Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT

TL;DR

We develop a 5d AdS$_5$ framework to compute one-loop corrections to the vacuum energy $E_c$ and 4d boundary conformal anomalies $a$ and a proposed form for $c$ for generic higher-spin representations $(\Delta;j_1,j_2)$. The results reproduce known CHS and supergravity data, yield exact relations for KK towers of type IIB on $S^5$, and provide a robust consistency check of adjoint and vectorial AdS/CFT dualities. By summing over KK levels with a symmetry-preserving regularization, we reproduce the expected adjoint duality $E_c,a,c$ relations and show anomaly cancellations upon combining conformal supergravity with Maxwell multiplets. The work also connects vectorial AdS/CFT consistency to the cancellation of conformal anomalies in 4d CHS theories, and discusses novel mixed-symmetry AdS$_5$ HS duals and supersymmetric generalizations. This framework strengthens the holographic understanding of higher-spin dynamics and their conformal boundary avatars.

Abstract

We consider general-symmetry higher spin fields in AdS_5 and derive expressions for their one-loop corrections to vacuum energy E and the associated 4d boundary conformal anomaly a-coefficient. We a propose a similar expression for the second conformal anomaly c-coefficient. We show that all the three quantities (E, a, c) computed for N=8 gauged 5d supergravity are -1/2 of the values for N=4 conformal 4d supergravity and also twice the values for N=4 Maxwell multiplet. This gives 5d derivation of the fact that the system of N=4 conformal supergravity and four N=4 Maxwell multiplets is anomaly free. The values of (E, a, c) for the states at level p of Kaluza-Klein tower of 10d type IIB supergravity compactified on S^5 turn out to be equal to those for p copies of N=4 Maxwell multiplets. This may be related to the fact that these states appear in the tensor product of p superdoubletons. Under a natural regularization of the sum over p, the full 10d supergravity contribution is then minus that of the Maxwell multiplet, in agreement with the standard adjoint AdS/CFT duality (SU(n) SYM contribution is n^2-1 of one Maxwell multiplet). We also verify the matching of (E, a, c) for spin 0 and 1/2 boundary theory cases of vectorial AdS/CFT duality. The consistency conditions for vectorial AdS/CFT turn out to be equivalent to the cancellation of anomalies in the closely related 4d conformal higher spin theories. In addition, we study novel example of vectorial AdS/CFT duality when the boundary theory is described by free spin 1 fields and is dual to a particular higher spin theory in AdS_5 containing fields in mixed-symmetry representations. We also discuss its supersymmetric generalizations.