Deformation Quantization of Kähler Manifolds and Their Twisted Fock Representation

Abstract

We introduce the notion of twisted Fock representations of noncommutative Kähler manifolds and give their explicit expressions. The so-called 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 deformation quantization of Kähler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the Kähler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, the twisted Fock representation of noncommutative Kähler manifolds is constructed.