Partition function of free conformal higher spin theory

TL;DR

This paper calculates the partition function of free conformal higher spin theories in various dimensions, revealing a factorized form, the total partition function, and the vanishing Casimir energy under specific regularizations, highlighting their conformal properties.

Contribution

It introduces a detailed operator counting method for conformal higher spin theories and relates the partition function to boundary conditions in AdS, providing new insights into their structure.

Findings

Partition function expressed as a sum over spins and dimensions.

Total partition function exhibits a simple factorized form.

Casimir energy vanishes with consistent regularization in even dimensions.

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.