Higher Symmetries of the Laplacian

TL;DR

The paper identifies the symmetry algebra of the Laplacian on Euclidean space as a quotient of the universal enveloping algebra $U(so(n+1,1))$, generated by conformal Killing tensors with a precise quadratic relation structure. Using ambient/conformal geometry and hints from the AdS/CFT correspondence, it proves a canonical decomposition $A_n \cong \bigoplus_{s\ge0} K_{n,s}$ where $K_{n,s}$ are conformal Killing tensors, and provides an explicit presentation of $A_n$ via generators and relations. It then extends these flat-space symmetries to curved conformal manifolds by constructing conformally invariant differential operators $\mathcal{D}_V$ with symbol given by a conformal Killing tensor $V$, including explicit $s=2$ formulas that involve curvature. Overall, the work connects Laplacian symmetries to conformal geometry and higher-spin ideas, offering a concrete algebraic framework and paving the way for curved generalizations and AdS/CFT-inspired insights.

Abstract

Using the AdS/CFT correspondence, we identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Proposition 1