dS$^4$ Metamorphosis

TL;DR

This paper analyzes the Euclidean path integral of higher spin gravity on S^4, deriving a gluing formula that relates it to an S^3 boundary theory, with implications for higher spin holography.

Contribution

It introduces a novel gluing formula for the S^4 path integral in higher spin gravity, connecting it to boundary theories on S^3 with specific symmetry properties.

Findings

Boundary theory on S^3 is the Sp(N) invariant sector of free scalars.

Leading contribution to the partition function is 2^N with exact one-loop cancellations.

The gluing formula applies to theories with both bosonic and fermionic higher spins.

Abstract

We study the Euclidean path integral of higher spin gravity on $S^4$. Based on a one-loop analysis, we are led to a gluing formula expressing the $S^4$ path integral in terms of an underlying $S^3$ path integral. We view the three-sphere as a boundary hypersurface splitting the four-sphere into two halves. For a higher spin spectrum containing even spins only, the resulting boundary theory living on the $S^3$ cut is the $\mathrm{Sp}(N)$ invariant sector of $N\in \mathbb{Z}^+$ anti-commuting, conformally coupled free scalars, with conformal higher spin sources mediating the gluing. This boundary $\mathrm{Sp}(N)$ theory was previously shown to compute the Hartle-Hawking wavefunction at $\mathcal{I}^+$ in the higher spin dS$_4$/CFT$_3$ correspondence. In contrast to the infinite spatial volume of $\mathcal{I}^+$, here the conformal fields populate a finite size $S^3$ hypersurface of $S^4$. For theories with both bosonic and fermionic higher spin fields, the gluing formula is instead built from an $\mathcal{N}=2$ superconformal boundary field theory coupled to $U(N)$ invariant superconformal sources. Under this assumption, the leading contribution to the four-sphere partition function is $2^N$, and we observe exact cancellations at one-loop.