Gravity/Non-Commutative Yang-Mills Correspondence and Doubletons

We discuss the gravity dual description for a non-commutative Yang-Mills theory, which reduces to that on AdS_{5} x S_{5} in the commutative limit. It is found that doubletons do not decouple in this dual gravity description unless one takes the commutative limit. The decoupling of the doubletons in AdS_{5} space implies that the dual gauge theory has SU(N) gauge symmetry. Our result implies that this gravity description is dual to non-commutative U(N) gauge theory. It is compatible with the claim that U(1) and SU(N) gauge symmetries can not separate in non-commutative U(N) gauge theory.

The mass spectrum of type IIB supergravity compactified on AdS 5 × S 5 was studied by Kim, Romans and van Nieuwenhuizen [10]. This theory gives (at low energy) the gravity dual description of the commutative 4d N = 4 Yang-Mills. It was argued that doubletons can be gauged away and decouple from the gravity and other fields, at least in the linearized level. In this paper, we compactify type IIB supergravity on M 5 × S 5 following [10], where M 5 is continuous deformation from AdS 5 . This theory is (at low energy) the gravity dual description of non-commutative Yang-Mills. We obtain 5dimensional linearized equations from the compactification.
To find the modes of non-decoupling doubletons, we compare our equations to those in [10]. Then we find two independent equations of scalar fields. These two equations have two independent solutions unless the parameter of non-commutativity a vanishes. But when this parameter vanishes, these two equations degenerate, and we obtain only one equation. Then this equation has infinite number of solutions. It is related to the appearance of local symmetry in a = 0, and modes in one of the two field equations can be gauged away as gauge modes of the local symmetry. These modes are exactly the scalars in doubleton multiplet. This is just what we expect. This paper is organized as follows : In section 2, we review the gravity dual description of non-commutative Yang-Mills theory, which is our starting point. In section 3, we consider linearized field equations in the gravity dual background, and expand these equations by 5-dimensional spherical harmonics. In section 4, which is the main part of this paper, we discuss the field equations associated with doubleton scalars. A short comment for doubleton 2-forms is also given. Finally, in section 5, implication to the dual non-commutative Yang-Mills is discussed.
Throughout the paper, we discuss the bosonic part only. It is sufficient to investigate this part for our purpose.
2 Gravity dual description of non-commutative Yang-

Mills theory
Our starting point is a gravity dual description of 4d non-commutative Yang-Mills theory [2,3]. 5 This is the string theory on the spacetime M 5 × S 5 with flux of external fields : ds 2 str = α ′ R 2 u 2 (−dx 2 0 + dx 2 1 ) + u 2 1 + a 4 u 4 (dx 2 2 + dx 2 3 ) + du 2 u 2 + dΩ 2 S 5 , B 23 = B ∞ a 4 u 4 1 + a 4 u 4 , B ∞ = α ′ R 2 a 2 , e 2Φ =ĝ 2 1 + a 4 u 4 , This solution is written in string metric. Φ is a dilaton, B 23 and A 01 are NS-NS and R-R 2-form fields, andF 0123u is a 5-form field strength which satisfies the self-duality condition. These backgrounds are obtained by taking decoupling and non-commutative scaling limit in the supergravity solution of D3-brane with the B 23 field. R is written by the scaled string coupling constantĝ and the number of D3-branes N ; R 4 = 4πĝN.
The parameter a represents non-commutativity. Non-commutativity appears in B∞ ∼ a 2 in the dual theory, because B 23 has a non-zero value in solution (1).
For au → 0, the gravity solution (1) becomes AdS 5 × S 5 solution. This means that the non-commutative theory reduces to commutative one in the IR scale.
3 Spherical harmonics expansion of bosonic field equations Now we expand bosonic field equations of type IIB supergravity in small fluctuations around the solution (1) in the previous section. Making this expansion to first order in the fluctuations, one finds linearized field equations. From now on, we use the notation and convention which are given in Appendix A.
Bosonic field equations in the string metric are where, C is R-R scalar. H (3) = dB (2) and F (3) = dA (2) are NS-NS and R-R 3-form field strength, respectively.F (5) is defined byF (5) We consider small fluctuations around the gravity solution (1): Mixing with metric fluctuation and dilaton comes from g (s) µν . This is the relation between the string metric g µν . 6 Substituting (8) into field equations (2)-(7), one obtains linearized field equations. All results are given in Appendix B.
Next, following the procedure in [10], let us expand linearized field equations by 5dimensional spherical harmonics. Then we obtain 5-dimensional equations which depend on coordinates x 0 , x 1 , x 2 , x 3 , u only. When a → 0, these results reduce to those of AdS 5 ×S 5 compactification of type IIB supergravity discussed by Kim, Romans and van Nieuwenhuizen.
Spherical harmonics expansions are as follows : where Y 's are 5-dimensional spherical harmonics (see Appendix A). These expansions are chosen to satisfy gauge conditions These conditions respect the invariance under (i) 5-dimensional diffeomorphism; . Those Y I (y) satisfy ∇ (m ∇ n) Y I = 0, which are called conformal scalars in [18].
Substituting expansions (9) into linearized equations in Appendix B, we obtain 5dimensional field equations on M 5 . All results are given in Appendix C.
These equations contain 5-dimensional physical fields but unphysical fields are also contained. In the linearized level, these unphysical fields are algebraically eliminated.
In fact, the 5-formF I 1 µνρστ is algebraically eliminated by (14), for all k ≥ 0. When k ≥ 1, we obtain the field equation for scalar fields d I 1 : from a combination of (14) and (15). Before we discuss the decoupling of doubletons in detail, here let us give the outline of the decoupling. Similarly toF I 1 µνρστ , we can algebraically eliminate the scalar fields h I 1 ρ ρ as unphysical modes for k ≥ 1 , and obtain two independent modes associated with equations for scalar fields π I 1 , d I 1 . This procedure breaks down when the non-commutative parameter a vanishes. At a = 0, scalar field equations for k = 1 have conformal diffeomorphism invariance. Then the scalar modes associated with one of the two scalar field equations become pure gauge modes of the local gauge symmetry. These pure gauge modes can be gauged away and disappear from 5d equations. This is the decoupling of doubleton scalars. Now, we discuss the decoupling mechanism in more detail. If k ≥ 2, we can eliminate scalar fields h I 1 ρ ρ from (11),(16) by using (13). Then we obtain two independent field equations for scalar fields π I 1 , d I 1 which have mixing with other tensors.
Next, we consider the case of k = 1. For k = 1, (13) is trivial. To eliminate h I 1 ρ ρ from (11), we use (16). Then one finds This is a field equation for π I 1 , d I 1 which have mixing with other tensors. Another field equation for π I 1 , d I 1 comes from (10), which is independent of (17). We use (12) and (16) to eliminate ∇ µ ∇ ν h I 1 ρ ρ and h I 1 ρ ρ from (10). Then one finds µνρστF µνρστ φ I 1 Therefore, even if (13) is trivial for k = 1, we can also eliminate h I 1 ρ ρ from field equations. We can obtain two independent solutions for π I 1 , d I 1 by using (17) and (18).
However, if the non-commutative parameter a becomes zero, this procedure breaks down. When a = 0, equations (17), (18) degenerate to the same one : We then find that (19) has infinite number of solutions. This is a consequence of conformal diffeomorphism invariance for arbitrary function λ(x). The reason for these infinite number of solutions originates from the fact that the local gauge symmetry (20) is not fixed.
In this case, we should return to equations (12), (11) and (16). When a = 0, these equations are [eq. (11) To fix freedom of the conformal diffeomorphism invariance (24), we set gauge fixing condition where α is gauge fixing parameter. 7 The gauge symmetry (24) is not fixed completely under gauge fixing condition (25). The residual gauge symmetry is generated by function λ which satisfy Substituting the gauge condition (25) into (22) and (23), one can obtain two scalar field equations. After diagonalization of mass matrix, one of the two equations becomes (19). The other equation becomes It has the same form as (26). The solutions of (27) are actually pure gauge modes! As a result, setting the gauge fixing condition (25), we obtain two diagonalized equations (19) and (27) at a = 0. One finds physical modes with mass m 2 = 45e 2 in (19) and pure gauge modes of conformal diffeomorphism with mass m 2 = 25 3 − 5α e 2 in (27). 7 More general gauge fixing condition; h I1 ρ ρ = α π I1 + β∇ 2 x π I1 + γ ed I1 + ε∇ 2 x ed I1 is also possible.
The pure gauge modes can be gauged away and decouple from other physical modes.
This is just what we expect. Doubleton scalars become pure gauge modes and decouple in the limit a → 0.
Finally, we mention the doubleton 2-form. For a = 0, 2-form field equations (56) and (57) in Appendix C can be written in a factorized form [10] : where * D a µν = ǫ ρστ µν ∇ ρ a στ . (32) implies that k = 0 mode is a pure gauge, and can be gauged away. This mode is the 2-form in the doubleton multiplet.
When a = 0, however, solutions of (32) are not the pure gauge mode only, because the right hand side of (32) is not zero. So the decoupling of 2-form dose not occurs.

Summary and discussion
In this paper, we consider the type IIB supergravity compactified on the gravity solution (1) at the linearized level.
5d field equations (in Appendix C) become complicated form with field mixing. The mixing (or mass) matrices obtained from the 5d field equations are functions of u which is a coordinate on M 5 . This complication derives from lack of isometry, like conformal symmetry. Unlike the AdS space, the particle mass is not preserved by symmetry in the gravity solution (1), due to the lack of the dilatation invariance. This property corresponds to the fact that dimension of operators may be ambiguous in non-commutative Yang-Mills.
We find that the doubletons do not decouple unless a = 0. At a = 0, doubletons couple with U(1) operators of commutative Yang-Mills : Φ I , F µν , λ α and λᾱ, on the boundary in gauge invariant way. But the gravity in the interior of boundary has no information about U(1) operators, because doubletons decouple from other fields in the interior of the boundary. On the other hand, if a = 0, we can obtain the information of these U(1) operators. 9 Therefore, the gravity description is dual to U(N) theory. It is compatible with the fact that U(1) and SU(N) gauge symmetries cannot be separated in U(N) noncommutative Yang-Mills theory.
Finally, we mention related questions. (i) 4d N = 2 super Yang-Mills theory was constructed by brane configuration [17]. In this construction, the gauge symmetry U(1) of U(N) ≃ U(1) × SU(N) is "frozen out". There is a question whether the frozen out occurs or not in presence of B-field.
(ii) Recently, non-commutative version of SU(N) Yang-Mills theory is proposed [19]. The D-brane interpretation of this theory may be interesting.

Appendix A.Notations
• x A with capital indices A, B, ... means 10d coordinates.
The Riemann curvature tensor is defend by : Spherical harmonics : • Y I 1 (y) is scalar.
• Y I 5 m (y) is vector.
These spherical harmonics are eigenfunctions of the Hodge-de Rham operator ∆ and satisfy where e 2 = 1/α ′ R 2 .

Appendix B. Linearized field equations
Linearized equations for small fluctuation around gravity solution (1) are given in this section.

Appendix C. Linearized equations expanded by Spherical harmonics
Results of harmonic expansion are given in this section.
Definitions of differential operator Max are as follows : Definitions of 4-formD I and 5-formF I are as follows B µν a I 10 ,