Twisted Fock representations of noncommutative Kähler manifolds

Abstract

We introduce twisted Fock representations of noncommutative Kähler 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 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, we construct the twisted Fock representation of noncommutative Kähler manifolds and give a dictionary to translate between the twisted Fock representations and functions on noncommutative Kähler manifolds concretely.

Hermitian-Einstein metrics from noncommutative U(1) instantons

“…There is a one to one correspondence between F and some subalgebra of C ∞ C 2 . For arbitrary noncommutative Kähler manifold obtained by deformation quantization with separation of variables [9], we can find the similar correspondence [8]. The following is the simplest example of the correspondence.…”

“…It is known from (4.4) and Proposition 4.2 how to translate the operators into functions. (See also Appendix B.2 and [8].) Then the k-instanton curvature tensor is expressed by concrete elementary functions as follows:…”

“…For this reason we made Ricci-flat metrics from instantons on F instead of the noncommutative C 2 described as a formal power series, in this article. Here F is a noncommutative algebra constructed in Section 4 following the method in [8] [44]. The product of the algebra F is given by the star product (A.4).…”

“…The Fock space is defined by Heisenberg algebra generated by noncommutative complex coordinates. There is a dictionary between the linear operators acting on the Fock space and usual functions [8]. The dictionary is applicable for arbitrary noncommutative Kähler manifold obtained by deformation quantization with separation of variables [9].…”

Noncommutative deformations of locally symmetric Kähler manifolds

“…Using new star products with separation of variables for Riemann surfaces given in this article, we can make such kind of Fock representation, similarly. Indeed, the recipe to construct the twisted Fock representation is already constructed for general Kähler manifolds in [30]. The second comment is about the star products for general Grassmann manifolds.…”