Gravity as Gauge Theory Squared: A Ghost Story

TL;DR

This paper places gravity, including the graviton, a Kalb-Ramond two-form, and the dilaton, in a BRST-covariant framework by representing it as the product of two Yang-Mills theories (gravity = gauge × gauge) at linear order. It constructs a field-theoretic product via a convolution with a spectator bi-adjoint field and derives a unique dictionary that maps YM equations of motion and BRST variations to their gravitational counterparts, with nonlocal $□^{-1}$ (and potentially $□^{-2}$) operators required for consistency. A novel feature is the automatic emergence of anti-BRST symmetry, with the condition $m_{(d)}=ξ_{(d)}$ reproducing a modified two-form structure; the mapping preserves gauge independence and provides explicit expressions for the dilaton, graviton, and two-form dictionaries and their ghost sectors. The work opens routes to higher-order extensions, field-antifield formalisms, and potential generalizations to supergravity or theories with asymmetric left/right gauge factors, offering a concrete field-theoretic realization of the gravity–gauge double-copy paradigm at the level of equations of motion and gauge fixing. Overall, the approach clarifies how gravity can be obtained as a BRST-covariant product of gauge theories and highlights the role of nonlocal operators and spectator fields in disentangling gravitational degrees of freedom.

Abstract

The Becchi-Rouet-Stora-Tyutin (BRST) transformations and equations of motion of a gravity-two-form-dilaton system are derived from the product of two Yang-Mills theories in a BRST covariant form, to linear approximation. The inclusion of ghost fields facilitates the separation of the graviton and dilaton. The gravitational gauge fixing term is uniquely determined by those of the Yang-Mills factors which can be freely chosen. Moreover, the resulting gravity-two-form-dilaton Lagrangian is anti-BRST invariant and the BRST and anti-BRST charges anti commute as a direct consequence of the formalism.