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Recent studies of holographic properties of massless higher-order gravities, whose linear spectrum contains only the (massless) graviton, yielded some universal relations in

Owing to the enlarged global symmetry, conformal field theories (CFTs) are much simpler than the general quantum field theories and many tantalizing features have attracted considerable attention. The subject has been further driven by the advancement of the powerful AdS/CFT correspondence, which may provide a new window for a deeper understanding of both classical and quantum gravity. An important feature of any CFT in even

The conformal symmetry has the power that many universal properties can be uncovered without having to know the details of a specific theory. The two-point function of the energy-momentum tensor is completely determined up to an overall coefficient

In four dimensions, the identity

While the identity

The purpose of this paper is to employ higher-order massless gravities in seven dimensions and to compute their holographic

We begin with a review of the structure of conformal anomalies in

The conformal anomalies in

In this section, we consider Einstein gravity with a negative cosmological constant, extended with higher-order invariant polynomials of curvature tensor and their covariant derivatives, up to and including six derivatives. We then follow the standard technique to compute the holographic anomalies and derive some universal relations.

The Lagrangian of general higher-order gravities, up to and including six derivatives, is

In literature, it is common to introduce dimensionless couplings

We now consider the linear perturbation on the AdS background. Owing to the higher-order derivative terms, the linear spectrum contains two massive scalars and two massive spin-2 modes, in addition to the usual (massless) graviton mode. We are interested in massless gravity where all these massive modes decouple from the spectrum. This requires a set of linear constraints on the coupling constants. In particular, the decoupling of the massive scalar modes requires

According to the AdS/CFT dictionary, the massless graviton is associated with the boundary stress tensor. Following the standard procedure, one can obtain its two-point function. The overall coefficient is then clearly proportional to

For given gravity with an AdS vacuum in odd

The higher-order gravities become all massless once we impose the conditions

Since the differential relation

For a well-defined CFT, one may expect that there exists an

We follow the technique of

Massless gravities are those whose linearized spectrum in a maximally symmetric background consists only of the graviton. Massless higher-order gravities provide a class of bulk theories for studying the holographic properties, where conformal anomalies, two-point functions and

The result

The holographic approach using massless higher-order gravities provides enormous insights to the properties of CFTs. It is of great interest to investigate whether these universal relations can be established by CFTs themselves. The differential relations, however, present an immediate challenge since they involve

We are grateful to Yue-Zhou Li for useful discussions. We are grateful to the PRD referee for the useful comments that lead to the paragraph of Eqn.