Features of the Partition Function of a $\Lambda >0$ Universe

Abstract

We consider properties of the gravitational path integral, $Z_{\text{grav}}$, of a four-dimensional gravitational effective field theory with $\Lambda >0$ at the quantum level. To leading order, $Z_{\text{grav}}$ is dominated by a four-sphere saddle subject to small fluctuations. Beyond this, $Z_{\text{grav}}$ receives contributions from additional geometries that may include Einstein metrics of positive curvature. We discuss how a general positive curvature Einstein metric contributes to $Z_{\text{grav}}$ at one-loop level. Along the way, we discuss Einstein-Maxwell theory with $\Lambda >0$, and identify an interesting class of closed non-Einstein gravitational instantons. We provide a detailed study for the specific case of $\mathbb{C}{P}^2$ which is distinguished as the saddle with second largest volume and positive definite tensor eigenspectrum. We present exact one-loop results for scalar particles, Maxwell theory, and Einstein gravity about the Fubini-Study metric on $\mathbb{C}{P}^2$.

26 pages plus appendices, v3 minor update on volumes of manifolds