Gauge $\times$ Gauge $=$ Gravity on Homogeneous Spaces using Tensor Convolutions

TL;DR

This paper introduces a convolution operation for tensor fields on homogeneous spaces, linking gauge theories to gravity by showing how gauge field transformations generate diffeomorphisms, with applications to specific spacetime models.

Contribution

It defines a tensor convolution on homogeneous spaces and demonstrates its role in deriving gravity from gauge fields via BRST transformations.

Findings

Convolution of tensor fields on homogeneous spaces is established.

Gauge BRST transformations induce diffeomorphisms in gravity.

Applicable to ultrastatic spacetimes with compact slices.

Abstract

A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of `gravity $=$ gauge $\times$ gauge'. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the `gauge $\times$ gauge' convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.