Explicit formula of deformation quantization with separation of variables for complex two-dimensional locally symmetric Kähler manifold

We give a complex two-dimensional noncommutative locally symmetric Kähler manifold via a deformation quantization with separation of variables. We present an explicit formula of its star product by solving the system of recurrence relations given by Hara–Sako. In the two-dimensional case, this system of recurrence relations gives two types of equations corresponding to the two coordinates. From the two types of recurrence relations, symmetrized and antisymmetrized recurrence relations are obtained. The symmetrized one gives the solution of the recurrence relation. From the antisymmetrized one, the identities satisfied by the solution are obtained. The star products for C² and CP² are constructed by the method obtained in this study, and we verify that these star products satisfy the identities.