Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds
Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds
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Calabi-Yau 3 倍相干滑轮计数不变量的全纯生成函数
DOI：
10.2140/gt.2007.11.667
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发表时间：
2006
影响因子：
2
通讯作者：
D. joyce
中科院分区：
文献类型：
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作者：
D. joyce
设X是Calabi-Yau 3-fold，T = D B（coh（X））是X上凝聚层的导出范畴，Sta B（T）是T上Bridgeland稳定性条件的复流形.证明了对于（Z，P）∈ Stab（T）和a ∈ K（T），可以定义不变量J α（Z，P）eQ，推广了Donaldson-Thomas不变量，它“计数”X上的（Z，P）-半稳定相干层（复体），并且它在（Z，P）变化下的变换规律是已知的.本文说明了如何将这样的不变量J α（Z，P）（如果它们存在的话）联合收割机组合成一族全纯生成函数F α：Stab（T）→ C，其中a ∈ K（T）.令人惊讶的是，要求F α是连续的和全纯的决定了它们本质上是唯一的，并意味着它们满足p.d.e.，它可以解释为Stab（T）上的联络的平坦性，其值在无限维李代数L中。作者认为，在这个数学的基础上，应该有一些新的物理学，如弦论和镜像对称。弦理论家被邀请来解决和解释这个新物理学。
Let X be a Calabi-Yau 3-fold, T = D b (coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions on T. It is conjectured that one can define invariants J α (Z, P) e Q for (Z, P) ∈ Stab(T) and a ∈ K(T) generalizing Donaldson-Thomas invariants, which "count" (Z, P)-semistable (complexes of) coherent sheaves on X, and whose transformation law under change of (Z, P) is known. This paper explains how to combine such invariants J α (Z, P), if they exist, into a family of holomorphic generating functions F α : Stab(T) → C for a ∈ K(T). Surprisingly, requiring the F α to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T) with values in an infinite-dimensional Lie algebra L. The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.