Residual Symmetries and BRST Cohomology of Schwarzschild in the Kerr-Schild Double Copy

TL;DR

The paper addresses whether residual symmetries survive the Kerr-Schild double copy by deriving the full gauge residual algebras for Abelian and non-Abelian KS fields and comparing them to the gravitational residual diffeomorphisms of the KS Schwarzschild solution. It finds that gauge residuals form infinite-dimensional algebras (Abelian: $C^^on(R)$; non-Abelian: 𝔤 ⊗ $C^^on(R)$) whereas the Killing-sector gravitational residuals collapse to the finite Schwarzschild isometry algebra 𝔰𝔬(3) ⊕ ℝ, revealing a structural mismatch. A BRST construction shows the residual symmetry in this Killing sector has only a trivial cohomology, providing a quantum-consistency check of the algebraic collapse. These results imply that symmetry preservation in the KS double copy is more subtle than exact solution matching and motivate a detailed study of the proper conformal Killing vector sector (CKV) in Part II, as well as extensions to other spacetimes and double-copy frameworks. The work highlights that while KS double copy is powerful for generating exact geometries, it may not preserve residual symmetry algebras in a straightforward, one-to-one way.

Abstract

The Kerr-Schild (KS) double copy is celebrated for producing exact gravitational spacetimes from gauge fields, yet the preservation of symmetry content remains largely unexplored. We investigate the fate of residual symmetries in the KS double copy, focusing on the Schwarzschild solution. On the gauge theory side, we derive the residual transformations that preserve the Abelian and non-Abelian KS ansatzë, finding they both form an infinite-dimensional Lie algebra parameterized by arbitrary null functions. On the gravity side, we analyze the resulting residual diffeomorphisms of the KS Schwarzschild metric. Restricting our focus to the Killing vector class of solutions, we find that the only surviving diffeomorphisms are the finite-dimensional global isometries of Schwarzschild, reducing the residual gauge algebra to the subalgebra generated by time translations and spatial rotations. This finding reveals a fundamental structural mismatch: the infinite-dimensional algebra of the gauge side admits no simple counterpart in this constrained gravitational sector. We formalize this by showing that the BRST operator for the residual symmetry is trivialized under the Killing condition. This result serves as a crucial consistency check, validating the kinematic algebraic collapse within a quantum field theoretic framework. This paper is the first of a two-part series. In the second paper, we complete this analysis by examining the more complex proper conformal Killing vector (CKV) class of solutions and formulating a unified BRST framework to definitively test the structural obstruction.