The quantised McKay correspondence

设X是一个奇异代数簇，Y是它的奇点的分解：一个基本（和中心）的例子是X是一个克莱因曲面奇点，Y是它的最小分解。在文献中有一个有影响力的猜想（Crepant Resolution Conjecture，简称CRC），通过一个合适的解析延拓和（部分）专门化他们的论点，预测了X和Y的格零Gromov-Witten（GW）理论之间的精确关系。在过去的十年中，已经开发了一系列新的工具来研究这些几何关系：这些工具使用镜像对称性来获得对orbifolds的量子上同调的分析控制，并且它们可以有效地用于证明和改进CRC的含义。最先进的技术关键在于使用Givental型镜像对称定理来证明复曲面的轨道和完全相交，并且没有通用的技术工具来建立类似的证明复曲面的情况。特别是，克莱因奇点的中心情况，这是标记的ADE Dynkin图，是非环面的D型和E型，并为CRC成为一个高度精炼的，量子版本的经典麦凯对应，已经超出了通常的镜像对称方法。问题1.证明Gorenstein表面奇异性的镜像对称性和量子McKay对应。问题2。将此扩展到更高的属，完全一般性的全后代McKay对应。

Let X be a singular algebraic variety, and Y a resolution of its singularities: a basic (and central) example is given by X being a Kleinian surface singularity, and Y its minimal resolution. An influential conjecture in the literature (the Crepant Resolution Conjecture, or CRC in short), predicts a precise relation between the genus zero Gromov-Witten (GW) theory of X and that of Y through a suitable analytic continuation and (partial) specialisation of their arguments. In the last decade, a host of new tools have been developed for studying these conjectural relations: these tools use mirror symmetry to gain analytical control on the quantum cohomology of orbifolds, and they can be effectively employed to prove and improve the implications of the CRC. The state-of-the-art hinges crucially on the use of Givental-type mirror symmetry theorems for toric orbifolds and complete intersections therein, and there are no general technical tools to build upon a similar proof as for the toric case. In particular, the central case of Kleinian singularities, which are labelled by ADE Dynkin diagrams, are non-toric in type D and E, and for which the CRC becomes a highly refined, quantum version of the classical McKay correspondence, is already out of reach of usual mirror symmetry methods.Problem 1. Prove mirror symmetry and the quantum McKay correspondence for Gorenstein surfacesingularities.Problem 2. Extend this to a higher genus, full-descendent McKay correspondence in full generality.