The quantised McKay correspondence

设X是一个奇异代数变量，Y是它的奇异点的分辨率：一个基本的（和中心的）例子是X是一个Kleinian曲面奇异点，Y是它的最小分辨率。文献中有一个很有影响力的猜想（Crepant Resolution conjecture，简称CRC），通过对X和Y的格零格罗莫夫-威腾（GW）理论进行适当的解析延拓和（部分）专一化，预测了它们之间的精确关系。在过去的十年中，人们开发了许多新的工具来研究这些猜想关系：这些工具使用镜像对称来获得对轨道的量子上同调的分析控制，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对环面轨道和其中的完全相交使用给定型镜像对称定理，并且没有通用的技术工具来建立与环面情况类似的证明。特别是，被ADE Dynkin图标记的Kleinian奇点的中心情况，在D型和E型中是非环面的，并且CRC成为经典McKay对应的高度精炼的量子版本，已经超出了通常镜像对称方法的范围。问题1。证明戈伦斯坦表面奇点的镜像对称和量子麦凯对应。问题2。将其推广到一个更高级的属，全后裔McKay对应的完全一般。

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.