Partition function of free conformal higher spin theory

TL;DR

This paper computes the canonical partition function for free conformal higher spin theories in even dimensions by two complementary routes: conformal operator counting in flat space and finite‑temperature determinants on S^1×S^{d−1}. It establishes a factorized, tractable form for the CHS operator on conformally flat backgrounds, derives explicit spin‑s partition functions Z_s, and shows that summing over all spins yields a total partition function with symmetric q↔1/q behavior and vanishing Casimir energy in any even d. The authors connect these results to AdS/CFT via shadow/conserved current pairings and provide a representation‑theoretic derivation using so(d,2) Verma modules and BGG resolutions. The work supports a consistent nonlinear CHS theory as an induced action and highlights the deep ties between conformal higher spins, Killing tensors, and higher‑spin holography.

Abstract

We compute the canonical partition function Z of non-interacting conformal higher spin (CHS) theory viewed as a collection of free spin s CFT's in R^d. We discuss in detail the 4-dimensional case (where s=1 is the standard Maxwell vector, s=2 is the Weyl graviton, etc.), but also present a generalization for all even dimensions d. Z may be found by counting the numbers of conformal operators and their descendants (modulo gauge identities and equations of motion) weighted by scaling dimensions. This conformal operator counting method requires a careful analysis of the structure of characters of relevant (conserved current, shadow field and conformal Killing tensor) representations of the conformal algebra so(d,2). There is also a close relation to massless higher spin partition functions with alternative boundary conditions in AdS_{d+1}. The same partition function Z may also be computed from the CHS path integral on a curved S^1 x S^{d-1} background. This allows us to determine a simple factorized form of the CHS kinetic operator on this conformally flat background. Summing the individual conformal spin contributions Z_s over all spins we obtain the total partition function of the CHS theory. We also find the corresponding Casimir energy and show that it vanishes if one uses the same regularization prescription that implies the cancellation of the total conformal anomaly a-coefficient. This happens to be true in all even dimensions d >= 2.