Abstract
The holographic Weyl anomaly for GJMS operators (or conformal powers of the Laplacian) are obtained in four and six dimensions. In the context of AdS/CFT correspondence, free conformal scalars with higher-derivative kinetic operators are induced by an ordinary second-derivative massive bulk scalar. At one-loop quantum level, the duality dictionary for partition functions entails an equality between the functional determinants of the corresponding kinetic operators and, in particular, it provides a holographic route to their Weyl anomalies. The heat kernel of a single bulk massive scalar field encodes the Weyl anomaly (type-A and type-B) coefficients for the whole tower of GJMS operators whenever they exist, as in the case of Einstein manifolds where they factorize into product of Laplacians.While a holographic derivation of the type-A Weyl anomaly was already worked out some years back, in this note we compute holographically (for the first time to the best of our knowledge) the type-B Weyl anomaly for the whole family of GJMS operators in four and six dimensions. There are two key ingredients that enable this novel holographic derivation that would be quite a daunting task otherwise: (i) a simple prescription for obtaining the holographic Weyl anomaly for higher-curvature gravities, previously found by the authors, that allows to read off directly the anomaly coefficients from the bulk action; and (ii) an implied WKB-exactness, after resummation, of the heat kernel for the massive scalar on a Poincaré-Einstein bulk metric with an Einstein metric on its conformal infinity.The holographically computed Weyl anomaly coefficients are explicitly verified on the boundary by exploiting the factorization of GJMS operators on Einstein manifolds and working out the relevant heat kernel coefficient.
Highlights
While a holographic derivation of the type-A Weyl anomaly was already worked out some years back, in this note we compute holographically the type-B Weyl anomaly for the whole family of GJMS operators in four and six dimensions
Graham [18] already noticed that the conformal invariance properties of the renormalized volume of a Poincare-Einstein metric are reminiscent of those for the functional determinants of conformally invariant differential operators, e.g. conformal Laplacian and higher-order GJMS operators, being conformal invariant in odd dimensions but having an anomaly in even dimensions and, on the other hand, those for the volume anomaly are similar to those for the constant term in the expansion of the integrated heat kernel for the conformally invariant differential operator, which vanishes in odd dimensions but in even dimensions is a conformal invariant obtained by integrating a local expression in curvature, namely the conformal anomaly
We have shown the way one bulk Laplacian rules the whole family of boundary GJMS operators and, in particular, the way the conformal anomaly is encoded in the bulk heat kernel
Summary
Let us start by examining the GJMS operators on an even d-dimensional compact manifold where the very existence of the “supercritical” ones, i.e. P2k with k > d/2, is not granted in general. The contribution of each Laplacian to the functional determinant, and to the anomaly, can be computed with standard heat kernel techniques As it has already been noticed and successfully put into use [51,52,53], the Einstein condition brings in many simplifications, the curvature invariants that enter the type-B Weyl anomaly remain independent and their coefficients can be efficiently obtained by this shortcut route. A direct way to work out the Weyl anomaly for the GJMS operators is to exploit their factorization on a generic compact Einstein manifold, look for the relevant heat kernel coefficient for each individual factor and add them all. The relevant heat-kernel coefficient of the individual Laplacians can be worked out (see e.g. [54]) and the raw result on a 6D Einstein metric, modulo trivial total derivatives, reads b(6i)
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