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

Understand

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.