Higher Spin de Sitter Holography from Functional Determinants

TL;DR

The paper advances higher spin de Sitter holography by computing the Hartle–Hawking wavefunctional for deformations of the bulk scalar and graviton using the free $Sp(N)$ model, employing the Dunne–Kirsten functional determinant method to handle radial deformations. It uncovers notable normalizability behavior: fixing the uniform $S^3$ scalar mode yields a wavefunctional that remains bounded in all other deformation directions, while other deformations can produce local maxima away from homogeneous geometries. The work interprets double-trace deformations as a basis change or convolution in the holographic dictionary and explores extensions to AdS$_4$ and dS$_4$ via Chern–Simons couplings and flavor structures, highlighting potential routes to richer HS holographic frameworks. Overall, the study provides concrete computational support for a holographic picture of de Sitter space in the higher spin context and broadens the landscape of possible dual descriptions and deformations relevant for cosmology.

Abstract

We discuss further aspects of the higher spin dS/CFT correspondence. Using a recent result of Dunne and Kirsten, it is shown how to numerically compute the partition function of the free Sp(N) model for a large class of SO(3) preserving deformations of the flat/round metric on R^3/S^3 and the source of the spin-zero single-trace operator dual to the bulk scalar. We interpret this partition function as a Hartle-Hawking wavefunctional. It has a local maximum about the pure de Sitter vacuum. Restricting to SO(3) preserving deformations, other local maxima (which exceed the one near the de Sitter vacuum) can peak at inhomogeneous and anisotropic values of the late time metric and scalar profile. Numerical experiments suggest the remarkable observation that, upon fixing a certain average of the bulk scalar profile at I^+, the wavefunction becomes normalizable in all the other (infinite) directions of the deformation. We elucidate the meaning of double trace deformations in the context of dS/CFT as a change of basis and as a convolution. Finally, we discuss possible extensions of higher spin de Sitter holography by coupling the free theory to a Chern-Simons term.

Figures (17)

• Figure 1: Examples of the radial deformations (\ref{['gauss']}) on the left, (\ref{['r2gauss']}) in the middle and (\ref{['dbgauss']}) on the right. We have suppressed the polar coordinate $\theta$ of the $S^2$ but kept the azimuthal direction.
• Figure 2: Plot of $|\Psi_{HH}(\lambda,A)|^2$ for $N=2$ for the Gaussian profile (\ref{['gauss']}) with $\lambda=1$ using $l_{max} = 45$. The solid blue line is an interpolation of the numerically determined points (shown in red). The wavefunction grows and oscillates in the negative $A$ direction.
• Figure 3: Left: Density plot of $|\Psi_{HH}(\lambda,a,A)|^2$ for $N=2$ for the profile (\ref{['r2gauss']}) as a function of A (vertical) and $\lambda$ (horizontal) for $a=5$ using $l_{max} = 45$. Again, the wavefunction grows and oscillates in the negative $A$ and positive $\lambda$ directions. Right: Plot of $|\Psi_{HH}(\lambda,a,A)|^2$ for the profile (\ref{['r2gauss']}) as a function of $\lambda$ for $A=-0.022$ and $a=5$.
• Figure 4: Left: Density plot of $|\Psi_{HH}(\lambda_i,a_i,A_i)|^2$ for $N=2$ for the double Gaussian profile (\ref{['dbgauss']}) as a function of $A_1$ ($x$-axis) and $A_2$ ($y$-axis) for $a_1=0$, $a_2=5$, $\lambda_1=\lambda_2=1$ using $l_{max} = 45$. The wavefunction grows and oscillates for negative $A_1$ and $A_2$. Right: Density plot of $|\Psi_{HH}(\lambda_i,a_i,A_i)|^2$ for the double Gaussian profile (\ref{['dbgauss']}) as a function of $\lambda_1$ ($x$-axis) and $\lambda_2$ ($y$-axis) with $A_1 = -1$, $A_2 = -1/100$, $a_1 = 0$ and $a_2 = 5$.
• Figure 5: The "balloon" deformation of $\mathbb{R}^3$, defined by (\ref{['evap']}) and (\ref{['balloonfunction']}), represented schematically for positive $\zeta$.