On the M5 and the AdS7/CFT6 Correspondence

The chiral primary operators of the D=6 superconformal (2,0) theory corresponding to 14 scalars of N=4 D=7 supergravity are obtained by expanding the world volume action for the M5-brane around an AdS_7 x S^4 background. In the leading order, the operators take their values in the symmetric traceless representation of the SO(5) R-symmetry group in consistency with the early conjecture on their structure based on the superconformal symmetry and Matrix-like model arguments.

The equations of motion for N=4 D=7 SO(5) gauged SUGRA [34] can be obtained from the D=11 SUGRA by the use of non-linear Kaluza-Klein S 4 reduction ansatz presented in [29], [30]. In the notation of [35], it is given by Here where A (1) ab are the 10 gauge fields of N = 4 D = 7 gauged supergravity. In (4) -(7) ǫ (7) is the volume form on the seven-dimensional space-time and T ab is a symmetric unimodular matrix of scalars in the 14' representation of SO(5) which admits the following representation Substitution of the ansatz forF (4) andF (7) = * F (4) into the Bianchi identity forF (4) and D=11 equation of motion (2) leads to the following D=7 equations of motion with These equations, which are the bosonic part of the field equations of the seven-dimensional supergravity, will be relevant for our discussions below.

CFT operators from the M5-brane world volume action
Now let us calculate the CFT operators by expanding the M5 brane action in the AdS 7 × S 4 background. To be concrete, we restrict our attention to the CFT operators that correspond to the SUGRA scalar only. The conformal dimension of these fields is equal to △ = 2 (see, e.g., [15]- [19]). Below we will, following [22], restrict to the subsectors of the scalar matrix T ab . The full case is discussed at the end of this section. The subsectors we consider are obtained by setting the gauge fields A (1) ab and C a to zero. Then, eqs. (9) -(12) reduces to Therefore, this setting allows one to choose a diagonal parameterization [30] for the matrix T ab : and Here φ is the vector defining four independent scalars appearing in the reduction from M 11 to AdS 7 × S 4 and b a are the weight vectors of the fundamental reps. of SL(5, R) which have the following properties, for an arbitrary vector u. 6 6 The explicit representations for the b ′ a s are as follows: Substituting the diagonal parameterization (15) into the metric ansatz (4) and expanding it in linear order of φ, we have To make the SO(5) covariance manifest one can rewrite (18) in a coordinate system of Cartesian type, (x i , x a ), Note that g is the inverse radius of the S 4 , i.e. g −1 = R. The space-time metric of BPS p-brane configurations has the form of (see, e.g., [5], [38]) where the coordinates, x i , are the brane coordinates and the coordinates, x a , are transverse to the brane with r 2 ≡ (x a ) 2 . In the near horizon region r ≪ R this metric simplifies to the geometry of an AdS p+2 × S D−p−2 For the M5 case the near-horizon region is AdS 7 × S 4 , with the metric given by Using this background metric, (18) can be rewritten as with After reconstruction of the Lagrangian and the equations of motions for the scalar fields, the n-point functions of the CFT operators can be computed, as discussed in [22], by use of the formulae in [36], [37].
There are additional terms coming from (18) which are of second order in φ, therefore neglected. Finally, we expand the action for the M5 [8], [9], [10] around the background defined by (24) in the small velocities approximation [38]. In order to do that we need to find the explicit forms of theÂ (6) andÂ (3) gauge fields from the expressions of their field strengths, (5) and (6). After some algebra one can derive the following equations, where we have split the index a = 0, 1, . . . , 4 into the set of (0, α). Note that as in [22] these are on-shell results because they hold only up to the equation of motion, (14). However, the on-shell results are sufficient for our purpose. The small velocity expansion 7 leads to where we have omitted the terms of higher order in φ or derivatives (of φ and x a as well). Several remarks are in order concerning how to obtain (30). The general form of the expansion is S ≈ d 6 ξ L (0,0) + L (0,1) + L (1,0) + L (1,1) + . . .
The superscript index, (p, q), indicates the order of φ and the number of derivatives acting on them and x a , respectively. Now we will prove that there are no other terms of the type (1, 0) than those we have already given in (30). To this end, note that the induced metric on the M5 worldvolume, which corresponds to the ansatz (4), has the following form 7 We have used det M = exp(T r ln M ), which in turn implies The action for the M5 also involves the inverse worldvolume metricĝ mn , which can be shown to beĝ Up to the terms of the order (2, 0) the equations (32) and (33) can be written asĝ mn ≈△ 1/3 g mn andĝ mn ≈△ −1/3 g mn respectively. The leading terms in H (3) , which come from the first line of eq. (27), are given by [39] S H ≈ They do not contribute to the (1, 0) part because (34) has the "weight"△ 0 that only contributes to the (0, 0), (2, 0) and higher order in φ with or without derivatives. As for the WZ terms, it is clear, from (29), that there is no contribution to the (1, 0) type terms from theÂ (6) part. A straightforward calculation also shows that the contributions of the second term in the WZ part of the action are solely to (0, 0), (2, 0) and higher order terms, which completes our proof. For the subsectors given by (15) and (16) we have achieved the goal because the CFT operator has appeared as the coefficient of φ. The coordinates x a are transverse to the M5 worldvolume and they are the ones that are identified with the scalars Φ a of the on-shell (2,0) (ultrashort) supermultiplet.
For the full sectors one should keep the fields, A ab and C a . After finding the complete ansatz forÂ 3 andÂ 6 and substituting them into the M5 brane action, one again only keeps the terms linear in S ab . 8 Finally one should set all the supergravity modes to zero after taking the derivative with respect to S ab : it is not difficult to see that the terms that involve A (1) ab and C (3) a will not be relevant for the final result. Therefore we deduce from (30) the relevant part of the action through the following chain of relations In the boundary region, r → ∞, S ab ∝ r −1 and therefore the boundary condition can be chosen as Taking the trace constraint on S ab into account we obtain the CFT operator,

Conclusions
Substituting the non-linear ansatz for the eleven-dimensional metric and gauge fields into the M5-brane action and expanding it around an AdS 7 × S 4 background we have obtained the CFT operators that correspond to 14 scalars of N=4 D=7 supergravity. The leading terms of the operators are in the symmetric traceless representation of the SO(5) R-symmetry group. Therefore, our result is consistent, in the leading order, with the conjecture based on the superconformal symmetry and Matrix-like model arguments. However, the CFT operators have subleading terms as well that include e.g., the (2,0) CFT scalar fields and their derivatives. Appearance of such terms has been discussed in [40] in the context of type IIB supergravity on AdS 5 ×S 5 . As noted in [22] the subleading terms appearing in the CFT operator could be viewed as in accordance with claim of [40] that supergravity modes are dual to the "extended" chiral primary operators. Or/and there could be some field redefinitions on the CFT side such as the one discussed in [41]. The interesting problem, therefore, is to compute the n-point correlators for scalar supergravity modes propagating on AdS 7 by the use of non-linear reduction ansatz 9 and to check explicitly this observation.
Another problem one can consider is to extend the results obtained here to another class of CFT operators that correspond to other supergravity modes and to compare with the results of [44] based on the primary superfields considerations.