A conformally invariant differential operator on Weyl tensor densities

TL;DR

This paper introduces a new fourth-order conformally invariant differential operator acting on Weyl tensor densities in 4-manifolds, revealing duality reversal and expanding understanding of conformal invariance in geometric analysis.

Contribution

It derives an explicit tensorial formula for a novel conformally invariant operator on Weyl tensor densities, highlighting its duality-reversing property in four-dimensional conformal geometry.

Findings

Operator reverses duality in oriented manifolds

Explicit formula for the conformally invariant operator

Connections to classification of conformally invariant operators

Abstract

We derive a tensorial formula for a fourth-order conformally invariant differential operator on conformal 4-manifolds. This operator is applied to algebraic Weyl tensor densities of a certain conformal weight, and takes its values in algebraic Weyl tensor densities of another weight. For oriented manifolds, this operator reverses duality: For example in the Riemannian case, it takes self-dual to anti-self-dual tensors and vice versa. We also examine the place that this operator occupies in known results on the classification of conformally invariant operators, and we examine some related operators.