Six-dimensional one-loop divergences in quantum gravity from the $\mathcal{N}=4$ spinning particle

TL;DR

This paper computes the one-loop divergences in quantum gravity in six dimensions using the worldline formalism and heat kernel method, determining the Seely-DeWitt coefficient $a_3(D)$ for Einstein manifolds, which was previously unknown.

Contribution

It introduces a novel calculation of the Seely-DeWitt coefficient $a_3(D)$ for quantum gravity with a cosmological constant using the $ abla$4 spinning particle model, advancing understanding of gauge-invariant divergences.

Findings

Computed the Seely-DeWitt coefficient $a_3(D)$ for Einstein manifolds.

Validated results using both worldline formalism and heat kernel method.

Compared six-dimensional divergences with existing literature.

Abstract

In this work, we investigate the computation of the counterterms necessary for the renormalization of the one-loop effective action of quantum gravity using both the worldline formalism and the heat kernel method. Our primary contribution is the determination of the Seleey-DeWitt coefficient $a_3(D)$ for perturbative quantum gravity with a cosmological constant, which we evaluate on Einstein manifolds of arbitrary $D$ dimensions. This coefficient characterizes quantum gravity in a gauge-invariant manner due to the on-shell condition of the background on which the graviton propagates. Previously, this coefficient was not fully known in the literature. We employ the $\mathcal{N}=4$ spinning particle model recently proposed to describe the graviton in first quantization and then use the heat kernel method to cross-check the correctness of our calculations. Finally, we restrict to six dimensions, where the coefficient corresponds to the logarithmic divergences of the effective action, and compare our results with those available in the literature.