Explicit formulas for noncommutative deformations of ${\mathbb C}{P^N}$CPN and ${\mathbb C}{H^N}$CHN

Abstract

We give explicit expressions of a deformation quantization with separation of variables for CP N and CH N . This quantization method is one of the ways to perform a deformation quantization of Kähler manifolds, which is introduced by Karabegov. Star products are obtained as explicit formulae in all order in the noncommutative parameter. We also give the Fock representations of the noncommutative CP N and CH N .

Noncommutative deformations of locally symmetric Kähler manifolds

“…In [27] explicit expressions of star products of noncomutative CP N and CH N were provided as the deformation quantization with separation of variables. 1 The reason that they were provided is that CP N and CH N are locally symmetric spaces.…”

“…The symmetry makes our problems be simple ones. In [27], star products on general locally symmetric Kähler manifolds are also discussed, but it is not enough to obtain explicit expression of the star products.…”

“…The purpose of this article is to derive algebraic recurrence relations to make concrete expression of star products on locally symmetric Kähler manifolds. The method in this article overcome the problems in [27]. Using the recurrence relations, star products on some locally symmetric Kähler manifolds, the Riemann surfaces and CP N , are obtained.…”

Twisted Fock representations of noncommutative Kähler manifolds

“…,z N ) to Φ(z,z) for convenience. In [22], it is shown that e −Φ/ corresponds to a vacuum projection operator | 0 0| for the noncommutative CP N . We extend this statement for general Kähler manifolds.…”

“…Example 4 : Fock representation of noncommutative CH N Here, we give an explicit expression of the Fock representation of noncommutative of CH N [22,23]. We choose a Kähler potential satisfies the condition (3.14)…”