Wave function of Vasiliev’s universe: A few slices thereof

Abstract

We study the partition function of the free $\mathrm{Sp}(N)$ conformal field theory (CFT) 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 ${\mathrm{S}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{1}$ geometry as a function of the relative size of ${\mathrm{S}}^{2}$ and ${\mathrm{S}}^{1}$. We find that the partition function is divergent at large negative mass in the first case, and for small ${\mathrm{S}}^{1}$ 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 ${\mathrm{S}}^{1}$ 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.