The phase of de Sitter higher spin gravity

TL;DR

The paper resolves the phase ambiguity of the one-loop sphere partition function in de Sitter higher spin gravity by summing over the full Vasiliev spectrum. Using two regularization schemes, dimensional and character regularization, the authors show that the non-minimal theory (all integer spins) yields a vanishing total phase and zero HS dimension, implying the sphere partition function can be interpreted statistically without an observer. In the minimal theory (even spins only), the total phase is nonzero and given by $P_{\rm min} = -\frac{\sqrt{\pi}}{2^d \Gamma\left(\frac{d+1}{2}\right) \Gamma\left(2-\frac{d}{2}\right)}$, vanishing only on $S^3$ among odd spheres, with $P_{\rm min} = -\dim G^{\rm min}_{\rm HS}$. The results strengthen the link between the sphere partition function and de Sitter entropy in higher spin holography, clarifying the role of the infinite spin tower in dS/CFT contexts and the potential need (or lack thereof) for an observer to restore a counting interpretation in the non-minimal case.

Abstract

The one-loop Euclidean partition function on the sphere is known to exhibit a nontrivial phase for massless fields of spin greater than one. Such a phase appears to be in tension with a state counting interpretation of the partition function and its relation to the de Sitter entropy. It has been recently argued that the phase associated with the gravitational path integral can be cancelled by including the contribution of an observer. In this note, we compute the total phase of Vasiliev higher spin gravity on the sphere by summing over the contributions of all spins. We evaluate the resulting infinite sum using two different regularization schemes, obtaining consistent results. We find that for the non-minimal Vasiliev theory, which includes massless fields of all integer spins, the total phase vanishes in all dimensions. This result suggests that the sphere partition function of these theories may be consistent with a counting interpretation, without explicitly including an observer.

Figures (1)

• Figure A.1: Contour deformation of the $z$ integral. The red line $C$ denotes the original contour within the unit disk, and the blue line $C'$ represents the deformed contour along the branch cut.