An invariant formula for a star product with separation of variables
Let \(M\) be a pseudo-Kähler manifold with metric tensor \(g_{k\overline{l}}\). Its inverse is a Kähler-Poisson tensor, denoted by \(g^{\overline{l}k}\). Let \(\{\cdot,\cdot\}\) be the Poisson bracket corresponding to the tensor \(g^{\overline{l}k}\). A \(\ast\)-product on \(M\) is an associative product on \(C^\infty(M)\nu\) such that \(f\ast g=fg+\sum_{k\geq1}\nu^kB_k(f,g)\) where \(B_1(f,g)-B_1(g,f)=i\{f,g\}\). A \(\ast\)-product on \(M\) is called to be with separation of variables if \(B_1(f,g)=g^{\overline{l}k}\frac{\partial f}{\partial\overline{z}^l}\frac{\partial g}{\partial z^k}\). Introducing a locally defined total symbol \(\sigma(A)\) of a differential operator \(A\) on \(M\), the author provides a closed formula for the total symbol \(\sigma(L_f)\), where \(L_f(g)=f\ast g\) is the left \(\ast\)-multiplication operator with respect to deformation quantization with separation of variables on \(M\), and then proves an invariant formula with respect to such a product.
deformation quantization with separation of variables
scientific article; zbMATH DE number 6086943
scientific article