Abstract
We present a first attempt to derive the full (type-A and type-B) Weyl anomaly of four dimensional conformal higher spin (CHS) fields in a holographic way. We obtain the type-A and type-B Weyl anomaly coefficients for the whole family of 4D CHS fields from the one-loop effective action for massless higher spin (MHS) Fronsdal fields evaluated on a 5D bulk Poincaré-Einstein metric with an Einstein metric on its conformal boundary. To gain access to the type-B anomaly coefficient we assume, for practical reasons, a Lichnerowicz-type coupling of the bulk Fronsdal fields with the bulk background Weyl tensor. Remarkably enough, our holographic findings under this simplifying assumption are certainly not unknown: they match the results previously found on the boundary counterpart under the assumption of factorization of the CHS higher-derivative kinetic operator into Laplacians of “partially massless” higher spins on Einstein backgrounds.
Highlights
There is a “kinematic” relation at the one-loop quantum level in the bulk theory and subleading in the large-N expansion of the boundary theory: a holographic formula relating both O(1)-corrected partition functions
The coupling at the linearized level should be of the Fradkin-Vasiliev type [62] involving higher derivatives of the fields compensated by powers of the cosmological constant. Another important feature that emerges when departing form conformally flat bulk and boundary backgrounds is the presence of mixing terms between different spins [60, 61] that are likely to be present on both sides of the holographic formula, so that the equality between one-loop partition functions may require the inclusion of the whole family of higher spin fields
We have presented a holographic derivation of the Weyl anomaly for Maxwell photon and Weyl graviton
Summary
Let us start by spelling out the details of the holographic derivation for the gauge vector. Let us first write down the WKB-exact heat expansion in AdS5 [51, 52]. With this information one can readily get the type-A, but since the bulk is conformally flat (as well as its boundary) any information on the Weyl tensor structure is washed away. All pure-Ricci terms will produce the very same answer as in AdS that we already know, so that the information relevant for the type-B anomaly is contained in the terms involving the bulk Weyl tensor and we only need to keep track on the bulk Weyl-square term, in conformity with the holographic recipe [63]. The heat coefficients for the bulk spin zero field contain Weylsquare contributions starting with the third termb(40) and it is universally given by spin zero: ˆb(40)
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