In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a principal fiber bundle and show how the deformation of associated vector bundles can be obtained.
In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative) Poisson algebras. Next, we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.
本文利用Ballesteros和Musso在“Quantum Algebras as Quantizations of Dual Poisson-Lie Groups”[J. Phys. A：Math.Theor.，46（2013），195203]。我们得到了一类QUE代数，它对应于其中两个形变参数趋近于零的$G_{NC}^{α,β,γ}$，这类QUE代数与量子Heisenberg群的已有结果一致.最后，我们对偶的基础QUE代数得到一个表达式的光滑函数之间的潜在的星积$G_{NC}^{α,β,γ}$。
In this paper we shall show the equivalence of algebraic and analytic localisation for algebras of smooth deformation quantization for several situations. The proofs are based on old work by Whitney, Malgrange and Tougeron on the commutative algebra of smooth function rings from the 60's and 70's.
