Formula for the projectively invariant quantization on degree three (Q2748295)

Let \(M\) be a \(n\)-dimensional manifold and \(\nabla \) a fixed linear connection on \(M\). An explicit formula for the projectively invariant quantization map between Pol\(^3_{\delta }(T^*M)\), the space of symbols of degree three, and \({\mathcal D}^3_{\lambda ,\mu}(M)\), the space of third-order linear differential operators, is given. Both spaces are viewed as modules over Diff\((M)\), the group of diffeomorphisms, and Vect\((M)\), the Lie algebra of vector fields on \(M\). An important remark is that for \(\delta=\frac{n+3}{n+1}, \frac{n+4}{n+1}, \frac{n+5}{n+1}\) the quantization map is not unique.