The phase of charged Nariai solutions

TL;DR

This paper calculates the phase of the one-loop Euclidean path integral around charged Nariai solutions in four dimensions, revealing how charge influences the phase and confirming results with a 2D reduction approach.

Contribution

It provides the first detailed analytical computation of the phase for charged Nariai solutions, including gauge field fluctuations, and introduces a method for handling residue zero modes.

Findings

Phase matches uncharged case for small charges

Phase becomes purely imaginary for large charges

Analytical results agree with 2D dilaton gravity reduction

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.