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Fromula for the Projectively Invariant Quantization on Degree Three

We give an explicit formula for the projectively invariant quantization map between the space of symbols of degree three and the space of third-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on a manifold.


Introduction
Let M be a manifold of dimension n. Fix an affine connection ∇ on M. Denote by F λ (M ) the space of λ−densities on M (i.e. sections of the bundle (∧ n T * M ) ⊗λ ). This space admits naturally a structure of module over the group of diffeomorphisms Diff(M ) and the Lie algebra of vector fields Vect(M ). Consider D λ,µ (M ) the space of linear differential operators acting from F λ (M ) to F µ (M ). This space is a module over Diff(M ) and Vect(M ) (see [1,3,6,7]). The action is given as follows: take f ∈ Diff(M ) and A ∈ D λ,µ (M ) then where f * λ is the standard action of a diffeomorphism on F λ (M ). Differentiating the action of the flow of a vector field, one gets the corresponding action of Vect(M ).
Denote by D 3 λ,µ (M ) the space of third-order linear differential operators endowed with the structure of module (1.1). The module D 3 λ,µ (M ) is viewed as a submodule of D λ,µ (M ). Consider now Pol(T * M ) the space of functions on the cotangent bundle T * M, polynomials on the fibers. This space is naturally isomorphic to the space of symmetric contravariant tensor fields on M. One can define a one-parameter family of Diff(M )−modules by taking Pol δ (T * M ) := Pol(T * M ) ⊗ F δ (M ). Let us give explicitly this action: take f ∈ Diff(M ) and P ∈ Pol δ (T * M ) then where f * is the natural action of a diffeomorphism on contravariant tensor fields, and J f is the Jacobian of f. Differentiating the action of the flow of a vector field, one gets the corresponding action of Vect(M ).
Denote by Pol 3 δ (T * M ) the space of symbols of degree three endowed with the module structure given by (1.2).
Suppose M := R n is endowed with a flat projective structure (coordinates change are projective transformations). In this case, Lecomte and Ovsienko in [6] construct a quantization map between the space Pol δ (T * R n ) and the space D λ,µ (R n ), equivariant with respect to the action of the Lie algebra sl n+1 (R) ⊂ Vect(R n ). Consider now any manifold M and fix an affine connection on it. It is interesting to ask if there exists a canonical quantization map associated to the given connection. On degree two, the author construct in [1] a quantization map depending only on the projective class of the connection (see also [3] for the conformal case). This approach generalizes Lecomte and Ovsienko's approach for the flat case. On higher order, the problem of existence of the projectively invariant quantization map is open.

Main result
The main result of this note is Theorem 2.1 For n > 1, and for δ = n+3 n+1 , n+4 n+1 , n+5 n+1 , there exits a quantization map Q : where R ij are the components of the Ricci tensor of the connection ∇, the constants α, β 1 , β 2 , η 1 , η 2 , η 3 , are given by and have the following properties (i) It depends only on the projective class of the connection ∇ (see [5]).
(ii) If M = R n is endowed with a flat projective structure the map (2.1) is the unique map that preserves the principal symbols, equivariant with respect to the action of the Lie algebra sl n+1 (R) ⊂ Vect(R n ).
Proof. Let us give an idea of the proof. Let∇ be another connection projectively equivalent to ∇. Denote by Q∇ the quantization map (2.1) written with the connectioñ ∇. We have to prove that Q∇ = Q ∇ . We need some formulae. Since∇ is projectively equivalent to ∇ there exists a 1-form ω on M such that the Christoffel symbols of the connections ∇ and∇ are related bỹ (see [5]). It follows that, for any φ ∈ F λ , one has ∇ k φ =∇ k φ + λ(1 + n)ω k . In the same manner we can express the tensors ∇ i ∇ j φ, ∇ i ∇ j ∇ k φ, with the tensors∇ i∇j φ,∇ i∇j∇k φ, respectively. Using also formula (2.2) one has ∇ i P ijk =∇ i P ijk + ((1 + n)δ − (n + 5))ω i P ijk . In the same manner we can express the tensors ∇ j ∇ i P ijk , ∇ k ∇ j ∇ i P ijk , R ij , ∇ k R ij , with the tensors∇ j∇i P ijk ,∇ k∇j∇i P ijk ,R ij ,∇ kRij . Replacing now these formulae into (2.1), we see that Q ∇ = Q∇ if and only if the constants α, β 1 , β 2 , η 1 , η 2 , η 3 are given as above.
To prove part (ii), recall that the Lie algebra sl n+1 (R) can be identified with the Lie sub-algebra of Vect(R n ) generated by the vector fields where (x i ) is the coordinates of the projective structure. The proof now is a simple computation (see [6]).
For the particular values of δ : , there exists a quantization map given by (2.1) with particular values of λ, µ, given in the following table Here t is a parameter.

Remark 2.3 (i)
For the particular values of δ, the quantization map (2.1) is not unique (it is given by the parameter t).
(ii) In the one dimensional case, the quantization map was given in [2,4].
(iii) Another approach to the quantization map equivariant with respect to the action of the conformal group in a Riemannian manifold was given in [3,7].