On conformally invariant differential operators in odd dimensions

Abstract

This is a brief announcement in which we identify all the conformally invariant differential operators acting on densities other than −(n/2) + k and other than l, where k is a positive integer and l is a nonnegative integer.

Announcement

Let (Mⁿ, [g]) be a conformal manifold. By a Riemannian differential operator, we mean a (possibly nonlinear) differential operator L_(M,g), associated to each Riemannian manifold, with the following properties.

1. There is a fixed polynomial P in variables ∂^αg_ij/∂x… ∂x, (det g_ij)⁻¹, and ∂^βf/∂x … ∂x so that for any Riemannian manifold (M, g) and any smooth function f on M we have in local coordinates the formula

   

2. If (M, g) and (M*, g*) are isometric by a map φ : M → M*, then L_(M,g)(f* ○ φ) = (L_(M*,g*)f*) ○ φ for any smooth function f* on M*.

By a conformally invariant differential operator of bidegree (a, b) we mean a Riemannian operator, which has the additional property of taking conformal densities of weight a into conformal densities of weight b on M. We will denote such an operator by P : E[a] → E[b]. In a conformal manifold (Mⁿ, [g]), given a metric g in the conformal class, we can view a w density (in E[w]) as a function f_g in C^∞(M) for each g ∈ [g], which under conformal rescalings ĝ = Ω²g transforms by f_ĝ = Ω^wf_g. Thus in this setting, for a differential operator P_g to be conformally invariant of bidegree (a, b) it is necessary and sufficient that

Of course, there are many examples of conformally invariant operators acting on bundles other than conformal densities, e.g. the Dirac operator on spinors. However, we shall not be dealing with such operators here.

The most classical example of a conformally invariant operator is the conformal Laplacian: Δ^c: E[−(n/2) + 1] → E[−(n/2) − 1]. In 1984, Paneitz (1) showed that for weights E[−(n/2) + 2] → E[−(n/2) − 2] one can add lower-order terms to Δ² and make it conformally invariant. Finally, Graham et al. (2) addressed the problem of extending the powers of the Laplacian, which are known to be invariant in Euclidean space, to arbitrary conformal manifolds. For odd dimensions, their result was that one can add lower-order terms to Δ^k, for any k, to obtain a conformally invariant operator P_2k of bidegree (−(n/2) + k, −(n/2) − k). P₂ is then Δ^c, and P₄ is the Paneitz operator.

In a given dimension, given a Δ^k there can only be one weight on which P_2k can act and be conformally invariant, namely (−(n/2) + k). It can be seen by elementary representation theory that in Euclidean space these are the only conformally invariant linear differential operators. On the other hand, one could ask the question of which operators acting on Riemannian manifolds are conformally invariant. This is different from the problem of extending Δ^k to curved space, because one can conceivably have a Riemannian differential operator with a symbol that vanishes on flat space but does not vanish in general.

It turns out that such operators do exist, and we have a theorem (below) that identifies all of them in odd dimensions and for weights w ≠ −(n/2) + k, k = 1, 2, … and w ≠ l, l = 0, 1, 2, … . For example, one can add lower-order terms to ∥W∥²Δ_gf or W^ijklW(∇)_lm(f) and make them conformally invariant for any weight not of the form (−(n/2) + k), k = 1, 2, … . By contrast, Δ can be corrected by lower-order terms to be conformally invariant only for one particular weight for each given order 2k and dimension n.

We construct these conformally invariant operators through the Fefferman–Graham ambient metric construction (this metric was used for ref. 2 too). This was developed originally to construct conformally invariant scalars, i.e. polynomials P in the curvature and its covariant derivatives, which rescale under conformal changes, i.e. P_Ω²g = Ω^kP_g. The idea in that paper was to imbed the whole conformal class (Mⁿ, [g]) into an ambient, formally defined, Riemannian Ricci-flat manifold (G̃ⁿ⁺², g̃) with curvature tensor R̃ and connection ∇̄. In the coordinates (t, x→, ρ) used in ref. 3 for G̃ we can define f_g ∈ E[w] to be a homogeneous function on G = {ρ = 0}:  f(t, x→, 0) = t^wf(1, x→, 0). Then, for w ≠ −(n/2) + k, k = 1, 2, … , one can formally extend this to a homogeneous function u on G̃:  u(t, x→, ρ) = t^wu(1, x→, ρ) so that Δ_g̃u = 0. Then, a complete contraction in the ambient space, of the form

is a differential operator that is conformally invariant. Observe that if Σl_i + Σ(k_i + 2) is fixed, then each complete contraction as described above takes densities of a given weight into densities of the same weight. Our result, then, is discussed below.

Theorem.

Given a number w, with w ≠ −(n/2) + k, and w ≠ l, wherek is a positive integer and l is a nonnegative integer, then all the conformally invariant differential operators acting on E[w] are linear combinations of contractions as above, where Σl_i + Σ(k_i + 2) is fixed.

Remarks:

1. Because u is defined to be harmonic and R̃ Ricci-flat, not all the contractions (see Eq. 1) are nonzero: e.g., contracting the first two indeces in ∇̄^l_iu or contracting i and k in ∇̄^k_mR̃_ijkl will cause the whole contraction to vanish. Not all of them are zero, however, and one can easily see that P_g ∗ Δf, where P_g is a conformally invariant scalar, can always admit lower-order terms and be made a conformally invariant operator.

2. The method of the proof essentially follows ref. 4, where an invariant theory is developed to deal with conformally invariant scalars (see also ref. 5).

3. Apart from the bundle E[w] of conformal densities one can ask for invariant differential operators between different bundles. See, for example, refs. 6–8 for an excellent survey of this issue.

4. Suppose we include a contraction in the sum with no R̃ term. Then, that will not necessarily vanish in flat space. Such contractions appear already in ref. 9 and are the only ones that do not vanish in flat space. They are nonlinear.