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Symmetry breaking in orbifold models
Volume 220, number 4
PHYSICS LETTERS B
13 April 1989
SYMMETRY BREAKING IN ORBIFOLD MODELS P. BANTAY Institute for Theoretical Physics, EStvOs University, H-1088 Budapest, Hungary
Received 13 January 1989
Gauge symmetry breaking to some physically interesting gauge groups is studied in the context of symmetric orbifolds.
Orbifold models [ 1,2 ] may provide a link between strings and particle physics. To do so, they must break down the spacetime (super)symmetries and the gauge symmetries of the string. Gauge symmetry breaking is achieved by embedding the space group of the orbifold into the gauge group of the string, which we will take to be E8 X E8. Then the unbroken gauge group is the centralizer of the space group in E8 X Es. In order to lower the rank in this process, the embedding has to be non-abelian [ 3 ]. The dimension of the centralizer is easily computed from the character of the embedding (as the gauge group is represented on its Lie algebra, so are its subgroups). But in general this is all the information we can get without a more detailed and cumbersome analysis. But of course we do want to generate all the possible orbifold models, but only those which can accomodate low-energy phenomenology. Thus we want the gauge group to be that of either the standard model or some G U T group. This gives severe constraints on the space group of the orbifold, as we will see in the following. The crucial remark is that the G U T groups-SU (5), SO(10) and E6 - are characterized uniquely (up to conjugacy) as the centralizers in Es of some SU (n) subgroup. Indeed, consider the subgroup chain Es ~ S U ( 9 ) = . . . ~ S U ( 2 ) , where SU (9) is a maximal subgroup of Es and SU (n) is embedded into SU (n + 1 ) in the standard way. Let us denote these subgroups of E8 by SU* (n) to distinguish them from other SU ( n ) s u b g r o u p s . The centralizer o f SU* (n) is E9 .... where for n < 6 the group 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
E,, belongs to the generalized E series, for example E 5 = S O ( 1 0 ) , E 4 = S U ( 5 ) and E 3 = S U ( 3 ) × S U ( 2 ) . Note that the subgroups isomorphic to E,, for n > 3 are all conjugate in E8. This is no longer true for E3, but there is a single conjugacy class that embeds into the G U T groups in the desired way. The outcome of these considerations is that, in order to break down E8 to some G U T group E,,, the space group of the orbifold has to lie in S U * ( 9 - n ) . So we have to embed the space group into E8 by firstly embedding it into some SU (n), and then by identifying this SU (n) with SU* (n). But embedding some group into S U ( n ) is just to find an n-dimensional unitary representation whose determinant representation - that is, the one-dimensional representation obtained by taking the determinants of the representation matrices - is trivial. Of course this n-dimensional representation - we will call it R, - has to satisfy additional constraints in order to break down the gauge group to E 9 _ n exactly, and not to some bigger group. An analysis of the branching rules for the SU* ( n )'s shows that these constraints require that no exterior power of R,, besides A "R,,=det R,, contain the trivial representation and that R,, be irreducible. As a first result, we note that it is not possible in this framework to get an S U ( 5 ) supersymmetric model, because the existence of a five-dimensional irreducible representation would imply the existence of a point group element o f order five which would break SUSY [ 1 ]. The most interesting possibility is that of breaking down Es to the standard model gauge group, which is E3 X U ( 1 ). In this case one has to embed the space 531
Volume 220, number 4
PHYSICS LETTERS B
group into S U ( 6 ) in such a way as to get an extra U ( 1 ) in the centralizer. It turns out that in this case the constraint on the representation R 6 is the same as before with the exception that it does not have to be irreducible but rather to be the direct sum o f two nonequivalent irreducible representations. So R 6 = A ~ B , where A and B are irreducible representations with non-trivial determinants. There are three possibilities: ( 1 ) dim A = 5 and B = (det A ) ~. These models are not supersymmetric, so they have tachyons [ 1 ]. ( 2 ) d i m A = 4 , d i m B = 2 , d e t B = ( d e t A ) - I , neither A nor B are quaternionic (pseudo-real) and ^ 2A does not contain B, the conjugate o f B. ( 3 ) d i m A = d i m B = 3, A and B are neither equivalent nor conjugates. The space group must have irreducible unitary representations fulfilling one o f these conditions to be able to generate a model with the s t a n d a r d m o d e l gauge group. Such groups will be exhibited in another paper [4]. It can also be shown that the models so o b t a i n e d do not have exotic representations in the massless spectrum, only the s t a n d a r d ones. One word about m o d u l a r invariance [1,5]. One m a j o r advantage to work with the E8 × E8 heterotic string is that one has to e m b e d the space group into
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13 April 1989
each factor separately. M o d u l a r invariance (at one l o o p ) tells us that once we have chosen a specific e m b e d d i n g into one o f the factors, the other one is (almost) uniquely fixed. But we do not have to worry about the first embedding, because one can always find a pair to it which makes the theory m o d u l a r invariant. Moreover, the gauge structure of the real world fixes almost uniquely the gauge structure o f the shadow world. In s u m m a r y , we have reduced the p r o b l e m o f characterizing orbifolds with some G U T or standard model gauge group to rather easy representation theoretic conditions on the space group. It must be stressed that we break E8 directly at the string level, without i n t e r m e d i a t e steps. I would like to thank Professor L. Palla and Professor Z. H o r v a t h for their help.
References [ 1] L. Dixon, J.A. Harvey, C. Vafa and F. Witten, Nucl. Phys. B 261 (1986) 678; B274 (1986) 285. [ 2 ] L.E. Ibfifiez, preprint CERN-TH 4769/87. [3] P. Bantay, Phys. Lett. B 203 (1988) 367. [4] P. Bantay, in preparation. [5] C. Vafa, Nucl. Phys. B 273 (1986) 592.
PHYSICS LETTERS B
13 April 1989
SYMMETRY BREAKING IN ORBIFOLD MODELS P. BANTAY Institute for Theoretical Physics, EStvOs University, H-1088 Budapest, Hungary
Received 13 January 1989
Gauge symmetry breaking to some physically interesting gauge groups is studied in the context of symmetric orbifolds.
Orbifold models [ 1,2 ] may provide a link between strings and particle physics. To do so, they must break down the spacetime (super)symmetries and the gauge symmetries of the string. Gauge symmetry breaking is achieved by embedding the space group of the orbifold into the gauge group of the string, which we will take to be E8 X E8. Then the unbroken gauge group is the centralizer of the space group in E8 X Es. In order to lower the rank in this process, the embedding has to be non-abelian [ 3 ]. The dimension of the centralizer is easily computed from the character of the embedding (as the gauge group is represented on its Lie algebra, so are its subgroups). But in general this is all the information we can get without a more detailed and cumbersome analysis. But of course we do want to generate all the possible orbifold models, but only those which can accomodate low-energy phenomenology. Thus we want the gauge group to be that of either the standard model or some G U T group. This gives severe constraints on the space group of the orbifold, as we will see in the following. The crucial remark is that the G U T groups-SU (5), SO(10) and E6 - are characterized uniquely (up to conjugacy) as the centralizers in Es of some SU (n) subgroup. Indeed, consider the subgroup chain Es ~ S U ( 9 ) = . . . ~ S U ( 2 ) , where SU (9) is a maximal subgroup of Es and SU (n) is embedded into SU (n + 1 ) in the standard way. Let us denote these subgroups of E8 by SU* (n) to distinguish them from other SU ( n ) s u b g r o u p s . The centralizer o f SU* (n) is E9 .... where for n < 6 the group 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
E,, belongs to the generalized E series, for example E 5 = S O ( 1 0 ) , E 4 = S U ( 5 ) and E 3 = S U ( 3 ) × S U ( 2 ) . Note that the subgroups isomorphic to E,, for n > 3 are all conjugate in E8. This is no longer true for E3, but there is a single conjugacy class that embeds into the G U T groups in the desired way. The outcome of these considerations is that, in order to break down E8 to some G U T group E,,, the space group of the orbifold has to lie in S U * ( 9 - n ) . So we have to embed the space group into E8 by firstly embedding it into some SU (n), and then by identifying this SU (n) with SU* (n). But embedding some group into S U ( n ) is just to find an n-dimensional unitary representation whose determinant representation - that is, the one-dimensional representation obtained by taking the determinants of the representation matrices - is trivial. Of course this n-dimensional representation - we will call it R, - has to satisfy additional constraints in order to break down the gauge group to E 9 _ n exactly, and not to some bigger group. An analysis of the branching rules for the SU* ( n )'s shows that these constraints require that no exterior power of R,, besides A "R,,=det R,, contain the trivial representation and that R,, be irreducible. As a first result, we note that it is not possible in this framework to get an S U ( 5 ) supersymmetric model, because the existence of a five-dimensional irreducible representation would imply the existence of a point group element o f order five which would break SUSY [ 1 ]. The most interesting possibility is that of breaking down Es to the standard model gauge group, which is E3 X U ( 1 ). In this case one has to embed the space 531
Volume 220, number 4
PHYSICS LETTERS B
group into S U ( 6 ) in such a way as to get an extra U ( 1 ) in the centralizer. It turns out that in this case the constraint on the representation R 6 is the same as before with the exception that it does not have to be irreducible but rather to be the direct sum o f two nonequivalent irreducible representations. So R 6 = A ~ B , where A and B are irreducible representations with non-trivial determinants. There are three possibilities: ( 1 ) dim A = 5 and B = (det A ) ~. These models are not supersymmetric, so they have tachyons [ 1 ]. ( 2 ) d i m A = 4 , d i m B = 2 , d e t B = ( d e t A ) - I , neither A nor B are quaternionic (pseudo-real) and ^ 2A does not contain B, the conjugate o f B. ( 3 ) d i m A = d i m B = 3, A and B are neither equivalent nor conjugates. The space group must have irreducible unitary representations fulfilling one o f these conditions to be able to generate a model with the s t a n d a r d m o d e l gauge group. Such groups will be exhibited in another paper [4]. It can also be shown that the models so o b t a i n e d do not have exotic representations in the massless spectrum, only the s t a n d a r d ones. One word about m o d u l a r invariance [1,5]. One m a j o r advantage to work with the E8 × E8 heterotic string is that one has to e m b e d the space group into
532
13 April 1989
each factor separately. M o d u l a r invariance (at one l o o p ) tells us that once we have chosen a specific e m b e d d i n g into one o f the factors, the other one is (almost) uniquely fixed. But we do not have to worry about the first embedding, because one can always find a pair to it which makes the theory m o d u l a r invariant. Moreover, the gauge structure of the real world fixes almost uniquely the gauge structure o f the shadow world. In s u m m a r y , we have reduced the p r o b l e m o f characterizing orbifolds with some G U T or standard model gauge group to rather easy representation theoretic conditions on the space group. It must be stressed that we break E8 directly at the string level, without i n t e r m e d i a t e steps. I would like to thank Professor L. Palla and Professor Z. H o r v a t h for their help.
References [ 1] L. Dixon, J.A. Harvey, C. Vafa and F. Witten, Nucl. Phys. B 261 (1986) 678; B274 (1986) 285. [ 2 ] L.E. Ibfifiez, preprint CERN-TH 4769/87. [3] P. Bantay, Phys. Lett. B 203 (1988) 367. [4] P. Bantay, in preparation. [5] C. Vafa, Nucl. Phys. B 273 (1986) 592.