The phase of charged Nariai solutions

TL;DR

This work computes the exact one-loop phase of the Euclidean gravitational path integral around magnetically charged Nariai saddles in $4$ dimensions, including both metric and $U(1)$ fluctuations, and confirms agreement with a corresponding $2$D dilaton gravity reduction. By performing a careful mode analysis in the de Donder gauge, identifying six ghost zero modes, and then lifting them via a controlled gauge deformation, the authors map out the negative-mode structure across the charged Nariai branch. A critical charge $Q_* = \frac{1}{4}\sqrt{\frac{3}{\Lambda}}$ marks a transition: for $0<Q<Q_*$ the phase is unity, while for $Q_*<Q<Q_{\max}$ the phase becomes $i^3$, aligning with expectations from related de Sitter constructions. The results reinforce the consistency between 4D and reduced-dimensional analyses and illuminate subtle issues in negative-mode counting and observer-related state counting in de Sitter quantum gravity.

Abstract

In this note, we compute the phase of the one-loop Euclidean path integral around charged Nariai solutions in 4 dimensions, including both metric and gauge field fluctuations. These solutions have a $S^{2} \times S^{2}$ geometry, and a magnetic flux in one of the spheres. For charges smaller than a critical value, the phase matches the result for the uncharged Nariai solution, and for charges bigger than that value, the phase is $i^{3}$. Our analytical calculation in the full 4D geometry matches the result obtained recently within a 2D dilaton gravity reduction. Along the way, we also develop a method of dealing with residue zero modes in the de Donder gauge.

Figures (1)

• Figure 1.1: We plot the allowed values of the mass and charge $(M, Q)$ for black holes in de Sitter for a given value of the cosmological constant $\Lambda>0$. The black hole solutions are points in the shaded region delimited by the blue and orange $+$ green curves. The blue curve corresponds to extremal black holes, that is, the black holes with the smallest value of mass for a given charge. The dashed red line stands for the so-called "lukewarm" solutions, which are black holes in thermal equilibrium with the cosmological horizon. The orange and green curves correspond to the "Nariai solutions", which are $dS_{2}\times S^{2}$ geometries. The orange and green colours stand for whether their charge is smaller or larger than a threshold value of $Q=Q_{*}=\frac{1}{4}\sqrt{\frac{3}{\Lambda}}$, respectively. The transition between the curves is marked with a red dot.