The Wave Function of Vasiliev's Universe - A Few Slices Thereof

TL;DR

The paper tests the dS/CFT proposal for Vasiliev’s higher-spin gravity by computing the Hartle-Hawking wavefunction of the universe via the partition function of the dual free Sp(N) CFT under finite deformations on S^3, a squashed S^3, and S^2×S^1. It develops a renormalization scheme to define finite Z, reveals a mass-parameter divergence at large negative σ and a small-S^1 divergence on S^2×S^1, and shows a global maximum of the wavefunction at the round S^3. By comparing with Einstein gravity, the authors find that some divergences persist beyond higher-spin specifics, suggesting a genuine instability of dS space in the higher-spin context. They also show nonperturbative issues with the critical Sp(N) theory on S^3, and discuss bulk interpretations, including potential big-bang–type cosmologies and topology-dependent effects, pointing to a rich but delicate structure for de Sitter holography in this setting.

Abstract

We study the partition function of the free Sp(N) conformal field theory recently conjectured to be dual to asymptotically de Sitter higher-spin gravity in four-dimensions. We compute the partition function of this CFT on a round sphere as a function of a finite mass deformation, on a squashed sphere as a function of the squashing parameter, and on an S2xS1 geometry as a function of the relative size of S2 and S1. We find that the partition function is divergent at large negative mass in the first case, and for small $S^1$ in the third case. It is globally peaked at zero squashing in the second case. Through the duality this partition function contains information about the wave function of the universe. We show that the divergence at small S1 occurs also in Einstein gravity if certain complex solutions are included, but the divergence in the mass parameter is new. We suggest an interpretation for this divergence as indicating an instability of de Sitter space in higher spin gravity, consistent with general arguments that de Sitter space cannot be stable in quantum gravity.

Figures (6)

• Figure 1: A plot of the probability distribution $|Z_{finite}|^2=|\Psi_{HH}|^2$ as a function of a constant mass perturbation $\sigma$ on the $S^3$, for $N=2$.
• Figure 2: A plot of $|\Psi|^2$ with $N=2$ as a function of $\rho$, with $\rho$ related to the squashing $\alpha$ as $\alpha = e^{2\rho} - 1$.
• Figure 3: Left figure: $Z_{finite}[T]$ for $T < \sqrt{N}$. Right figure: $Z_{finite}[T]$ for $T > \sqrt{N}$ with $N = 1$. For larger $N$ the curves become increasingly steep.
• Figure 4: Important features of the $\sigma$ plane, and a plot of our large $N$ result for $\exp\left[\frac{1}{N} \log Z_{crit}[\widetilde{\sigma}]-\frac{f \pi^2\widetilde{\sigma}^2}{16}\right]$ as a function of $\widetilde{\sigma}$.
• Figure 5: The first few diagrams that contribute at order $N$ to the $\phi^2$ two point function.