Gauge $\times$ Gauge on Spheres

TL;DR

This paper demonstrates how gauge theories on a sphere can be combined using a convolution product, revealing that gravity coupled to matter fields can be derived from Yang-Mills theories, extending to a cosmological background.

Contribution

It introduces a convolution on the 2-sphere and shows how gravity coupled to matter fields arises from Yang-Mills theories via this product, including extension to a cosmological setting.

Findings

Convolution on a 2-sphere is defined and utilized.

Gravity and gauge fixing conditions follow from Yang-Mills products.

Extension of the product to a 1+2 dimensional Einstein-static universe.

Abstract

We introduce a convolution on a 2-sphere and use it to show that the linearised Becchi-Rouet-Stora-Tyutin transformations and gauge fixing conditions of Einstein-Hilbert gravity coupled to a two-form and a scalar field, follow from the product of two Yang-Mills theories. This provides an example of the convolutive product of gauge theories on a non-trivial background. By introducing a time direction the product is shown to extend to the $D=1+2$ Einstein-static universe.