On higher spin partition functions

TL;DR

The paper establishes that summing one-loop partition functions over entire towers of free higher-spin fields in flat, AdS, and conformally-flat backgrounds yields $Z_{ m tot}=1$ under a regularization compatible with higher-spin gauge symmetry, signaling a topological-like cancellation across spins. It analyzes three families—massless higher spins (MHS), conformal higher spins (CHS), and conformal symmetric tensors (CST)—across flat, Ricci-flat, and conformally-flat spaces, uncovering cancellations in flat and AdS settings and revealing intricate regulator-dependent relations (notably CHS$_s$ vs MHS in AdS$_5$ via dimensional regularization). In particular, explicit results for $s=2$ CST (a and c anomaly coefficients) are derived and shown to match AdS$_5$ shadow representations, while general-spin CST/CHS cases illustrate deeper structure and AdS/CFT consistency. The findings support a picture of enhanced gauge symmetry driving cancellations across infinitely many spins, paralleling features of supersymmetric/topological theories and reinforcing the vectorial AdS/CFT duality framework.

Abstract

We observe that the partition function of the set of all free massless higher spins s=0,1,2,3,... in flat 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 reflects 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 particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z=1 is also true in the conformal higher spin theory (with higher-derivative d^{2s} kinetic terms) expanded near flat or conformally flat S^4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has a Weyl-invariant action in curved background and corresponds to "partially massless" field in AdS_5. We discuss in detail the special case of s=2 (or "conformal graviton"), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.