Residual Symmetries and Their Algebras in the Kerr-Schild Double Copy

TL;DR

This paper investigates the residual symmetry structures in the Kerr-Schild double copy correspondence, revealing a mismatch between gauge and gravity symmetries that is resolved through cohomological analysis.

Contribution

It demonstrates that the Kerr-Schild double copy does not map residual symmetries directly, and introduces a cohomological framework to understand symmetry reductions in gravity.

Findings

Residual symmetries in gauge theory form an infinite-dimensional algebra.

Gravity residual symmetries include a conformal algebra on S^2.

Cohomological analysis shows physical symmetries reduce to global isometries.

Abstract

The Kerr-Schild double copy (KSDC) is well-known for relating exact classical solutions between Yang-Mills theory and theories of gravity. However, whether this correspondence provides a more fundamental mapping between the underlying symmetries of gauge theory and gravity remains an underdeveloped area of research in the contemporary double copy program. In this paper, we demonstrate that the KSDC correspondence does not provide a mapping between the residual symmetry structures of the Kerr-Schild ansatz in Yang-Mills theory and gravity. On the gauge theory side, residual symmetries form an infinite-dimensional algebra of functions along null directions. On the gravitational side, residual diffeomorphisms preserving the Kerr-Schild form of the Schwarzschild metric generate a conformal algebra on $S^2$, which decomposes into Killing vectors and proper conformal Killing vectors (CKVs). While the Killing sector reproduces the expected global isometries, the CKV sector yields an infinite-dimensional algebra after imposing asymptotic flatness and horizon regularity. This appears to contradict the fact that the Schwarzschild solution admits no proper conformal symmetries. We resolve this apparent contradiction by constructing a Weyl-compensated BRST complex, showing that the CKV sector is BRST-exact and therefore trivial in cohomology, so that the physical symmetry algebra reduces to the global isometries of Schwarzschild. This demonstrates that the KSDC introduces an enlarged symmetry structure at the level of the ansatz, but preserves physical symmetries after a cohomological reduction, revealing a fundamental mismatch between Yang-Mills and gravity at the level of residual symmetries.