dS$^4$ Metamorphosis

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.

Figures (2)

• Figure 1: Interplay between the Euclidean $S^4$ partition function and the Lorentzian Hartle-Hawking wavefunction of a $\Lambda>0$ higher spin theory. The one-loop analysis motivates the gluing formula (\ref{['eq:HS_S4']}), which suggests that the $S^4$ partition function is obtained by gluing two hemispheres with a common three-sphere boundary. On the boundary we place the $\mathrm{Sp}(N)$ invariant sector of a $\mathrm{Sp}(N)$ vector model, coupled to conformal higher spin sources. Strikingly, the $\mathrm{Sp}(N)$ theory which appears in this bilinear pairing also encodes the Lorentzian Hartle-Hawking wavefunction computed in the context of higher spin dS$_4$/CFT$_3$. This suggests that the sphere partition function captures aspects of the wavefunction norm (\ref{['eq: Norm HH']}).
• Figure 2: The sphere partition function of four-dimensional higher spin theory with $\Lambda>0$ can be obtained by gluing together two hemispheres with an $S^3$ boundary. The underlying theory living on the $S^3$ boundary is built from the $\mathrm{Sp}(N)$ invariant sector of the $N$ anti-commuting, conformally coupled real scalars $\chi^I$, with $I=1,\ldots ,N$. The glue, indicated by the blue lines, are conformal higher spin gauge fields.