 To2TI1TI3 To3TIoTI3 To4TIoTI2
UI [/2 ~73 UITo2T02 UITolT03 ~f2TI oT30 U2T20720 ~2r3TI 3T31 ~[3T22T22 TolTI2T31 TolTI3T30 TolT21T22 To2TIIT31 To2TI2T30 To2T20T22 To2T21T21 To3TIoT31 To3TIIT30 703TzoT2L TIoT]2T22 TjoTI3T21 TIITI2T21 TIITIIT22 T~ITI3T20 T~2T12T20
Z3X~-6 UI U2U3 UITolT05 elTo2T04 UITo3T03 e2TioT20 U3TI4T22 TolTI3T22 TojTI4T21 To2TI2T22 To2TI3T21 TO2TI4T20 To3TI i ]r'22 To3TI2T21 To3TI3T20 To4TIoT22 To4TIIT21 To4T12T20 TosTIoT21 TosTIIT20 TIoTI2T14 TIoTI3TI3 TIITIITI4 TIITI2Tt3 Tj2TI2TI2 Z6×Z6
Uit½U~ TolTI4Ts1 To2T31T33 To4TI2Tso TIoTI4742 TI2TI2T42 TI3T22~131
UtTo,T05 Tol TI 5Tso
To2T32T32 To4T20T42 T, oTIsT41 T,2TI3T41 TI3T23T30
UITo2T04 U1To3T03 U2TIoTso ]r'olT24T41 TolT32T33 To3TI2TsI To3TI3Tso To3T21T42 To4T21T41 To4T22T40 To4T30T32 TIoT23T33 TIoT24T32 TIITI3T41 TIETI4T40 TI2T21T33 TI2T22T32 T14T20T32 TI4T21T31 Tj4~22T30 Tol ~23T42
U2T20T40
/~½T30T30
To2TI3TsI To3T22T41 TosTIoTsI TIITI4T41 TI2T23T31 TIsT20T31
To2TI4Tso To3T23T40 TosTiiTso TjjTIsT40 TI2T24T30 TIt,T21T30
U3TIsTsI To2T22T42 To3T30T33 TosT20T41 TIIT22T33 ToTI3T40
U3T24T42
To2T23T41 To3T31T32 TosT21T40 TuTz3T32 TL3T20T33 720T22T24 T20T23T23
U3T33T33 To2T24T40 To4TI1TsI
TosT30T31 TIIT24T31 TI3T2,T32 T21T21T24
T2~Tz2T2~ T22T22T22 427
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points of a P-twist ofT/4 have the reversed vectors 7o = (0, 0) and 71 = (0, - 1 ) based on SO(5). Those o f a 02twist are the same as those ofT/2. Two of them, c~o= (0, 0) and c~2= ( 1 1 ), are simultaneous fixed points of the P-twist and correspond to eigenstates of the P-twist as well. Since the remaining fixed points (1, 0) and (0, 1 ) of the 02-twist transform into each other under the P-twist, linear combinations of twisted states associated with the fixed points, [o~1 ) = ( 1 / x f 2 ) [ I (1, 0 ) ) + [ (0, 1 ) ) ] and I c ~ - ) = (1/,,/2)[1 (1, 0 ) ) - I (0, 1 ) ) ], a r e t h e e i genstates of the 0-twist with eigenvalues 1 and - 1, respectively. The 7/6 rotation has only one fixed point (0, 0) under the 0-twist. The 02-twist has the same fixed points as those of 7/3. Among the twisted states attached on them I (0, 0) ) and ( 1/xf2) [I (0, - 1 ) ) _+ I (0, - 2 ) ) ] based on G2 are eigenstates of the 0-twist with eigenvalues 1 and _+ 1, respectively, the fixed points of the three-twist are the same as those of 7/2. The eigenstates ] (0, 0) ) and ( 1/x/3) [ I ( 1, 0) ) + e)[ (0, 1 ) ) + ~021 ( 1, 1 ) ) ], are both of the 0-twist with eigenvalues of 1 and o, and of the 02-twist with eigenvalues of 1 and ~o2, respectively, where oJ is a triple root of unity. One can obtain twisted eigenstates of the 7/,4,× 7/M orbifolds by multiplying the above two-dimensional twisted eigenstates. For example, the exponents 8 = ( 1, 0, - 1 ) / 4 and e)= (0, 1, - 1 ) / 4 of the 24×7/4 orbifold are realized by six-dimensional rotations (C, 1, C - l ) and ( 1, C, C 1), respectively, where C is the Coxeter element of SO (5). Then we get twisted eigenstates under 0 as I:'p, 0, 7r), under &o as l Yp, 7q, oek) and l Tp, 7q, oz ), etc., where p, q, r = 0 , 1 and k = 0 , l, 2. Physical states are selected by a generalized GSO projection [ l 1 ] : 1 N- 1 1 '~ 1 G(k, l ) = ~_h__~° Mm=o e~k" m)2(k' l; h, m)A(k, l; h, m) ,
(6)
A(k, l; h, in) =p(k.z) exp{2zti [ - ½(hVj +m V2) (kVl +lV2) + ½(hv, + my2) (kvl +lv2) + (hV, + m V 2 ) ( P + k V L +IV2)- (hVl-]-my2) (p+k/21 +l/22)]},
(7)
where P(k,t~ indicates a contribution of oscillators. The degeneracy factor ,~(k, l; h, m) is the number of (k, l) fixed points which remain fixed by the (h, m )-twist. The extra phase ~ is not determined by a one-loop modular invariance, but by a two-loop modular invariance. It is related to a background antisymmetric tensor and is quantized such that e x = 1. The m o m e n t u m P is a root vector on an E s × E s root lattice and VI and V2 are shift vectors on it. The standard embedding, namely the (2, 2) compactification, is realized by setting VI =/21 and V2= v2 at the first three components and the remaining thirteen components to vanish. By making use of the generalized GSO projection, one can immediately derive the degeneracy of a physical twisted state of a representation. The fixed points of the physical twisted sector can be determined by the values of A and e (see ref. [ 15 ] ). As an example, table 3 lists the physical twisted state of 27 and 27 representations of E6 in the standard embedding with e= 1, - 1 and i ofT/4XT/4orbifolds. Adding three 27 of its untwisted sector, we can confirm that their total numbers in table 3 are coincident with those of ref. [6 ]. It is remarkable that some origins of the twodimensional tori are projected out when e # 1. On the contrary, all the origins ofZ~, orbifolds survive under the generalized GSO projection, even if Wilson lines are introduced. Following similar considerations as in a previous work, three twisted states associated with fixed points (0k'e/'; V, ) ( i = 1, 2 3 ) denoted by a twist ok*~o/'and a reversed vector/2i are permitted to couple, if Okl+k2+k3[t) 11+12+13= 1 ,
/23 -t-0k3(j)13/)2 -]-ok3+k2(j)13+12Vl = ( 1 --Okao)l~)A ,
where a is allowed to be 1, 2 or 3. The former is called the point group selection rule that is satisfied by the threepoint couplings of table 2. The former and the latter are generically called the space group selection rule. It is found from the selection rules that physical states attached at the fixed points on the SO( 5 ) torus, as an example, can interact through triple-couplings such as I1;0)10;?'p)[03;7~,),
11;0)]02;a,)]02;o~i),
[O;Tp)lO;yq)]02;o~,)
for
428
p+q+i=Omod2,
ll;0)[02;c~-)102;c~ -),
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Table 3 Fixed points in the 7~4X7~4 orbifold. Eigenstates of physical twisted sectors are represented by reversed vectors of their fixed points: 10)=1(0, 0)), [y0)=l(0, 0)), l y ~ ) = l ( 0 , - 1 ) ) , l a o ) = [ ( 0 , 0)), I c q ) = ( 1 / x / 2 ) [ l ( 0 , 1 ) ) + l ( 1 , 1 ) ) ] , l a 2 ) = l ( l , 0)) and I o~- ) = ( 1/ x//2) [ I (0, 1 ) ) - I ( 1, 1) ) ], whose bases are SO ( 5 ) simple roots. The subscripts p, q and r run from 0 to 1, and i, j and k run from 0 to 2. Twisted sector
Shift kv~ -F1~,2
Momentum p~ + kv~ + Ivz
e= 1 27
e= - 1 27
~= i E
27 To, To2 To3 T,o T,i T,2 T, 3 T2o T21 722 T30 T3~
(0, 1, - 1 )/4 (0, 2, - 2 ) / 4 (0, 3, - 3 ) / 4 (1,0, - 1 ) / 4 (1, 1, - 2 ) / 4 ( 1, 2, - 3 ) / 4 (1, 3, --4)/4 (2, 0, - 2 ) / 4 (2, 1, --3)/4 (2, 2, --4)/4 (3, 0, --3)/4 (3, 1, --4)/4
(0, 1, 3)/4 (0, 2, 2)/4 (0, 3, 1 )/4 (1,0,3)/4 (1, 1,2)/4 (1, 2, 1 )/4 ( 1, 3, 0)/4 (2,0,2)/4 (2, 1, 1)/4 (2, 2, 0)/4 (3, 0, 1 )/4 (3, 1,0)/4
total
10, ~)q,~r> 10, %, ag> 10, ~/q,7r) lTp, 0, 7r) lyp, yq, a,> lye, aa, yr) 1}%?q, 0) [ai, O, ak) la,, 7q, Yr) [a,, %, 0) lYp,0, 7~> lTp, yq, O) 87
27 I O, yq, y~)
10, as, ~k>
10, a - , a - >
]Tp,yo, a > lye, a--, 7r> l Yp,Yq,0) lai, O, aj,) l a - , yq, Y~> la,, a s, 0)
l a - , 0, oz ) l a - , a--, 0) lYp,0, ~2r>
39
3
12
where ~ is a 774 rotation, p, q = 0 , 1 a n d i = 0 , 1, 2. E x t e n d i n g this e s t i m a t i o n to the (2, 2) 774X774 orbifold with e = 1, we get i n d e p e n d e n t ( 2 7 ) ( 2 7 ) ( 2 7 ) couplings o f 1 for UI U2 U3,
9 for U 1 To2 To2 ,
4 for Ul Toj Tos,
4 for U2 Tlo T3o ,
9forU2T2oT2o,
4 for U3TI3T31 ,
9 f o r U3T22T22 ,
24 for TolT12T31,
8 for To ITI3 T3o,
36 for To~ T2, T22, 3 6 f o r T o 2 T ~ T 3 1 , 27 for To2T2oT22,
36 for To2 TI2 T3o ,
78 for To2T21T21,
8 for To3TloT31 ,
24 for To3 TII T3o , 36 for To3 T20T21 ,
36 for Tlo TI2 7"22 ,
24 for TloT13T21 , 78 for TII Tll T22 ,
216 for Tij Tl2 T21 ,
36 for Tll T13 T2o , 78 for TI2TI2T2o . T h i s Y4X774 orbifold m o d e l has the s a m e n u m b e r o f 27 as the 26 m o d e l of G e p n e r ' s coset c o n s t r u c t i o n [ 11,17,18 ], where the n i n e t y 27 are r e p r e s e n t e d as ( 2 2 0 0 0 0 ) , ( 2 1 1 0 0 0 ) a n d ( 1 1 1 1 0 0 ) by U ( 1 ) charges q~ (i = 1, 2, ..., 6 ) o f the 26 model. A n o n - v a n i s h i n g t h r e e - p o i n t c o u p l i n g m u s t satisfy q}~) + q} 2) --Fq} 3) = 2 for a n y i [ 7,19 ]. W i t h this restriction, we get the i n d e p e n d e n t couplings o f 15for(220000)(O02200)(O00022),
90for(220000)(O02110)(O00112),
15for(220000)(O01111)(O01111),
90 f o r ( 2 1 1 0 0 0 ) ( O 0 0 2 1 1 ) ( O l l O 1 1 ) ,
120for(211000)(O01120)(O10102),
60for(211000)(O01111)(OlOlll),
15for(llllO0)(llOOll)(O01111). T h e total n u m b e r 405 does n o t c o i n c i d e with 821 o f the 774X/7 4 orbifold. 429
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The 7/3 X Z3 orbifold and the 19 model are also equivalent with respect to the number of 27 ~1. Through similar estimations, we can get 280 and 469 allowed three-point couplings in the 19 model and 7/3 X Z3 orbifolds, respectively. The 19 model permits only one type of three-point couplings such as (111000000) (000111000) (000000111 ), because eighty-four 27 of the 19 model are represented as (111000000) by the U ( 1 ) charges qi ( i = 1, 2 .... , 9 ) of the 19 model and a non-vanishing three-point coupling must satisfy q}~) +q}2) _t_q}3) = 1 for any i [7,19]. For the ?73× 7/3 orbifolds, we can get independent (27) (27) (27) couplings of 1 for UI U2U3, 9 for
U3TI2T21
27 f o r T o 2 T I o T 2 1
9forUiTmTo2,
9forU2T, oT2o,
81 for Tol Tll T21 , 27 for Tm Tl2T2o,
, ,
81 for To2Tll T2o , 81 for TloTll TI2 ,
144 for Tll Tll TIj . We shall study the relations of the 7/3 X 7/3 and the 19 to make the discrepancies clear. The 27 representations of both models can correspond to each other as follows [ 20 ]: U 1 for ( 111 000 000) ,
U2 for (000 111 0 0 0 ) ,
Tlo for (100 000 110) ,
T2o for (110 000 1 0 0 ) ,
To~ for (000 100 1 1 0 ) ,
To2 for (000 1 lO 1 0 0 ) ,
T12 for (100 110 000) ,
Tzj for (110 1O0 000) ,
U3 for (000 000 111 ) ,
Ti i for ( 1O0 1O0 100) .
Under this correspondence, the number of four couplings, Toj T~I T2~, To~ Tl2T20, T~oTI~TI2 and TjjT~I T,~ of the 23×7/3 do not agree with those of 19 , which are 54, 54, 54 and 36, respectively. It is most remarkable that the three-point self-couplings of each twenty-seven 27 in the TI ~T~ ~T~ ~ coupling are not prohibited in the Z3 X Z3 orbifold but in 19. If one forbids three-point couplings of three fixed points having at least one identical complex coordinate in the 7/3 X 7/3, their allowed numbers of the four couplings become the same as those of the 19. Therefore the 19 may possess some additional constraints compared with the 7/3 × 773. In a similar manner one can find the correspondence between the physical states of the 7/4 X 7/4 and 2 6 and a forbidden rule to the couplings of the 7/4 X 7/4. The 19 model has larger U ( 1 ) gauge symmetries and more singlets than the 7/3 × 7/3- It has been shown that the 7/3 X 7/3 acquires the same massless spectra as 19 when the 7/3 X 7/3 orbifold model is enhanced so as the gauge group becoming E8 × Es X SU (3)3 at a multicritical point [12,20]. It may be possible to forbid the extra couplings of the 7/3X7/3 for the enhanced U( 1 ) gauge symmetries or the deficit three-point couplings of the 19 may be filled up by non-renormalizable couplings including singlets with non-vanishing vacuum expectation values which break the extra U ( 1 ) symmetries and make the extra singlets heavy. In this letter we have investigated the structures of the 7/N× ZM orbifolds, especially for the SO (10) momentum and the fixed points. We have acquired the method for obtaining the non-vanishing couplings of the 7/NX 7/M orbifold models. The allowed three-point Yukawa couplings have been studied for the 7/4×774 orbifold as a concrete example. They have been compared with those of the 26 model in the SU ( 2 ) / U ( 1 ) coset construction, which has the same numbers of the 27. It has been found that their total numbers of allowed couplings do not coincide with each other. This observation has been also supported by the comparison of the 7/3 X 7/3 orbifold and the 19 model. In order to verify the identifications of the models via distinct constructions, in general, comparisons of discrete symmetries are required other than the massless spectra and the Yukawa couplings, as shown in the identifications of some Gepner's models and some Calabi-Yau models [6-8 ]. Although our examinations have been limited to the three-point couplings of the (2, 2) models, the above ~ The 17.4 and 13.43 models have the same number of 27.
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m e t h o d c a n b e a p p l i e d to n o n - r e n o r m a l i z a b l e c o u p l i n g s a n d t h e (0, 2 ) m o d e l s d e r i v e d f r o m n o n - s t a n d a r d e m b e d d i n g s o f t h e 0kco t r o t a t i o n a n d e x t r a e m b e d d i n g s o f t h e r e v e r s e d l a t t i c e v e c t o r v ( W i l s o n l i n e ) i n t o E8 × Es. T h e g a u g e i n v a r i a n c e , w h i c h h a s n o t b e e n c o n s i d e r e d explicitly, i m p o s e s a d d i t i o n a l c o n s t r a i n t s o n t h e Y u k a w a c o u p l i n g s o f t h e (0, 2 ) m o d e l s . I f o n e t r i e s t o s e a r c h for r e a l i s t i c f o u r - d i m e n s i o n a l s t r i n g t h e o r i e s f r o m t h e (0, 2 ) m o d e l s o f t h e ~NX/ff~/orbifold,t h e i n v e s t i g a t i o n s o f t h e Y u k a w a c o u p l i n g s will b e c o m e a p o w e r f u l c r i t e r i o n .
Acknowledgement T h e a u t h o r s w o u l d like to t h a n k t h e m e m b e r s o f t h e p a r t i c l e p h y s i c s g r o u p o f K a n a z a w a U n i v e r s i t y . T h e y also a c k n o w l e d g e P r o f e s s o r E. Y a m a d a a n d D r . D. S u e m a t s u for r e a d i n g t h e m a n u s c r i p t a n d h e l p f u l c o m m e n t s .
References [ 1 ] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678; B 274 (1986) 285. [2] H. Kawai, D. Lewellyn and A.H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. B 288 (1987) 1; I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87. [3] K.S. Narain, Phys. Lelt. B 169 (1986) 41; W. Lerche, D. Lfist and A.N. Schellekens, Nucl. Phys. B 287 (1987) 477. [4] D. Gepner, Phys. Lett. B 199 (1987) 380; Nucl. Phys. B 296 (1987) 757; Y. Kazama and H. Suzuki, Nucl. Phys. B 321 ( 1989 ) 232. [5] Y. Katsuki, Y. Kawamura, T. Kobayashi and N. Ohlsubo. Phys. Lett. B 212 (1988) 339; Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo, Y. Ono and K. Tanioka, Phys. Lett. B 218 (1989) 169; B 227 (1989) 381 ; preprint DPKU-8904; Nucl. Phys. B 341 (1990) 611; Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo and K. Tanioka, Prog. Theor. Phys. 82 (1989) 171. [6] J. Distler and B. Greene, Nucl. Phys. B 309 (1988) 295. [7] D. Gepner, Nucl. Phys. B 311 (1988/89) 191. [8] A. Kato and Y. Kitazawa, Nucl. Phys. B 319 (1989) 474. [9] D. Suematsu, Phys. Rev. D 38 (1988) 3128. [ 10] P. Zoglin, Phys. Lett. B 228 (1989) 47. [ 11 ] A. Font, L.E. Ibfifiez and F. Quevedo, Phys. Lett. B 217 (1989) 272. [ 12] A, Font, L.E. Ib~ifiez, F. Quevedo and A. Sierra, Nucl. Phys. B 337 (1990) 119. [ 13 ] S. Hamidi and C. Vafa, Nucl. Phys. B 279 ( 1987 ) 465. [ ! 4 ] L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 282 ( 1987 ) 13. [ 15 ] T. Kobayashi and N. Ohtsubo, Phys. Lett. B 245 (1990) 441. [ 16] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 17] C.A. Liitken and G.G. Ross, Phys. Lett. B 213 (1988) 152. [ 18 ] M. Lynker and R. Schimmrigk, Phys. Lett. B 215 ( 1988 ) 681. [19] Z. Qui, Phys. Lett. B 188 (1987) 207. [20] E.J. Chun and J.E. Kim, Phys. Lett. B 238 (1990) 265.
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Allowed Yukawa couplings
of7/NX
77M
27 June 1991
orbifold models
Tatsuo Kobayashi a and Noriyasu Ohtsubo b a Department of Physics, Kanazawa University, Kanazawa 920, Japan b Kanazawa Institute of Technology, lshikawa 921, Japan
Received 3 August 1990; revised manuscript received 8 April 1991
SO(10) momenta and fixed points are studied for obtaining non-vanishing Yukawa couplings of ZN× YMorbifold models. As an example, we investigate explicitly three-point couplings of 714M774 and Z3×Z3 orbifolds with the standard embedding and no discrete torsion. Comparing with the 26 and 19 model by Gepner's coset construction, respectively, we find discrepancies with respect to the numbers of (27) (27) (27) couplings, while the numbers of 27 are identical.
U p to now, m a n y four-dimensional string vacua have been found through orbifold compactification [ 1 ], fermionic string constructions, [ 2 ], self-dual lattice construction [ 3 ], coset construction [ 4 ] and so on. They gave us various kinds o f gauge groups and m a t t e r contents through (0, 2) world-sheet supersymmetric models. F o r example, 4, 12, 58, 61, 39, 246, 248, 3026 and 3013 i n d e p e n d e n t models have been o b t a i n e d for 2r3, 224, 2261, 226-II, 227, 718-I, 7/8-11, 212 -I and 22~2-II orbifolds without Wilson lines, respectively [5]. Even if one limits the consideration to the (2, 2 ) model, there exist m a n y possibilities. Their gauge groups and m a t t e r representations are restricted so as to include E 6 X E8 and 27 and 27, but the numbers of the representations and the allowed interactions d e p e n d on the topologies o f internal spaces. It has been found that some o f G e p n e r ' s models have the same numbers o f m a t t e r representations and the same types o f Yukawa couplings as those o f models on the Calabi-Yau manifold [ 6 - 8 ] . These facts indicate that the C a l a b i - Y a u manifolds have the same topological structures as the c = 9 internal sectors o f G e p n e r ' s models. It is supported by a more detailed analysis o f fourgeneration models o f Y(4; 5)/275×7/5 and 35/22/5×225 [9,10]. A coincidence between some orbifold models and G e p n e r ' s models has been reported with respect to their matter contents as well [11,12]. R e m a r k a b l e examples are given by 7IN×Z× orbifold models with standard embeddings and no discrete torsion [ 1 1 ]. All the models correspond to at least one G e p n e r model with respect to the numbers o f 27 and 27. However, the correspondence among the Yukawa interactions has never been investigated explicitly. Although the general features o f Yukawa interactions on the orbifolds have been discussed [ 13,1 4 ], concrete investigations have been restricted to the case o f the 223 orbifold. In ref. [ 1 5 ], structures o f physical twisted states have been clarified and then calculations o f the Yukawa couplings have been m a d e possible in the case o f the other 22zvorbifold models. In this letter we extend the previous work to the ZN×22M orbifold models, and compare them with the corresponding G e p n e r models. First o f all, we study a constraint for the Yukawa couplings due to SO ( I 0 ) invariance o f the N S R fermion. In table 1 we list all the three-point couplings allowed by the SO ( 10) invariance. Next, we study structures (e.g., fixed p o i n t s ) o f six-dimensional Lie lattices possessing 22N× 22at rotations as their automorphisms. A generalized G S O projection is given, with which one can obtain a degeneracy factor and derive physical twisted states belonging to a representation. Space group conditions are also investigated in the 224× 224 orbifold as an example. Finally we examine a correspondence between the 22~vX 22~1orbifold and a G e p n e r coset construction through concrete examples o f 774×22 4 and 2 6, and 223×223and 1 9. 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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Table 1 Twisted sectors in/Z,,,,X7/,,~r orbifolds. The fourth column lists one example of the six-dimensional Lie lattices possessing the point group ZNXZ~ as the aulomorphisrn. The twisted sector Tkt in the last column denotes that the massless matter superfields of the sector are allowed under the shift kvl + lv2 of SO ( 10 ). Point group v~
vz
6D Lielattice
Twisted sector
272X~2 Z3XYs Z2×Z4 Z2XZ; ~2)~Z6 ~-4X/74 ~TSXZ6 ~6)~6
(0, 1, - 1)/2 (0, 1 , - 1 ) / 3 (0, 1,--1)/4 (1,1,-2)/6 (0, 1,--1)/6 (0, 1,--1)/4 (0, 1,--1)/6 (0, 1 , - 1 ) / 6
SO(4) 3 SU(3) 3 SO(4) X S O ( 5 ) 2
Tob T~o, TH Tol, To2, Tio, Tll, T.2, ]20, 121 Tol, To2, To3, Tlo, Tj b TI2 lo~, Toz, To3, T~o,T~, T~s, T~4 Tot, To2, To3, 704, To5, Tlo, TII, TI2, TI3 Tol, To2, To3, T,o, TII, TI2, TI3, T20, T21, T22, 730, Tst Tol, To2, 703, 704, ]/05, Tlo, TII, TI2, TI3, Ti4, T20, T21, T22 ]rOb 702, To3, To4, To5, Tlo, TII, TI2, Tt3, Ti4, TIs, T20, T2b 7"22, T23, T24, T3o, T31, T32, T3s, T4o, T41, T,2, I20, Ts~
( 1, 0, - 1)/2 (1,0,-I)/3 (1,0,-1)/2 (1,0,-1)/2 (1,0,--1)/2 (1,0,--1)/4 (1,0,--1)/3 (1,0~-1)/6
G~ SO(4)×G 2 SO(5) 3 SU(3)xG~ G3
The NS and R fermions in the right-moving sector o f the heterotic string can be represented by vectors and spinors o f S O ( 1 0 ) , respectively, in the bosonized version. Their m o m e n t a are Pv = ( -+ 1, 0, 0, 0, 0), where the underline means permutations, and ps = ( _+ 1, _+ 1, _+ 1, _+ 1, _+ 1 ) / 2 with an even number o f + signs. The ~NX~M rotation becomes shifts of a S O ( 1 0 ) root lattice under the bosonized expression. They are written as v~ = (a~, b~, q , O, O ) / N and v2= (a2, b2, c2, 0, O ) / M , where a~, b~ and c~ (a2, b2 and c2) must be a c c o m m o d a t e d with exponents o f the ZN (ZM) rotation due to preserving the right-moving N = 2 world-sheet SUSY. We impose on them the conditions a ~+ b~ + c~ = 0 and a2 + b2 + c2 = 0, which guarantee conservation o f at least one spacetime SU SY. (The values o f the first three c o m p o n e n t s of v~ and v2 o f each 7/x × 7/M orbifold, which conserve only one ( N = 1 ) spacetime SUSY, are given in the second and the third column of table 1, respectively.) Since the N = 1 spacetime supercharge Q must satisfy p ~ vt = p ~ / ) 2 = 0 , it is represented by p ~ = ( 1, 1, 1, _+ 1, _+ 1 ) / 2 , where + ( - ) denotes the upper (lower) c o m p o n e n t s o f the four-dimensional spinor. We define four-dimensional chiral m a t t e r of the untwisted sectors by Pv = ( 1, 0, 0, 0, 0) and their superpartners with p + =Pv - P ~ , where the sign conventions of p~ are same as those of Po- The three untwisted sectors are denoted by their vector m o m e n t a as U,:pv=(1,0,0),
U 2 : P v = (0 , 1 , 0 ) ,
U 3 : P v = (0, 0, 1 ) .
Here and hereafter, the fourth and fifth components o f the S O ( 1 0 ) m o m e n t a are ignored, because they are preserved automatically. The m o m e n t a o f a (k, l)-twisted sector are represented by Pv + kvj + l/)2 and Ps + kv~ + l/)2 for a boson and a fermion, respectively, where p+ + k/)~ + Iv2 =Pv + k/)~ + [/)2 - P ~ . The massless conditions o f chiral matter superfields for the (k, /)-twisted sector are given by 3
3
(p~ + k/)~ + l/)~2)2= ~ t=
I
~=
1
(p's + kV~ + l/)tz)2 + ½= l -- 2Ckl ,
(1)
1
3
c k l = ~ ~.= ( I k / ) ] + l / ) ~ l - I n t I k / ) ~ + l v ~ [ ) ( 1 - 1 k / ) ] - l / ) ~ l + I n t
Ikv]+l/)~l).
(2)
The last column o f table 1 denotes the twisted sectors satisfying the massless condition ( 1 ) in each ZNX ZM orbifold. The SO (10) m o m e n t a must be conserved in order to give a non-vanishing scattering amplitude. This means that
p!,) ..Fp+(2)+ps+ (3)=pv~l) +p!2)+pv~3)--Pf2 --PQ = 0
(3)
for three-point interactions o f one boson and two fermions with S O ( 1 0 ) m o m e n t a p~v~), ps~2) and p}3), respec426
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tively. Conditions for more than three point interactions are obtained by means of a picture changing operator [16]. When rn bosons have m o m e n t a pv~), ..., pv~") and 2n fermions have momenta p~m+ ~) , -.., ps(m+2n) , the coupling condition for these states is that m+2n i=1
P ~vO+ E
j=m+l
P~)=
m+n--2 Z (+1,0,0) k=l
.
(4)
From (3) we can conclude that only U~UzU3 is allowed for the three-point couplings of the pure untwisted sectors of all the 7N × ZM orbifolds. The (k, l)-twisted sector requires that the momenta p~ (p + ) of (3) and (4) are replaced by p~ + kv~ + lv~ (p+ + kv~ + lv:). Thus we can get the allowed three-point couplings including the twisted sectors. They are summarized in table 2. Next we shall consider the condition due to the six-dimensional orbifold. Fixed pointsf~ of the (k, /)-twisted sector should satisfy T~=O%~
+A ,
(5)
where A is a lattice vector of the six-dimensional torus and 0~o ~is an automorphism of the lattice. The 7/U× 7/M rotations can be constructed from a product of three Coxeter elements of Lie lattices with rank 2. The Z2 rotation of a two-dimensional space is generated by Coxeter elements of SO (4) and SU (2) × SU (2), a squared Coxeter element of SO ( 5 ) and a cubed one of G2. There exist four fixed points respecting eq. ( 5 ). Their reversed vectors A are written as (0, 0), ( 1, 0), (0, 1 ) and ( 1, 1 ) under bases of simple roots of S O ( 4 ) . The 7/3 rotation is realized by a Coxeter element of SU ( 3 ) and squared ones of SU (3) [ 2 ] (including an outer automorphism of order two) and G2. Their three fixed points have the reversed vectors of (0, 0), (1, 0) and ( 1, 1 ) based on S U ( 3 ) . TheZ4 (?76) rotation is realizedby Coxeter e l e m e n t s o f S O ( 5 ) a n d Sp2 (G2 and S U ( 3 ) [ 2 ] ) . T w o f i x e d Table 2 Three-point couplings allowed by SO ( 10 ) momentum conservation. Orbifold Three-point coupling ~2X~2
UI ~[2U3
UiTolTol U2TjoTlo U3TIITII
ToiTioTjl
'~3X~3 U1~72~73
UITolT02 U2TIoT20 U3TI2T21 TolTIIT21 TolTI2T20 To2TloTzl TozT11T20 TloTliTl2
UIU2~;3
UITolT03 elTo2T02 UzTIoTIo U3TI2T12 TolTIITI2 To2TioTl2 To3TIoTjj TllTliTo2
~2X~4
712XZ~ UI U2~f3
UITI3TI3 U2TloTlo U3To3T03 TolTozT03 TolTiiTi4
~2Xff6
el ~72U3
el ~02To4
To4Tll TI,
TosTjoT,~
~4X~4
TllTllTll
To2TozT02 To2TIoTI4 To2TIITI3 To3TIoTI3
elTo3T03 UITolT05 UzTloTlo U3TI3TI3 Tol TI2TI3 To2TI1TI3 To3TIoTI3 To4TIoTI2
UI [/2 ~73 UITo2T02 UITolT03 ~f2TI oT30 U2T20720 ~2r3TI 3T31 ~[3T22T22 TolTI2T31 TolTI3T30 TolT21T22 To2TIIT31 To2TI2T30 To2T20T22 To2T21T21 To3TIoT31 To3TIIT30 703TzoT2L TIoT]2T22 TjoTI3T21 TIITI2T21 TIITIIT22 T~ITI3T20 T~2T12T20
Z3X~-6 UI U2U3 UITolT05 elTo2T04 UITo3T03 e2TioT20 U3TI4T22 TolTI3T22 TojTI4T21 To2TI2T22 To2TI3T21 TO2TI4T20 To3TI i ]r'22 To3TI2T21 To3TI3T20 To4TIoT22 To4TIIT21 To4T12T20 TosTIoT21 TosTIIT20 TIoTI2T14 TIoTI3TI3 TIITIITI4 TIITI2Tt3 Tj2TI2TI2 Z6×Z6
Uit½U~ TolTI4Ts1 To2T31T33 To4TI2Tso TIoTI4742 TI2TI2T42 TI3T22~131
UtTo,T05 Tol TI 5Tso
To2T32T32 To4T20T42 T, oTIsT41 T,2TI3T41 TI3T23T30
UITo2T04 U1To3T03 U2TIoTso ]r'olT24T41 TolT32T33 To3TI2TsI To3TI3Tso To3T21T42 To4T21T41 To4T22T40 To4T30T32 TIoT23T33 TIoT24T32 TIITI3T41 TIETI4T40 TI2T21T33 TI2T22T32 T14T20T32 TI4T21T31 Tj4~22T30 Tol ~23T42
U2T20T40
/~½T30T30
To2TI3TsI To3T22T41 TosTIoTsI TIITI4T41 TI2T23T31 TIsT20T31
To2TI4Tso To3T23T40 TosTiiTso TjjTIsT40 TI2T24T30 TIt,T21T30
U3TIsTsI To2T22T42 To3T30T33 TosT20T41 TIIT22T33 ToTI3T40
U3T24T42
To2T23T41 To3T31T32 TosT21T40 TuTz3T32 TL3T20T33 720T22T24 T20T23T23
U3T33T33 To2T24T40 To4TI1TsI
TosT30T31 TIIT24T31 TI3T2,T32 T21T21T24
T2~Tz2T2~ T22T22T22 427
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points of a P-twist ofT/4 have the reversed vectors 7o = (0, 0) and 71 = (0, - 1 ) based on SO(5). Those o f a 02twist are the same as those ofT/2. Two of them, c~o= (0, 0) and c~2= ( 1 1 ), are simultaneous fixed points of the P-twist and correspond to eigenstates of the P-twist as well. Since the remaining fixed points (1, 0) and (0, 1 ) of the 02-twist transform into each other under the P-twist, linear combinations of twisted states associated with the fixed points, [o~1 ) = ( 1 / x f 2 ) [ I (1, 0 ) ) + [ (0, 1 ) ) ] and I c ~ - ) = (1/,,/2)[1 (1, 0 ) ) - I (0, 1 ) ) ], a r e t h e e i genstates of the 0-twist with eigenvalues 1 and - 1, respectively. The 7/6 rotation has only one fixed point (0, 0) under the 0-twist. The 02-twist has the same fixed points as those of 7/3. Among the twisted states attached on them I (0, 0) ) and ( 1/xf2) [I (0, - 1 ) ) _+ I (0, - 2 ) ) ] based on G2 are eigenstates of the 0-twist with eigenvalues 1 and _+ 1, respectively, the fixed points of the three-twist are the same as those of 7/2. The eigenstates ] (0, 0) ) and ( 1/x/3) [ I ( 1, 0) ) + e)[ (0, 1 ) ) + ~021 ( 1, 1 ) ) ], are both of the 0-twist with eigenvalues of 1 and o, and of the 02-twist with eigenvalues of 1 and ~o2, respectively, where oJ is a triple root of unity. One can obtain twisted eigenstates of the 7/,4,× 7/M orbifolds by multiplying the above two-dimensional twisted eigenstates. For example, the exponents 8 = ( 1, 0, - 1 ) / 4 and e)= (0, 1, - 1 ) / 4 of the 24×7/4 orbifold are realized by six-dimensional rotations (C, 1, C - l ) and ( 1, C, C 1), respectively, where C is the Coxeter element of SO (5). Then we get twisted eigenstates under 0 as I:'p, 0, 7r), under &o as l Yp, 7q, oek) and l Tp, 7q, oz ), etc., where p, q, r = 0 , 1 and k = 0 , l, 2. Physical states are selected by a generalized GSO projection [ l 1 ] : 1 N- 1 1 '~ 1 G(k, l ) = ~_h__~° Mm=o e~k" m)2(k' l; h, m)A(k, l; h, m) ,
(6)
A(k, l; h, in) =p(k.z) exp{2zti [ - ½(hVj +m V2) (kVl +lV2) + ½(hv, + my2) (kvl +lv2) + (hV, + m V 2 ) ( P + k V L +IV2)- (hVl-]-my2) (p+k/21 +l/22)]},
(7)
where P(k,t~ indicates a contribution of oscillators. The degeneracy factor ,~(k, l; h, m) is the number of (k, l) fixed points which remain fixed by the (h, m )-twist. The extra phase ~ is not determined by a one-loop modular invariance, but by a two-loop modular invariance. It is related to a background antisymmetric tensor and is quantized such that e x = 1. The m o m e n t u m P is a root vector on an E s × E s root lattice and VI and V2 are shift vectors on it. The standard embedding, namely the (2, 2) compactification, is realized by setting VI =/21 and V2= v2 at the first three components and the remaining thirteen components to vanish. By making use of the generalized GSO projection, one can immediately derive the degeneracy of a physical twisted state of a representation. The fixed points of the physical twisted sector can be determined by the values of A and e (see ref. [ 15 ] ). As an example, table 3 lists the physical twisted state of 27 and 27 representations of E6 in the standard embedding with e= 1, - 1 and i ofT/4XT/4orbifolds. Adding three 27 of its untwisted sector, we can confirm that their total numbers in table 3 are coincident with those of ref. [6 ]. It is remarkable that some origins of the twodimensional tori are projected out when e # 1. On the contrary, all the origins ofZ~, orbifolds survive under the generalized GSO projection, even if Wilson lines are introduced. Following similar considerations as in a previous work, three twisted states associated with fixed points (0k'e/'; V, ) ( i = 1, 2 3 ) denoted by a twist ok*~o/'and a reversed vector/2i are permitted to couple, if Okl+k2+k3[t) 11+12+13= 1 ,
/23 -t-0k3(j)13/)2 -]-ok3+k2(j)13+12Vl = ( 1 --Okao)l~)A ,
where a is allowed to be 1, 2 or 3. The former is called the point group selection rule that is satisfied by the threepoint couplings of table 2. The former and the latter are generically called the space group selection rule. It is found from the selection rules that physical states attached at the fixed points on the SO( 5 ) torus, as an example, can interact through triple-couplings such as I1;0)10;?'p)[03;7~,),
11;0)]02;a,)]02;o~i),
[O;Tp)lO;yq)]02;o~,)
for
428
p+q+i=Omod2,
ll;0)[02;c~-)102;c~ -),
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Table 3 Fixed points in the 7~4X7~4 orbifold. Eigenstates of physical twisted sectors are represented by reversed vectors of their fixed points: 10)=1(0, 0)), [y0)=l(0, 0)), l y ~ ) = l ( 0 , - 1 ) ) , l a o ) = [ ( 0 , 0)), I c q ) = ( 1 / x / 2 ) [ l ( 0 , 1 ) ) + l ( 1 , 1 ) ) ] , l a 2 ) = l ( l , 0)) and I o~- ) = ( 1/ x//2) [ I (0, 1 ) ) - I ( 1, 1) ) ], whose bases are SO ( 5 ) simple roots. The subscripts p, q and r run from 0 to 1, and i, j and k run from 0 to 2. Twisted sector
Shift kv~ -F1~,2
Momentum p~ + kv~ + Ivz
e= 1 27
e= - 1 27
~= i E
27 To, To2 To3 T,o T,i T,2 T, 3 T2o T21 722 T30 T3~
(0, 1, - 1 )/4 (0, 2, - 2 ) / 4 (0, 3, - 3 ) / 4 (1,0, - 1 ) / 4 (1, 1, - 2 ) / 4 ( 1, 2, - 3 ) / 4 (1, 3, --4)/4 (2, 0, - 2 ) / 4 (2, 1, --3)/4 (2, 2, --4)/4 (3, 0, --3)/4 (3, 1, --4)/4
(0, 1, 3)/4 (0, 2, 2)/4 (0, 3, 1 )/4 (1,0,3)/4 (1, 1,2)/4 (1, 2, 1 )/4 ( 1, 3, 0)/4 (2,0,2)/4 (2, 1, 1)/4 (2, 2, 0)/4 (3, 0, 1 )/4 (3, 1,0)/4
total
10, ~)q,~r> 10, %, ag> 10, ~/q,7r) lTp, 0, 7r) lyp, yq, a,> lye, aa, yr) 1}%?q, 0) [ai, O, ak) la,, 7q, Yr) [a,, %, 0) lYp,0, 7~> lTp, yq, O) 87
27 I O, yq, y~)
10, as, ~k>
10, a - , a - >
]Tp,yo, a > lye, a--, 7r> l Yp,Yq,0) lai, O, aj,) l a - , yq, Y~> la,, a s, 0)
l a - , 0, oz ) l a - , a--, 0) lYp,0, ~2r>
39
3
12
where ~ is a 774 rotation, p, q = 0 , 1 a n d i = 0 , 1, 2. E x t e n d i n g this e s t i m a t i o n to the (2, 2) 774X774 orbifold with e = 1, we get i n d e p e n d e n t ( 2 7 ) ( 2 7 ) ( 2 7 ) couplings o f 1 for UI U2 U3,
9 for U 1 To2 To2 ,
4 for Ul Toj Tos,
4 for U2 Tlo T3o ,
9forU2T2oT2o,
4 for U3TI3T31 ,
9 f o r U3T22T22 ,
24 for TolT12T31,
8 for To ITI3 T3o,
36 for To~ T2, T22, 3 6 f o r T o 2 T ~ T 3 1 , 27 for To2T2oT22,
36 for To2 TI2 T3o ,
78 for To2T21T21,
8 for To3TloT31 ,
24 for To3 TII T3o , 36 for To3 T20T21 ,
36 for Tlo TI2 7"22 ,
24 for TloT13T21 , 78 for TII Tll T22 ,
216 for Tij Tl2 T21 ,
36 for Tll T13 T2o , 78 for TI2TI2T2o . T h i s Y4X774 orbifold m o d e l has the s a m e n u m b e r o f 27 as the 26 m o d e l of G e p n e r ' s coset c o n s t r u c t i o n [ 11,17,18 ], where the n i n e t y 27 are r e p r e s e n t e d as ( 2 2 0 0 0 0 ) , ( 2 1 1 0 0 0 ) a n d ( 1 1 1 1 0 0 ) by U ( 1 ) charges q~ (i = 1, 2, ..., 6 ) o f the 26 model. A n o n - v a n i s h i n g t h r e e - p o i n t c o u p l i n g m u s t satisfy q}~) + q} 2) --Fq} 3) = 2 for a n y i [ 7,19 ]. W i t h this restriction, we get the i n d e p e n d e n t couplings o f 15for(220000)(O02200)(O00022),
90for(220000)(O02110)(O00112),
15for(220000)(O01111)(O01111),
90 f o r ( 2 1 1 0 0 0 ) ( O 0 0 2 1 1 ) ( O l l O 1 1 ) ,
120for(211000)(O01120)(O10102),
60for(211000)(O01111)(OlOlll),
15for(llllO0)(llOOll)(O01111). T h e total n u m b e r 405 does n o t c o i n c i d e with 821 o f the 774X/7 4 orbifold. 429
Volume 262, number 4
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27 June 1991
The 7/3 X Z3 orbifold and the 19 model are also equivalent with respect to the number of 27 ~1. Through similar estimations, we can get 280 and 469 allowed three-point couplings in the 19 model and 7/3 X Z3 orbifolds, respectively. The 19 model permits only one type of three-point couplings such as (111000000) (000111000) (000000111 ), because eighty-four 27 of the 19 model are represented as (111000000) by the U ( 1 ) charges qi ( i = 1, 2 .... , 9 ) of the 19 model and a non-vanishing three-point coupling must satisfy q}~) +q}2) _t_q}3) = 1 for any i [7,19]. For the ?73× 7/3 orbifolds, we can get independent (27) (27) (27) couplings of 1 for UI U2U3, 9 for
U3TI2T21
27 f o r T o 2 T I o T 2 1
9forUiTmTo2,
9forU2T, oT2o,
81 for Tol Tll T21 , 27 for Tm Tl2T2o,
, ,
81 for To2Tll T2o , 81 for TloTll TI2 ,
144 for Tll Tll TIj . We shall study the relations of the 7/3 X 7/3 and the 19 to make the discrepancies clear. The 27 representations of both models can correspond to each other as follows [ 20 ]: U 1 for ( 111 000 000) ,
U2 for (000 111 0 0 0 ) ,
Tlo for (100 000 110) ,
T2o for (110 000 1 0 0 ) ,
To~ for (000 100 1 1 0 ) ,
To2 for (000 1 lO 1 0 0 ) ,
T12 for (100 110 000) ,
Tzj for (110 1O0 000) ,
U3 for (000 000 111 ) ,
Ti i for ( 1O0 1O0 100) .
Under this correspondence, the number of four couplings, Toj T~I T2~, To~ Tl2T20, T~oTI~TI2 and TjjT~I T,~ of the 23×7/3 do not agree with those of 19 , which are 54, 54, 54 and 36, respectively. It is most remarkable that the three-point self-couplings of each twenty-seven 27 in the TI ~T~ ~T~ ~ coupling are not prohibited in the Z3 X Z3 orbifold but in 19. If one forbids three-point couplings of three fixed points having at least one identical complex coordinate in the 7/3 X 7/3, their allowed numbers of the four couplings become the same as those of the 19. Therefore the 19 may possess some additional constraints compared with the 7/3 × 773. In a similar manner one can find the correspondence between the physical states of the 7/4 X 7/4 and 2 6 and a forbidden rule to the couplings of the 7/4 X 7/4. The 19 model has larger U ( 1 ) gauge symmetries and more singlets than the 7/3 × 7/3- It has been shown that the 7/3 X 7/3 acquires the same massless spectra as 19 when the 7/3 X 7/3 orbifold model is enhanced so as the gauge group becoming E8 × Es X SU (3)3 at a multicritical point [12,20]. It may be possible to forbid the extra couplings of the 7/3X7/3 for the enhanced U( 1 ) gauge symmetries or the deficit three-point couplings of the 19 may be filled up by non-renormalizable couplings including singlets with non-vanishing vacuum expectation values which break the extra U ( 1 ) symmetries and make the extra singlets heavy. In this letter we have investigated the structures of the 7/N× ZM orbifolds, especially for the SO (10) momentum and the fixed points. We have acquired the method for obtaining the non-vanishing couplings of the 7/NX 7/M orbifold models. The allowed three-point Yukawa couplings have been studied for the 7/4×774 orbifold as a concrete example. They have been compared with those of the 26 model in the SU ( 2 ) / U ( 1 ) coset construction, which has the same numbers of the 27. It has been found that their total numbers of allowed couplings do not coincide with each other. This observation has been also supported by the comparison of the 7/3 X 7/3 orbifold and the 19 model. In order to verify the identifications of the models via distinct constructions, in general, comparisons of discrete symmetries are required other than the massless spectra and the Yukawa couplings, as shown in the identifications of some Gepner's models and some Calabi-Yau models [6-8 ]. Although our examinations have been limited to the three-point couplings of the (2, 2) models, the above ~ The 17.4 and 13.43 models have the same number of 27.
430
Volume 262, number 4
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27 June 1991
m e t h o d c a n b e a p p l i e d to n o n - r e n o r m a l i z a b l e c o u p l i n g s a n d t h e (0, 2 ) m o d e l s d e r i v e d f r o m n o n - s t a n d a r d e m b e d d i n g s o f t h e 0kco t r o t a t i o n a n d e x t r a e m b e d d i n g s o f t h e r e v e r s e d l a t t i c e v e c t o r v ( W i l s o n l i n e ) i n t o E8 × Es. T h e g a u g e i n v a r i a n c e , w h i c h h a s n o t b e e n c o n s i d e r e d explicitly, i m p o s e s a d d i t i o n a l c o n s t r a i n t s o n t h e Y u k a w a c o u p l i n g s o f t h e (0, 2 ) m o d e l s . I f o n e t r i e s t o s e a r c h for r e a l i s t i c f o u r - d i m e n s i o n a l s t r i n g t h e o r i e s f r o m t h e (0, 2 ) m o d e l s o f t h e ~NX/ff~/orbifold,t h e i n v e s t i g a t i o n s o f t h e Y u k a w a c o u p l i n g s will b e c o m e a p o w e r f u l c r i t e r i o n .
Acknowledgement T h e a u t h o r s w o u l d like to t h a n k t h e m e m b e r s o f t h e p a r t i c l e p h y s i c s g r o u p o f K a n a z a w a U n i v e r s i t y . T h e y also a c k n o w l e d g e P r o f e s s o r E. Y a m a d a a n d D r . D. S u e m a t s u for r e a d i n g t h e m a n u s c r i p t a n d h e l p f u l c o m m e n t s .
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