Analytic methods in deformation quantization

变形量子化是量子力学中“逆问题”的数学化身：如何将一个经典系统与一个以给定的经典系统作为其大尺度极限的量子力学系统联系起来？由于量子力学的结构更丰富，人们通常不期望这个问题有单一的解，因此很自然地将所有可能的量子化的集合视为一个家族，由它们共同的经典极限联合起来。虽然受到物理学的启发，但这个“量子化问题”在纯数学中产生了令人着迷的问题和发现：量子系统自然会产生非交换代数（因为观察者不能与系统相互作用而不影响它），而经典系统是交换的，因此可以自然地被视为几何空间上的函数。如果一个非交换代数在数学上可以实现为量子系统，那么计算它的经典极限会产生一个几何对象，它可能仍然反映了原始系统的许多重要性质。因此，它产生了一个桥梁之间的代数和几何已被证明是非常有启发性的。Kontsevich的著名工作在20世纪90年代表明，每一个泊松结构（即数学对象相关的经典力学系统）有一个量化和实际上如何理解家庭的可能量化。这导致了爆炸的活动在该地区，而Kontsevich工作的背景下，微分几何，适当的类似物，他的结果在代数几何现在的理解。这导致了几何表示论的许多方面的一个美丽的重铸：著名的贝林森-伯恩斯坦局部化定理可以被看作是一个关于旗簇的余切丛的量子化的陈述，旗簇是一个辛簇，它的性质是它的仿射奇异性的分解。在这种情况下，一个以前似乎是旗帜品种的一个非常不寻常的性质的定理成为一个令人兴奋的新领域“辛表示论”的动机。上述所有的发展都集中在“形式”量子化：变形参数是形式变量的变形。然而，我们也可以考虑收敛问题，也就是说，研究量子化和形式量子化的分析性质。在过滤量子化的相关设置中，这已经通过解析微局部化理论进行了研究，至少在复流形的余切丛的情况下。然而，这个项目将寻求在一些新的背景下研究分析量子化：例如，在任意辛（或更雄心勃勃的泊松）簇的背景下发展理论已经很有趣了，然而，在非阿基米德设置中研究分析技术也同样很自然，也就是说，在刚性解析几何的背景下。在如何实施趋同要求方面，存在着更丰富的可能性。最近的工作Ardakov和其他刚性解析D-模，这可以被认为是一种过滤量化的余切丛如上所述，证据表明，这是一个富有成果的研究领域。在变形量子化的更一般背景下寻求理解这些结果的一个自然目标是理解有限W-代数在刚性解析环境中的局部化，遵循Dodd-Kremnitzer在形式环境中的工作。这项工作将适合EPSRCs在代数，几何和数论领域的兴趣。

Deformation quantization is a mathematical incarnation of the "inverse problem" in quantum mechanics: how can one associate to a classical system a quantum mechanical one which has the given classical system as its large-scale limit? Because of the richer structure in quantum mechanics, one does not normally expect this problem to have a single solution, and it is thus natural to consider the collection of all possible quantizations as a family, united by their common classical limit. Though prompted by physics, this "quantization problem" has produced fascinating questions and discoveries in pure mathematics: Quantum systems naturally give rise to noncommutative algebras (as an observer cannot interact with the system without effecting it) whereas classical systems are commutative and hence can naturally be viewed as functions on a geometric space. If one can realize a noncommutative algebra in mathematics as quantum system, then computing its classical limit yields a geometric object which is likely to still reflect many important properties of the original system. It thus produces a bridge between algebra and geometry which has proved to be very illuminating.Kontsevich's famous work in the 1990s showed that every Poisson structure (I.e mathematical object associated to classical mechanical system) has a quantization and indeed how to understand the family of possible quantizations. This led to an explosion of activity in the area, and while Kontsevich worked in the context of differentiable geometry, the appropriate analogues of his results in algebraic geometry are now understood. This has led to a beautiful recasting of many aspects of geometric representation theory: the celebrated Beilinson-Bernstein localization theorem can be viewed as a statement about the quantization of the cotangent bundle of a flag variety, a symplectic variety which has the property that it is a resolution of singularities of its affinization. Placed in this setting, a theorem which had previously seemed a very unusual property of flag varieties became the motivation for an exciting new area of "symplectic representation theory". All the developments above have focused on "formal" quantizations: deformations where the deformation parameter is formal variable. One can also, however, consider convergence questions, that is, study analytic properties of quantizations as well as formal ones. In the related setting of filtered quantizations, this has been studied via the theory of analytic microlocalization, at least in the case of cotangent bundles of complex manifolds. This project however would seek to study analytic quantization in a number of new contexts: for example, it would already be interesting to develop the theory in the context of an arbitrary symplectic (or more ambitiously Poisson) variety, However, it would equally be natural to investigate analytic techniques in the non-Archimedean setting, that is, in the context of rigid analytic geometry. Here there is a richer array of possibilities in how one can impose convergence requirements. Recent work of Ardakov and others on rigid analytic D-modules, which can be thought of as a kind of filtered quantisation of cotangent bundles as mentioned above, gives evidence that this is a fruitful area of research. One natural goal in seeking to understand these results in the more general context of deformation quantisation would be to understand the localisations of finite W-algebras in the rigid analytic setting, following the work of Dodd-Kremnitzer in the formal setting. This work will fit into EPSRCs areas of interest in Algebra, Geometry and Number theory.