Generalized Curvature and Ricci Tensors for a Higher Spin Potential and the Trace Anomaly in External Higher Spin Fields in AdS_{4} Space

TL;DR

This work develops a geometric framework for linearized trace anomalies of conformally coupled scalars in AdS$_4$ with external higher-spin fields by exploiting a generalized curvature, the deWit–Freedman curvature, for higher-spin gauge fields. The authors explicitly analyze the spin-4 case, constructing all curvature traces (the HS analogues of Ricci tensors) and deriving primary and secondary Bianchi identities, then express these traces in terms of the Fronsdal operator acting on the HS potential. A key result is that, for s=4 in AdS$_4$, the trace anomaly contribution from the two-point function can be canceled completely by a suitable linear combination of local counterterms built from contracted squares of the curvature and its traces; three independent Weyl variations, $\delta K^{\Delta}$, $\delta K^{\gamma}$, and $\delta K^{\beta}$, suffice to remove the anomaly, with a Weyl-invariant combination reducing the effective count. This stands in contrast to the spin-2 case, where a remaining topological Euler-density term persists in a background curvature, underscoring a distinctive HS geometry in the absence of a dynamical background metric. The results outline an algorithmic route to HS anomaly cancellation at the quadratic (two-point) level and pave the way for higher-spin generalizations and applications to three-point functions.

Abstract

The curvature of a higher spin potential as constructed in a previous article of the same authors arXiv:0705.3528 is applied to the analysis of the linearized trace anomaly obtained from the quadratic part of the effective action for a conformally coupled scalar with linearized interaction with the external higher spin fields arXiv:hep-th/0602067. The spin is restricted to four to profit from technical simplifications but without reducing the problem in principle. The issue includes the calculation of all Ricci tensors as multiple traces of the curvature, the derivation of all primary and secondary Bianchi identities, expressing all Ricci tensors as differential operators applied to the Fronsdal term, calculating the Weyl variation of these, and showing finally that Weyl variations of integrals over contracted squares of Ricci tensors can be used to eliminate the anomaly completely. This peculiarity is discussed in detail. As tools we use the formalism of bisymmetric tensor fields whose space is equipped with a local bilinear invariant form, the *-form.