Curve counting via stable pairs in the derived category
Curve counting via stable pairs in the derived category
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DOI：
10.1007/s00222-009-0203-9
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发表时间：
2009-11-01
影响因子：
3.1
通讯作者：
Thomas, R. P.
中科院分区：
文献类型：
--
作者：
Pandharipande, R.;Thomas, R. P.
对于一个非奇异射影三维簇\(X\)，我们定义了一些整数不变量，它们实际上是对\((C,D)\)进行计数，其中\(C\subset X\)是一条嵌入曲线，\(D\subset C\)是一个除子。通过将一对\((C,D)\)视为\(X\)的导出范畴中的一个对象，在相关的模空间上构造了一个虚拟类。推测经过通用变换后，所得的不变量与\(X\)的格罗莫夫 - 温特（Gromov - Witten）理论和唐纳森 - 托马斯（Donaldson - Thomas）理论是等价的。对于卡拉比 - 丘三维簇，后一种等价性应被视为导出范畴中的一个壁穿越公式。我们进行了一些新不变量的计算。在法诺（Fano）情形下，找到了非奇异嵌入曲线的局部贡献。在局部环面卡拉比 - 丘情形下，描述了一种全新形式的拓扑顶点。对\((C,D)\)对的虚拟计数与戈帕库玛（Gopakumar）和瓦法（Vafa）的BPS态计数所依据的几何密切相关。我们证明了我们对格罗莫夫 - 温特不变量的整性预测与BPS整性是一致的。反之，BPS几何对\((C,D)\)对的计数施加了很强的条件。
For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where CaS,X is an embedded curve and DaS,C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category.Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.