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Constraints on asymmetric orbifolds
Volume 197, number 3
PHYSICS LETTERS B
29 October 1987
CONSTRAINTS ON ASYMMETRIC ORBIFOLDS A. G I V E O N Racah Institute of Physics, Hebrew University, 91904 Jerusalem, Israel
Received 24 June 1987
It is claimed that besides the left-right matching condition, one has to check the relative phases dictated by modular invariance for an asymmetric orbifold to give rise to an acceptable model.
Orbifold compactification o f the heterotic string [ 1] can lead to phenomenologically interesting models. To be physically interesting the twisted theory should be compatible with unitarity, and contain supergravity (gravity) and gauge bosons. In general the orbifold does not have to be standard (symmetric), as the left- and right-movers o f the heterotic string are separated. The left-moving degrees of freedom o f a string theory can live on one orbifold, while the right-movers live on another orbifold. The idea o f such asymmetric orbifolds has been described recently [ 2 ]. The spectrum of the theory is given by the partition function. In the path-integral formulation, we have to sum over untwisted and twisted sectors. Alternatively, in the hamiltonian approach, one projects the untwisted and twisted Hilbert spaces on states with definite eigenvalues o f the twist operator 0. The eigenvalues are fixed by the relative phases given to different sectors in the path-integral formulation. For the heterotic orbifold, modular invariance is not automatic. One has to check that the orbifold satisfies the left-right matching condition in order to be able to give rise to a modular invariant partition function. The condition of modular invariance also fixes some o f the phases o f the various sectors in the path integral. The purpose of this note is to stress that the left-right matching condition is not enough for an acceptable model. It is true that once the left-right matching condition is satisfied, one can fix the relative phases o f the sectors to have a modular invar-
iant partition function. However, these fixed phases may correspond to a model where the states which are invariant under the symmetry operator are projected out and only the non-symmetric states remain in the spectrum. The model is still consistent from the two-dimensional point of view but its spacetime interpretation is sick. Tree-level unitarity is lost ~1, and in particular there is no four-dimensional graviton in it. To study the question whether a theory contains gravity and gauge bosons or not, we must find the projection in the untwisted Hilbert space forced by modular invariance. Whenever we are forced to project on states with 0 ~ 1, the theory does not contain gravity (since the four-dimensional graviton does not excite the internal degrees o f freedom and has therefore 0 = 1 ). Also in such a case if after the orbifold identification some gauge group is left unbroken, the corresponding four-dimensional gauge bosons will be projected out. To clarify our point we shall discuss models in which we twist only left-moving degrees of freedom by a second-order twist. The twist o f the 16 left-movers xl, I = 9, ..., 24, which are compactified on E8 × E8 [or spin(32)/Z2] is equivalent to a second-order shift of the left-handed " m o m e n t u m " L ~2. The shift is accompanied by a Z2 modification of some other 2
~l The fact that tree-level unitarity is lost was pointed out by N. Seiberg. ~z An exceptional case is the outer automophism of E8× Es, which exchanges the two E8'sto give the model E8 [3,4].
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
347
Volume 197, number 3
PHYSICS LETTERS B
left-moving coordinates x a, a = 8 - 2 + 1.... ,8, by taking each coordinate to its minus. The left-right level matching condition, ~o =Lo, is a necessary and sufficient condition to guarantee modular invariance to one-loop order in orbifold compactification of the heterotic string [5]. The left-right matching condition in our case is 52=2 1
1 s ~-z ( 1 - ~ - z ) 1 mod 2/n, 1 - ~,,~,
(1)
where n = 2 is the twist's order, Z~=0 for a = 1, ..., 8 - 2 , z h = l for b = 8 - 2 + 1 ..... 8, (the factor
29 October 1987
tion leads to the non-supersymmetric heterotic string models [3]. The models with 2=4, 5 : e Z + 1/2 are new ten-dimensional SUSY models. Accompanying the left-handed twist by a 2rt spatial rotation gives rise to new non-supersymmetric models without tachyons. However, the projection of the untwisted Hilbert space of these models is such that we lose sixdimensional spacetime supergravity (gravity) and gauge symmetry. As an example we shall give the arguments that lead to the modular-invariant partition function of the model with 2=4, 82=3/2 for the spin(32)/Z2 heterotic string. In the twisted sector Lo = N + ½( L + 5 ) 2 - ] = N + ½L2 + L S ,
(4)
is the left-handed normal-ordering constant change in the twisted sector), thus
where Nis the left-moving sector occupation number,
2 = - 8 ~ 2 mod 8,
Lespin(32)/Z2 , 5= ( ( 1 / 2 ) 6 0 1 ° ) ,
(2)
Lo =Lo, therefore LoeZ. The modes in the twisted directions a = 5 , ..., 8 are half-integral, b_ra, re2~+ 1/2, therefore left-right matching allows only states for which
thus
62e 7/ i f f 2 = 0 o r 8, 6 2 e Z + 1/2
iff 2 = 4 .
(3) LdeZ
To classify all such models ~3 we can limit ourselves to 52~<2. In table 1 we describe all possible models. The models with 2=0, 52eZ give nothing new [twist Es×E8 and spin(32)/Z2 to themselves]. But accompanying the left shift by a 2~z spatial rota- It ~s also easy to classify n-order pure left-handed asymmetric orbifolds, n> 2.
iff
L~eT_+l/2
NeZ ,
iff N e Z + I / 2 .
(5)
Therefore, the phases in the twisted sector are fixed by the left-right matching condition. The left-moving partition function of the twisted sector is ZLW=l~'['2--1 (27r/01) ~ 8 ( 0 280 312 "l-t,' ~QI2,o8_I..JQ8jQI2 2 t.' 3 . t . ' 2 t J 4 - - v~12,O8~ 2 V41 , (6)
Table 1
348
52
5
2
Unbroken group
Untwisted Hilbert-space projection
0
0
0,8
Es×E8 spin(32)/Z2
+
1/2
((1/2)206; 08 ) ((1/2)20 ]4)
4
E8 X Es--}E7XSU2 X E 8 spin(32)/Z2-oO(4) × 0 ( 2 8 )
1
ref. [3]
0,8
ref. [31
3/2
((1/2)206; 107) ((1/2)601° )
4
EsXEs-*ETXSU2 ×O(16) spin(32)/Z,-~O(12) × 0 ( 2 0 )
2
(107; 107) ((1/2)808 )
0,8
Es×E8~O(16)×0(16) spin(32)/Z2~O(16) )
+
+
Volume 197, number 3
PHYSICS LETTERS B
where 0 i = 0 i ( r 10) is the Jacobi theta function in the notations o f ref. [ 6 ]. z = z ~+ z2 is the one-loop m o d ular parameter. The twist operator is 0 = yfl, where y -- ( - 1 )2L~ acts on the 16 left-moving s p i n ( 3 2 ) / Z 2 lattice, fl acts on the left-moving degrees o f freedom x a, a = 5 ..... 8, via f l ( x a ) - - - - x ", therefore f l ( a _ , ~ ) = - a _ , ", where c~_,, ~ are the untwisted m o d e s in the directions a = 5, ..., 8. The left-moving p a r t i t i o n function o f the untwisted sector is
Z~_t = ~ (2n/O',)8[T2-2( ~16v2_1_ " ~ "-" 316 -t_ r 2- I 0 380 48( 0 3 -43 1 " - 044 ) 1 ,
.~_ 016)
29 October 1987
ZRight = ZRight ut tw + ZRight , ZUt Right
~
(9)
7R/"\ _t_ _~ ) _~_z R (
t-,
--
_ ) ...~z N S (
--
_)
-Jr z N S ( -~- ~- ) -}- z R ( -Jr -- ) .Jr- Z R ( _ -~- )
+ZNS(- +) +zNS(+ --),
(10)
where R are R a m o n d and NS are N e v e u - S c h w a r z fermions. The first ( s e c o n d ) + / - s i g n is the periodicity in the o"2 diretion o f the four u n r o t a t e d ( r o t a t e d ) fermions. The world-sheet path integrals are
(7)
ZR(++)=q(R;++)Yrexp(irHR)
(__)F,
(11)
with the + sign if 0 = 1 and the - s i g n if 0 = - 1,
z R ( -- -- ) = t/(R; -- - ) Tr e x p ( i z H R ) ,
(12)
ZL = Z ~ ~+ Z L w .
zNS( - - - ) = q ( N S ; - - - )
(13)
(8)
M o d u l a r transformations o f (6) force a minus sign in (7). Therefore, the untwisted states which survive the projection have an o d d n u m b e r o f a n a, a = 5, ..., 8, together with L3sTZ, or an even n u m b e r o f a J , a = 5 , . . . , 8 , together with L 3 ~ + 1 / 2 . The graviton a n d the gauge bosons are thrown out o f the spectrum. The massless states in the untwisted sector are in (12, 20) o f O ( 1 2 ) x O ( 2 0 ) . The other massless states are O (12) × O( 20)'s singlets. In the twisted sector we get four copies (32, 1 ). The reason we get four copies is that the degeneracy in the twisted sector is 4 (the square root of 24), due to fixed points. In contrast to the second-order pure left-handed orbifold, in the case o f a second-order pure righth a n d e d orbifold not all phases are fixed by m o d u l a r invariance. An a s y m m e t r i c pure right-handed orbifold is such that only right-moving degrees o f freed o m are twisted. A second-order twist 0 is a Z2 m o d i f i c a t i o n o f 2 right-moving coordinates. The left-right matching c o n d i t i o n forces 2 = 0, 4 or 8. The action o f 0 on the N S R fermions is given by 0 rotation (this is the r e q u i r e m e n t o f preserving world-sheet s u p e r s y m m e t r y [7,2]). Including the G S O projection [8] the N S R fermions are twisted by a Z~ × Z2 " r o t a t i o n " . W h e n 2 = 4 there are 16 sectors. 8 sectors are not twisted in the ~ direction a n d are referred to the untwisted Hilbert space. The other 8 sectors are twisted in the a l diretion a n d are referred to the twisted Hilbert space, therefore
T r exp(iZHNs),
zNS( + + ) = ~/(NS; + + ) T r exp(iZHNs) ( -- )F, (14) z R ( + -- ) = ~/(R; + - ) Tr e x p ( i r H R ) ( - )Fo,
(15) ZR( -- + ) = ~/(R; -- + ) Tr exp(iZHR) 0 ,
(16)
zNS( -- + ) = q(NS; - + ) Tr exp(iZHNs) 0 ,
(17)
ZNs( + -- ) = q(NS; + - ) Tr exp(iZHR) ( - ) F 0 ,
(18) ( _ ) F is the o p e r a t o r that counts world-sheet ferm i o n s m o d u l o two. H y s / H R is the right-handed N S / R a m o n d hamiltonian. The q's are the phases. Following ref. [9], we arbitrarily define •(NS; - - ) = 1. Then by m o d u l a r transformations, a n d in o r d e r to describe a projection onto certain states in Hilbert space, the phases have to satisfy q(R; + - ) = - q ( R ; -
+)q(R; ++),
(19)
r/(NS; + - ) = q ( N S ; -
+)q(NS; ++),
(20)
r/(NS;++)=r/(R;--)=-I.
(21)
r/(R; + + ) cannot be d e t e r m i n e d by m o d u l a r invariance. The partition function o f the untwisted sector is therefore
349
Volume 197, number 3
PHYSICS LETTERS B
Z ut = T r exp(iHNs) X½[1 -- ( -
)F]
~ [ l + q(NS; -- + )01
- Y r e x p ( i r H R ) ½[ 1 - q ( R ; + + )( - )V] × ½[ l - ~ / ( R ; - + ) 0 ] .
(22)
It can be easily shown that T r exp(izHNs) ½[ I -- ( -- )F] 0 = T r exp(iTHR) ½[ 1 - * / ( R ; + + )( -- ) r ] 0 = 0 . (23) Therefore, q(NS; - + ) cannot be d e t e r m i n e d by m o d u l a r invariance. I f q(NS; - + ) = 1 the theory contains supergravity. However, if q(NS; - + ) = - 1 the theory does not contain supergravity. Both models have the same p a r t i t i o n function, therefore they are both consistent with m o d u l a r invariance. The relation (23) is a result o f a special s y m m e t r y o f the second-order pure right-handed a s y m m e t r i c orbifold we have described: The world-sheet ferm i o n s are organized in two groups o f four fermions, each group is either periodic or antiperiodic. This is the reason m o d u l a r invariance permits two different projections o f the Hilbert space o f states. However, i f the twist o f the right-moving sector is m o r e complicated, or if the twist acts on the left-moving degrees o f freedom, then the phases in the path integral are fixed by m o d u l a r invariance.
350
29 October 1987
In conclusion, we have found that besides the left-right matching condition, which is a necessary a n d sufficient condition to guarantee m o d u l a r invarlance, one has to check the relative phases. It is possible that the phases dictated by m o d u l a r invariance are such that the model is not acceptable since it does not contain gravity a n d gauge symmetry, a n d treelevel unitarity is lost. I would like to thank Shmuel Elitzur for useful discussions.
References [ 1] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678; B 274 (1986) 285. [2] K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric orbifolds, preprint HUTP-86/A089. [3] L.J. Dixon and J.A. Harvey, Nucl. Phys. B 274 (1986) 93. [4 ] S. Elitzur and A. Giveon, Hebrew University preprint (1986). [5] C. Vafa, Nucl. Phys. B 273 (1986) 592. [ 6 ] A. Erdelyi et al., Higher transcendental functions (McGrawHill, New York, 1953). [ 7 ] M. Meuller and E. Winen, Princeton preprint (1986). [8] F. Gliozzi, J. Scherck and D.I. Olive, Phys. Lett. B 65 (1976) 282; Nucl. Phys. B 122 (1977) 253. [9] N. Seiberg and E. Witten, Nucl. Phys. B 276 (1986) 272.
PHYSICS LETTERS B
29 October 1987
CONSTRAINTS ON ASYMMETRIC ORBIFOLDS A. G I V E O N Racah Institute of Physics, Hebrew University, 91904 Jerusalem, Israel
Received 24 June 1987
It is claimed that besides the left-right matching condition, one has to check the relative phases dictated by modular invariance for an asymmetric orbifold to give rise to an acceptable model.
Orbifold compactification o f the heterotic string [ 1] can lead to phenomenologically interesting models. To be physically interesting the twisted theory should be compatible with unitarity, and contain supergravity (gravity) and gauge bosons. In general the orbifold does not have to be standard (symmetric), as the left- and right-movers o f the heterotic string are separated. The left-moving degrees of freedom o f a string theory can live on one orbifold, while the right-movers live on another orbifold. The idea o f such asymmetric orbifolds has been described recently [ 2 ]. The spectrum of the theory is given by the partition function. In the path-integral formulation, we have to sum over untwisted and twisted sectors. Alternatively, in the hamiltonian approach, one projects the untwisted and twisted Hilbert spaces on states with definite eigenvalues o f the twist operator 0. The eigenvalues are fixed by the relative phases given to different sectors in the path-integral formulation. For the heterotic orbifold, modular invariance is not automatic. One has to check that the orbifold satisfies the left-right matching condition in order to be able to give rise to a modular invariant partition function. The condition of modular invariance also fixes some o f the phases o f the various sectors in the path integral. The purpose of this note is to stress that the left-right matching condition is not enough for an acceptable model. It is true that once the left-right matching condition is satisfied, one can fix the relative phases o f the sectors to have a modular invar-
iant partition function. However, these fixed phases may correspond to a model where the states which are invariant under the symmetry operator are projected out and only the non-symmetric states remain in the spectrum. The model is still consistent from the two-dimensional point of view but its spacetime interpretation is sick. Tree-level unitarity is lost ~1, and in particular there is no four-dimensional graviton in it. To study the question whether a theory contains gravity and gauge bosons or not, we must find the projection in the untwisted Hilbert space forced by modular invariance. Whenever we are forced to project on states with 0 ~ 1, the theory does not contain gravity (since the four-dimensional graviton does not excite the internal degrees o f freedom and has therefore 0 = 1 ). Also in such a case if after the orbifold identification some gauge group is left unbroken, the corresponding four-dimensional gauge bosons will be projected out. To clarify our point we shall discuss models in which we twist only left-moving degrees of freedom by a second-order twist. The twist o f the 16 left-movers xl, I = 9, ..., 24, which are compactified on E8 × E8 [or spin(32)/Z2] is equivalent to a second-order shift of the left-handed " m o m e n t u m " L ~2. The shift is accompanied by a Z2 modification of some other 2
~l The fact that tree-level unitarity is lost was pointed out by N. Seiberg. ~z An exceptional case is the outer automophism of E8× Es, which exchanges the two E8'sto give the model E8 [3,4].
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
347
Volume 197, number 3
PHYSICS LETTERS B
left-moving coordinates x a, a = 8 - 2 + 1.... ,8, by taking each coordinate to its minus. The left-right level matching condition, ~o =Lo, is a necessary and sufficient condition to guarantee modular invariance to one-loop order in orbifold compactification of the heterotic string [5]. The left-right matching condition in our case is 52=2 1
1 s ~-z ( 1 - ~ - z ) 1 mod 2/n, 1 - ~,,~,
(1)
where n = 2 is the twist's order, Z~=0 for a = 1, ..., 8 - 2 , z h = l for b = 8 - 2 + 1 ..... 8, (the factor
29 October 1987
tion leads to the non-supersymmetric heterotic string models [3]. The models with 2=4, 5 : e Z + 1/2 are new ten-dimensional SUSY models. Accompanying the left-handed twist by a 2rt spatial rotation gives rise to new non-supersymmetric models without tachyons. However, the projection of the untwisted Hilbert space of these models is such that we lose sixdimensional spacetime supergravity (gravity) and gauge symmetry. As an example we shall give the arguments that lead to the modular-invariant partition function of the model with 2=4, 82=3/2 for the spin(32)/Z2 heterotic string. In the twisted sector Lo = N + ½( L + 5 ) 2 - ] = N + ½L2 + L S ,
(4)
is the left-handed normal-ordering constant change in the twisted sector), thus
where Nis the left-moving sector occupation number,
2 = - 8 ~ 2 mod 8,
Lespin(32)/Z2 , 5= ( ( 1 / 2 ) 6 0 1 ° ) ,
(2)
Lo =Lo, therefore LoeZ. The modes in the twisted directions a = 5 , ..., 8 are half-integral, b_ra, re2~+ 1/2, therefore left-right matching allows only states for which
thus
62e 7/ i f f 2 = 0 o r 8, 6 2 e Z + 1/2
iff 2 = 4 .
(3) LdeZ
To classify all such models ~3 we can limit ourselves to 52~<2. In table 1 we describe all possible models. The models with 2=0, 52eZ give nothing new [twist Es×E8 and spin(32)/Z2 to themselves]. But accompanying the left shift by a 2~z spatial rota- It ~s also easy to classify n-order pure left-handed asymmetric orbifolds, n> 2.
iff
L~eT_+l/2
NeZ ,
iff N e Z + I / 2 .
(5)
Therefore, the phases in the twisted sector are fixed by the left-right matching condition. The left-moving partition function of the twisted sector is ZLW=l~'['2--1 (27r/01) ~ 8 ( 0 280 312 "l-t,' ~QI2,o8_I..JQ8jQI2 2 t.' 3 . t . ' 2 t J 4 - - v~12,O8~ 2 V41 , (6)
Table 1
348
52
5
2
Unbroken group
Untwisted Hilbert-space projection
0
0
0,8
Es×E8 spin(32)/Z2
+
1/2
((1/2)206; 08 ) ((1/2)20 ]4)
4
E8 X Es--}E7XSU2 X E 8 spin(32)/Z2-oO(4) × 0 ( 2 8 )
1
ref. [3]
0,8
ref. [31
3/2
((1/2)206; 107) ((1/2)601° )
4
EsXEs-*ETXSU2 ×O(16) spin(32)/Z,-~O(12) × 0 ( 2 0 )
2
(107; 107) ((1/2)808 )
0,8
Es×E8~O(16)×0(16) spin(32)/Z2~O(16) )
+
+
Volume 197, number 3
PHYSICS LETTERS B
where 0 i = 0 i ( r 10) is the Jacobi theta function in the notations o f ref. [ 6 ]. z = z ~+ z2 is the one-loop m o d ular parameter. The twist operator is 0 = yfl, where y -- ( - 1 )2L~ acts on the 16 left-moving s p i n ( 3 2 ) / Z 2 lattice, fl acts on the left-moving degrees o f freedom x a, a = 5 ..... 8, via f l ( x a ) - - - - x ", therefore f l ( a _ , ~ ) = - a _ , ", where c~_,, ~ are the untwisted m o d e s in the directions a = 5, ..., 8. The left-moving p a r t i t i o n function o f the untwisted sector is
Z~_t = ~ (2n/O',)8[T2-2( ~16v2_1_ " ~ "-" 316 -t_ r 2- I 0 380 48( 0 3 -43 1 " - 044 ) 1 ,
.~_ 016)
29 October 1987
ZRight = ZRight ut tw + ZRight , ZUt Right
~
(9)
7R/"\ _t_ _~ ) _~_z R (
t-,
--
_ ) ...~z N S (
--
_)
-Jr z N S ( -~- ~- ) -}- z R ( -Jr -- ) .Jr- Z R ( _ -~- )
+ZNS(- +) +zNS(+ --),
(10)
where R are R a m o n d and NS are N e v e u - S c h w a r z fermions. The first ( s e c o n d ) + / - s i g n is the periodicity in the o"2 diretion o f the four u n r o t a t e d ( r o t a t e d ) fermions. The world-sheet path integrals are
(7)
ZR(++)=q(R;++)Yrexp(irHR)
(__)F,
(11)
with the + sign if 0 = 1 and the - s i g n if 0 = - 1,
z R ( -- -- ) = t/(R; -- - ) Tr e x p ( i z H R ) ,
(12)
ZL = Z ~ ~+ Z L w .
zNS( - - - ) = q ( N S ; - - - )
(13)
(8)
M o d u l a r transformations o f (6) force a minus sign in (7). Therefore, the untwisted states which survive the projection have an o d d n u m b e r o f a n a, a = 5, ..., 8, together with L3sTZ, or an even n u m b e r o f a J , a = 5 , . . . , 8 , together with L 3 ~ + 1 / 2 . The graviton a n d the gauge bosons are thrown out o f the spectrum. The massless states in the untwisted sector are in (12, 20) o f O ( 1 2 ) x O ( 2 0 ) . The other massless states are O (12) × O( 20)'s singlets. In the twisted sector we get four copies (32, 1 ). The reason we get four copies is that the degeneracy in the twisted sector is 4 (the square root of 24), due to fixed points. In contrast to the second-order pure left-handed orbifold, in the case o f a second-order pure righth a n d e d orbifold not all phases are fixed by m o d u l a r invariance. An a s y m m e t r i c pure right-handed orbifold is such that only right-moving degrees o f freed o m are twisted. A second-order twist 0 is a Z2 m o d i f i c a t i o n o f 2 right-moving coordinates. The left-right matching c o n d i t i o n forces 2 = 0, 4 or 8. The action o f 0 on the N S R fermions is given by 0 rotation (this is the r e q u i r e m e n t o f preserving world-sheet s u p e r s y m m e t r y [7,2]). Including the G S O projection [8] the N S R fermions are twisted by a Z~ × Z2 " r o t a t i o n " . W h e n 2 = 4 there are 16 sectors. 8 sectors are not twisted in the ~ direction a n d are referred to the untwisted Hilbert space. The other 8 sectors are twisted in the a l diretion a n d are referred to the twisted Hilbert space, therefore
T r exp(iZHNs),
zNS( + + ) = ~/(NS; + + ) T r exp(iZHNs) ( -- )F, (14) z R ( + -- ) = ~/(R; + - ) Tr e x p ( i r H R ) ( - )Fo,
(15) ZR( -- + ) = ~/(R; -- + ) Tr exp(iZHR) 0 ,
(16)
zNS( -- + ) = q(NS; - + ) Tr exp(iZHNs) 0 ,
(17)
ZNs( + -- ) = q(NS; + - ) Tr exp(iZHR) ( - ) F 0 ,
(18) ( _ ) F is the o p e r a t o r that counts world-sheet ferm i o n s m o d u l o two. H y s / H R is the right-handed N S / R a m o n d hamiltonian. The q's are the phases. Following ref. [9], we arbitrarily define •(NS; - - ) = 1. Then by m o d u l a r transformations, a n d in o r d e r to describe a projection onto certain states in Hilbert space, the phases have to satisfy q(R; + - ) = - q ( R ; -
+)q(R; ++),
(19)
r/(NS; + - ) = q ( N S ; -
+)q(NS; ++),
(20)
r/(NS;++)=r/(R;--)=-I.
(21)
r/(R; + + ) cannot be d e t e r m i n e d by m o d u l a r invariance. The partition function o f the untwisted sector is therefore
349
Volume 197, number 3
PHYSICS LETTERS B
Z ut = T r exp(iHNs) X½[1 -- ( -
)F]
~ [ l + q(NS; -- + )01
- Y r e x p ( i r H R ) ½[ 1 - q ( R ; + + )( - )V] × ½[ l - ~ / ( R ; - + ) 0 ] .
(22)
It can be easily shown that T r exp(izHNs) ½[ I -- ( -- )F] 0 = T r exp(iTHR) ½[ 1 - * / ( R ; + + )( -- ) r ] 0 = 0 . (23) Therefore, q(NS; - + ) cannot be d e t e r m i n e d by m o d u l a r invariance. I f q(NS; - + ) = 1 the theory contains supergravity. However, if q(NS; - + ) = - 1 the theory does not contain supergravity. Both models have the same p a r t i t i o n function, therefore they are both consistent with m o d u l a r invariance. The relation (23) is a result o f a special s y m m e t r y o f the second-order pure right-handed a s y m m e t r i c orbifold we have described: The world-sheet ferm i o n s are organized in two groups o f four fermions, each group is either periodic or antiperiodic. This is the reason m o d u l a r invariance permits two different projections o f the Hilbert space o f states. However, i f the twist o f the right-moving sector is m o r e complicated, or if the twist acts on the left-moving degrees o f freedom, then the phases in the path integral are fixed by m o d u l a r invariance.
350
29 October 1987
In conclusion, we have found that besides the left-right matching condition, which is a necessary a n d sufficient condition to guarantee m o d u l a r invarlance, one has to check the relative phases. It is possible that the phases dictated by m o d u l a r invariance are such that the model is not acceptable since it does not contain gravity a n d gauge symmetry, a n d treelevel unitarity is lost. I would like to thank Shmuel Elitzur for useful discussions.
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