Weyl manifolds and deformation quantization

Noncommutative Yang-Mills from equivalence of star products

“…We are, however, interested in the existence of a natural star product for any Poisson manifold, which is guaranteed according to Kontsevitch. In this letter we technically only use the corresponding result for symplectic manifolds [16,17,18].…”

On the Dequantization of Fedosov's Deformation Quantization

“…. , where B r , r ≥ 1, are differential operators on M. The existence and classification problem for deformation quantization was first solved in the non-degenerate (symplectic) case (see [7], [21], [11] for existence proofs and [12], [19], [8], [2], [23] for classification) and then Kontsevich [18] showed that every Poisson manifold admits a deformation quantization and that the equivalence classes of deformation quantizations can be parameterized by the formal deformations of the Poisson structure. It turns out that all the explicit constructions of star-products enjoy the following property: for all r ≥ 0 the bidifferential operator C r in (1) is of order not greater than r in both arguments (the most important examples are Fedosov star-products on symplectic manifolds and the Kontsevich star-product on R n endowed with an arbitrary Poisson bracket).…”

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

“…After [1], several ways of deformation quantization were proposed [2,3,4,5]. In particular, deformation quantizations of Kähler manifolds were provided in [6,7,8,9] 1 .…”