The ternary invariant differential operators acting on the spaces of weighted densities

TL;DR

This paper classifies invariant ternary differential operators on weighted densities over n-dimensional manifolds, revealing new operators in one dimension and describing conformal invariants in higher dimensions.

Contribution

It provides a complete classification of invariant ternary differential operators, including new operators in one dimension and conformal invariants in higher dimensions.

Findings

Four new operators in one dimension.

Explicit list of conformal transvectors for n>1.

Identification of operators related to classical structures like Poisson bracket.

Abstract

Over n-dimensional manifolds, I classify ternary differential operators acting on the spaces of weighted densities and invariant with respect to the Lie algebra of vector fields. For n=1, some of these operators can be expressed in terms of the de Rham exterior differential, the Poisson bracket, the Grozman operator and the Feigin-Fuchs anti-symmetric operators; four of the operators are new, up to dualizations and permutations. For n>1, I list multidimensional conformal tranvectors, i.e.,operators acting on the spaces of weighted densities and invariant with respect to o(p+1,q+1), where p+q=n. Except for the scalar operator, these conformally invariant operators are not invariant with respect to the whole Lie algebra of vector fields.