Conformally invariant powers of the ambient Dirac operator

Recent work in even-dimensional conformal geometry [1],[3], [4], [6] has revealed the importance of conformally invariant powers of the Laplacian on a conformal manifold; that is, of operators Pk whose principal part is the same as ∆ k with respect to a representative of the conformal structure. These invariant powers of the Laplacian were first defined in [5] in terms of the Fefferman-Graham [2] ambient Lorentzian structure. The present paper defines conformally invariant powers of the Dirac operator by generalizing the construction of [5]. The authors’ primary motivation in introducing this generalization is as follows. One of the primary objects of study for conformal geometry in recent years has been Branson’sQ-curvature, which appears in higher dimensional conformal geometry as a natural analog of the scalar curvature in two dimensions. However, the Q-curvature is an elusive object to define. It was first defined in [1] via analytic continuation in a dimensional parameter, later redefined in [6] in terms of analytic continuation in a spectral parameter, defined directly in terms of formal asymptotics in [3], and computed explicitly by [4] using conformal tractor calculus. Roughly speaking, the Q curvature is a zeroth order approximation to the operator Pn/2 where n is the dimension of the manifold. It is natural to ask whether the Q curvature can be generalized to odd dimensions. Fefferman and Graham [3] give one generalization to odd dimensions using pseudodifferential operators ∆. The disadvantage of this approach is that it allows one to define Q only as a global object. However, there is another classical way to take the square root of the Laplacian, namely to consider the Dirac operator %∂. The original purpose of the present paper was to see if there was any way to approach the problem of defining the Q curvature in odd dimensions using a Dirac-type operator. The authors’ original interest in this problem stems from a variational problem in relativity. Frauendiener and Sparling [7] have attempted to find a Lagrangian L4(g) for the conformal field equations on M =M , and to derive the conformal field equations by variation of the action