Twisted Fock Representations of Noncommutative K\"ahler Manifolds

We introduce twisted Fock representations of noncommutative K\"ahler manifolds and give their explicit expressions. The twisted Fock representation is a representation of the Heisenberg like algebra whose states are constructed by acting creation operators on a vacuum state. "Twisted" means that creation operators are not hermitian conjugate of annihilation operators in this representation. In deformation quantization of K\"ahler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the K\"ahler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, we construct the twisted Fock representation of noncommutative K\"ahler manifolds and give a dictionary to translate between the twisted Fock representations and functions on noncommutative K\"ahler manifolds concretely.