On partition function and Weyl anomaly of conformal higher spin fields

Abstract

We study 4-dimensional higher-derivative conformal higher spin (CHS) fields generalising Weyl graviton and conformal gravitino. They appear, in particular, as "induced" theories in the AdS/CFT context. We consider their partition function on curved Einstein-space backgrounds like (A)dS or sphere and Ricci-flat spaces. Remarkably, the bosonic (integer spin s) CHS partition function appears to be given by a product of partition functions of the standard 2nd-derivative "partially massless" spin s fields, generalising the previously known expression for the 1-loop Weyl graviton (s=2) partition function. We compute the corresponding spin s Weyl anomaly coefficients a_s and c_s. Our result for a_s reproduces the expression found recently in arXiv:1306.5242 by an indirect method implied by AdS/CFT (which relates the partition function of a CHS field on S^4 to a ratio of known partition functions of massless higher spin field in AdS_5 with alternate boundary conditions). We also obtain similar results for the fermionic CHS fields. In this half-integer spin s case the CHS partition function on (A)dS background is given by a product of squares of "partially massless" spin s partition functions and one extra factor corresponding to a special massive conformally invariant spin s field. It was noticed in arXiv:1306.5242 that the sum of the bosonic a_s coefficients over all spins s is zero when computed using the zeta-function regularization, and we observe that the same property is true also in the fermionic case, suggesting that the corresponding conformal higher spin theory may be consistent at the quantum level.