论文标题
$ \ mathbb {c}^3 $（第I部分）的分类和K理论Donaldson-Thomas理论
Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^3$ (part I)
论文作者
论文摘要
我们开始研究Donaldson-Thomas不变的分类，与三维仿射空间上的Hilbert方案相关，我们称之为DT类别。 DT类别被定义为非交通性希尔伯特方案的基质因素化的类别，其关键基因座是希尔伯特的点方案。本文的第一个主要结果是构建DT类别的半三相分解，这可以被视为框架三循环颤动的分类墙壁交叉公式。每个求和由某些矩阵因法化的子类别的分类霍尔产品（称为quasi-bps类别）给出。它们是Špenko-van den bergh考虑的非交通性分辨率的曲折版本的矩阵因法化的类别，并被第一作者用于证明K Weoretic Hall代数的PBW定理。我们下一个通过Koszul二元性等价构建了准BPS类别的显式对象，并表明它们在圆环本地化的K理论中构成了基础。这些计算可以被视为希尔伯特（Hilbert）点方案的麦凯（McKay）对应关系的数值k理论类似物。特别是，DT类别的曲折局部K理论的基础是其基数是平面分区的数量，从而给出了Macmahon公式的k理论类似物。
We begin the study of categorifications of Donaldson-Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space, which we call DT categories. The DT category is defined to be the category of matrix factorizations on the non-commutative Hilbert scheme with a super-potential whose critical locus is the Hilbert scheme of points. The first main result in this paper is the construction of semiorthogonal decompositions of DT categories, which can be regarded as categorical wall-crossing formulae of the framed triple loop quiver. Each summand is given by the categorical Hall product of some subcategories of matrix factorizations, called quasi-BPS categories. They are categories of matrix factorizations on twisted versions of noncommutative resolutions of singularities considered by Špenko-Van den Bergh, and were used by the first author to prove a PBW theorem for K-theoretic Hall algebras. We next construct explicit objects of quasi-BPS categories via Koszul duality equivalences, and show that they form a basis in the torus localized K-theory. These computations may be regarded as a numerical K-theoretic analogue in dimension three of the McKay correspondence for Hilbert schemes of points. In particular, the torus localized K-theory of DT categories has a basis whose cardinality is the number of plane partitions, giving a K-theoretic analogue of MacMahon's formula.
