arXiv 0802.39482008-02math.COhep-thmath.AGmath.RT

三维着色Young图的Generating functions和轨道折叠的Donaldson-Thomas不变量

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

Benjamin Young, Jim Bryan

备注：: 38 pages, 10 figures. v2:fixed errors in defn 1.1, defn of e_n and h_n, the first displayed equation in the appendix (by Jim Bryan), and the sign for orbifold DT invariants of [C3/Z_3] in Remark A.5. Added example in section 3, updated bibliography, and made slight clarifications and changes in wording throughout
发表信息：: Duke Math. J. 152, no. 1 (2010), 115-153
DOI：: 10.1215/00127094-2010-009

摘要：

我们推导出两个多元生成函数的三维杨氏图（也称为平面分区）。变量对应于根据SO（3）的有限阿贝尔子群G的盒子的着色。我们使用Okounkov-Reshetikhin-Vafa的顶点算子方法为简单的情况G = Z/n;处理相当困难的情况G=Z/2 x Z/2，我们还将使用作者最近的q-枚举金字塔分区的改进。在附录中，我们将图母函数与轨道C^3/G的Donaldson-Thomas分拆函数联系起来。我们发现了轨道的Donaldson-Thomas分拆函数和它的G-Hilbert方案分辨率之间的关系。本文对满足Hard Lefschetz条件的Donaldson-Thomas定域轨道理论提出了一个可信的归结猜想。

英文摘要：

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

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摘要：

英文摘要：

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