CONSISTENCY CONDITIONS FOR AdS/CFT EMBEDDINGS

June 2003 Consistency Conditions for AdS/CFT Embeddings Paul H. Frampton(a) and Thomas W. Kephart(b) arXiv:hep-th/0306207v1 20 Jun 2003 (a)Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599. (b)Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37325. Abstract The group embeddings used in orbifolding the AdS/CFT correspondence to arrive at quiver gauge field theories are studied for both supersymmetric and non-supersymmetric cases. For an orbifold AdS5 × S 5 /Γ the conditions for embeddings of the finite group Γ in the SU (4) ∼ O(6) isotropy of S 5 are stated in the form of consistency rules, both for Abelian and Non-Abelian Γ. Typeset using REVTEX 1 Introduction Independently of the important question of whether or not superstring theory, or M theory, is the correct theory of quantum gravity, superstring theory can be used through the AdS/CFT correspondence [1] to arrive at interesting gauge field theories in four spacetime dimensions. According to the AdS/CFT correspondence, a Type IIB superstring compactified on an AdS5 × S 5 background is dual to an N = 4 supersymmetric gauge field theory in flat 4dimensional spacetime. The N = 4 theory is much too symmetric to be of phenomenological interest. What makes the AdS/CFT correspondence far more interesting is the possibility of orbifolding S 5 → S 5 /Γ where Γ is a finite group embedded in the isometry SU(4) ∼ O(6) of S 5 in the 3-dimensional complex space C3 ∼ R6 . Although O(6) and SU(4) have the same local structure, i.e. Lie algebra, globally O(6) is doubly-covered by SU(4) and this distinction will be important to us here. Another consideration, already discussed fully in [2], is the survival of chiral fermions. The correct statement of the necessary and sufficient condition is that the 4 of SU(4) must be neither real nor pseudoreal for chiral fermions to be present in the quiver gauge field theory. It is often stated that the number of surviving supersymmetries is N = 2, 1, 0 according to whether Γ ⊂ SU(2), SU(3), SU(4). While this statement essentially survives our analysis, the statement itself needs sharpening, in particular by a more careful definition of the embedding. The present article is devoted to addressing that issue. N = 1 Embeddings of Abelian Γ in SU(4) and O(6). To preserve N = 1 supersymmetry one must keep exactly one invariant spinor under the joint action of the finite symmetry Γ and the quiver gauge group, which for Abelian Γ is SU(N)p where p is the order of Γ. This implies that one component of the 4 of SU(4) is the trivial singlet representation of Γ. 2 The most general Abelian Γ of order g = o(Γ) is [3,4] made up of the basic units Zp , the order p cyclic groups formed from the pth roots of unity. It is important to note that the the product Zp Zq is identical to Zpq if and only if p and q have no common prime factor. If we write the prime factorization of g as: g= Y pki i (1) i where the product is over primes, it follows that the number Na (g) of inequivalent abelian groups of order g is given by: Na (g) = Y P (ki ) (2) ki where P (x) is the number of unordered partitions of x. For example, for order g = 144 = 24 32 the value would be Na (144) = P (4)P (2) = 5 × 2 = 10. For g ≤ 31 it is simple to evaluate Na (g) by inspection. Na (g) = 1 unless g contains a nontrivial power (ki ≥ 2) of a prime. These exceptions are: Na (g = 4, 9, 12, 18, 20, 25, 28) = 2; Na (8, 24, 27) = 3; and Na (16) = 5. This confirms that: 31 X Na (g) = 48 (3) g=1 Let us define α = exp(2πi/p) and write the embedding as 4 = (αA1 , αA2 , αA3 , αA4 ). For N = 1 we have agreed that one Aµ must vanish so let us put A4 = 0 and denote the embedding by the shorthand A ≡ (A1 , A2 , A3 ). Consider now the 6 of SU(4) which is the antisymmetric part in (4 × 4). The 6 is necessarily real, and in the double covering of O(6) is identified with the defining ”vector” representation of O(6). In the shorthand notation already adopted for the 4, the 6 is 6 = (A1 , A2 , A3 , A1 + A2 , A2 + A3 , A3 + A1 ). For general A, the 6 is not real but we must now impose the condition that the diagonal 3 × 3 matrix with diagonal elements αA be a transformation belonging to SU(3), i.e. a matrix which is unitary and of determinant one. This requires that P i Ai = 0 (mod p). But with that condition we can rewrite (up to mod p) 6 ≡ (A1 , A2 , A3 , −A3 , −A2 , −A3 ) which is manifestly real and hence we have a consistent embedding. 3 Thus, in this the simplest case, the embedding is uniquely defined for the 4 of SU(4) by the three components (recall A4 ≡ 0) of A ≡ (A1 , A2 , A3 ) which for consistency must satisfy A1 + A2 + A3 = 0 (mod p). This will always lead to a consistent N = 1 chiral quiver gauge theory. N = 0 Embeddings of Abelian Γ in SU(4) and O(6). For N = 0 the discussion is similar to the above except that A4 does not vanish. Therefore the considerations involve not A with three components but Aµ = (A1 , A2 , A3 , A4 ) with four components. Consider now the 6 of SU(4) which is the antisymmetric part in (4 × 4). The 6 can be written 6 = (A1 + A4 , A2 + A4 , A3 + A4 , A1 + A2 , A2 + A3 , A3 + A1 ). But here for consistency the 4 × 4 diagonal matrix with diagonal elements αAµ must be an SU(4) transformation which requires P µ Aµ = 0 (mod p) in which case the 6 is again manifestly real. For a consistent N = 0 quiver gauge field theory, therefore, the 4 of SU(4) is defined by four non-vanishing integers Aµ satisfying A1 + A2 + A3 + A4 = 0 (mod p). This is the necessary and sufficient condition for consistent Abelian orbifolding to N = 0. Similar results hold for products of abelian groups, e.g., for Γ = Zp ZQ we can write 4 = (eA1 eB1 , eA2 eB2 , eA3 eB3 , eA4 eB4 ) where we require P µ Aµ = 0 (mod p) and P µ Bµ = 0 (mod q). N = 1 Embeddings of Non-Abelian Γ in SU(4) and O(6). As usual, going from Abelian Γ to Non-Abelian Γ makes everything more complicated. This is why paper [2] is necessarily much longer than e.g. paper [5]! The complication arises because for Non-Abelian Γ, the irreducible representations ρi can have dimensions di greater than one. If the order of Γ is g the dimensions must satisfy Σi d2i = g. The resultant gauge group for the quiver theory is 4 N i SU(ndi ). Because of this more complex quiver theories emerge from Non-Abelian Γ than are possible for Abelian Γ. On the other hand, as shown in [2], the constraints imposed phenomenologically are also more complex to such an extent that for all g ≤ 31 only one acceptable theory occurs amongst many hundreds of candidates. For the Non-Abelian embeddings we must again study whether the 6 ≡ (4 × 4)a is consistently real. Let us, for the N = 1 case write 4 = ( P i′ ρi′ , 1) where P di′ = 3 and ′ where none of the ρi′ is the identity representation. (Here the set {i } is a subset of the {i}). This will generically lead to an N = 1 supersymmetric quiver gauge theory. The consistency requirements for this Non-Abelian embedding are very demanding and may be stated as the three rules for N = 1: 1. Exactly one component of the 4 of SU(4) must be the identity of Γ. That is, 4=( P P i′ ρi′ , 1) where the ρi′ are irreducible representations (not the identity) of Γ satisfying di′ = 3. 2. Written as a block-diagonal matrix the determinant of the 3 × 3 matrix ⊗i′ ρi′ must be real and equal to one. Note that this reduces to the vanishing trace condition if we were to break Γ to an abelian subgroup. To see just how demanding these three rules are consider even the simplest Non-Abelian finite group Γ = S3 ≡ D3 . The order is g = 6 and the available irreducible representations ′ are ρ = 1, 1′, 2. The only choice satisfying Rules 1 and 2 is 4 = (1 , 2, 1) and this satisfies Rule 2 because 2 is an SU(2) matrix which necessarily has determinant equals one. 5 N = 0 Embeddings of Non-Abelian Γ in SU(4) and O(6). The consistency requirements for Non-Abelian embedding to give N = 0 are similar to but different from those given for N = 0 and may be stated as the two new rules: 1. No component of the 4 of SU(4) may be the identity of Γ. That is, 4=( P P i′ ρi′ ) where the ρi′ are irreducible representations (not the identity) of Γ satisfying di′ = 4. 2. Written as a block-diagonal matrix the determinant of the 4 × 4 matrix ⊗i′ ρi′ must be real and equal to one. To check how these rules work, consider again the simplest Non-Abelian finite group Γ = S3 ≡ D3 . The only choice satisfying Rules 1 and 2 is 4 = (2, 2). Therefore Γ = S3 can be used to obtain either an N = 1 or an N = 0 quiver gauge field theory. [The only other ′ ′ consistent embeddings of S3 in SU(4) are 4 = (1, 1, 1 , 1 ) which leads to N = 2 and the trivial embedding which leaves N = 4.] 6 Summary and Discussion. Let us summarize the consistency conditions for the four cases considered: Abelian Γ; N = 1. A4 = 0 (mod p) and A1 + A2 + A3 = 0 (mod p). Abelian Γ; N = 0 A1 + A2 + A3 + A4 = 0 (mod p). Non-Abelian Γ; N = 1 1. Exactly one component of the 4 of SU(4) must be the identity of Γ. 2. The determinant of the 3 × 3 matrix ⊗i′ ρi′ must be real and equal to one. Non-Abelian Γ; N = 0 1. No component of the 4 of SU(4) may be the identity of Γ. 2. The determinant of the 4 × 4 matrix ⊗i′ ρi′ must be real and equal to one. Fortunately, these consistency conditions are respected by most of the earlier work on Abelian Γ. For example, the TeV unificiation model in [6] which provides an alternative to SUSYGUTs [7] is consistent. Most of the Abelian models in [8] are consistent but 7 of the 60 which are called non-partition are not. Even though the embeddings of these seven models are inconsistent, the models are themselves consistent gauge theories and of interest in their own right. Despite the fact that they are not derivable from orbifolding AdS × S 5 , they have many of the features of orbifolded AdS × S 5 models, e.g., vanishing first-order (and perhaps higher order) β functions, similar boson-fermion counting providing N = 1 SUSY, etc. It can be shown that these are the only models besides the partition models with a real 6, and that they fall into three classes. Recall first that the N = 1 partition models for Γ = Zp can be written as Mpp1 ,p2,p3 where p1 +p2 +p3 = 0(modp). In this notation the nonpartition models are 3p 6p 9p Mp,p,2p , Mp,2p,4p , and Mp,4p,7p . Along with their own phenomenological interest, these models can probably be used in comparison with properly embedded partition orbifolded models to investigate violation of conformal invariance, etc. In the N = 0 case, again one finds partition models Mpp1 ,p2,p3 ,p4 with p1 + p2 + p3 + p4 = 0(modp) are properly embedded. But 7 again there exist improperly-embedded, but phenomenologically-interesting, non-partition models with complex 4 and real 6 [9]. For Non-Abelian Γ the most extensive work is in our earlier paper [2]. The only phenomenologically-acceptable model out of the many considered in [4] was for the choice Γ ≡ 24/7, in the notation of Thomas and Wood [3] for D4 × Z3 . The embedding used is ′ 4 = (1α, 1 , 2α) and this satisfies Rules 1 and 2 for Non-Abelian Γ and N = 0. It is therefore a quiver gauge theory which arises from a consistent embedding of Γ in AdS5 × S 5 /Γ. 8 Acknowledgements TWK thanks the Department of Physics and Astronomy at UNC Chapel Hill and PHF thanks the Department of Physics and Astronomy at Vanderbilt University for hospitality while this work was in progress. This work was supported in part by the US Department of Energy under Grants No. DE-FG02-97ER-41036 and No. DE-FG05-85ER40226. 9 REFERENCES [1] J. Maldacena, Adv. Theor. Math. 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