Gauge $\times$ Gauge on Spheres

TL;DR

The paper develops a covariant convolution on the 2-sphere $S^2$ and demonstrates that the linearised BRST transformations and gauge-fixing conditions for gravity coupled to a Kalb-Ramond 2-form and a dilaton can be derived from the product of two Yang-Mills theories. By introducing a time direction, the authors extend the construction to the $D=1+2$ Einstein-static universe, providing a curved-background instance of the gravity $=$ gauge $\times$ gauge program. Central to the approach are the scalar left-convolution on $S^2$ and a tensor convolution on $SO(3)$, whose Fourier-space factorisation enables a precise mapping of Yang-Mills BRST data to the gravity sector, including ghost/ghost-for-ghost structure and gauge-fixing functionals. The resulting gravity/gauge dictionaries are unique for local derivative operators, while gauge-fixing often involves nonlocal derivatives, and the framework smoothly reduces to the flat-space results in the appropriate limit. The work points to natural generalisations to group manifolds (such as $S^3$) and to higher dimensions, with potential applications to classical double-copy constructions and supersymmetric extensions.

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.