On higher spin partition functions

We observe that the partition function of the set of all free massless higher spins s = 0, 1, 2, 3,... in fl at space is equal to one: the ghost determinants cancel against the ‘ physical ’ ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This re fl ects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z = 1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a parti- cular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We fi nd that Z =1 is true also in the conformal higher spin theory (with higher-derivative s 2 ∂ kinetic terms) expanded near fl at or conformally fl at S 4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor fi eld which has 2-derivative kinetic term but only scalar gauge invariance in fl at 4d space. This non-unitary theory has Weyl-invariant action in curved background and it corresponds to ‘ partially massless ’ fi eld in AdS 5 . We discuss in detail the special case of s = 2 (or ‘ conformal graviton ’ ), compute the corresponding conformal anomaly coef fi cients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.