Triangulated categories of singularities and D-branes in Landau-Ginzburg models
Triangulated categories of singularities and D-branes in Landau-Ginzburg models
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发表时间：
2003-02
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通讯作者：
Dmitri Orlov
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作者：
Dmitri Orlov
尽管标题中有物理术语，但这篇论文是纯数学的。其目的是引入与代数变体奇点相关的三角化范畴，并在Landau-Ginzburg模型中建立这些范畴与d膜的联系。似乎有两种不同类型的范畴可以与奇点（或映射的奇点）相关联。第一种类型的范畴与消失循环有关，并与[24]中引入的关于辛Picard-Lefschetz铅笔的范畴密切相关。第二类范畴是纯代数的，来自相干束的派生范畴。这类分类将是这项工作的中心。这里的一个重要概念是完美复合体的概念，它是在2010年引入的。完美复形是一个局部拟同构于有限型局部自由轴的有界复形（一个很好的参考是[25]）的轴的复形。对任何代数变量X，都可以附加相干束的有界派生范畴D（coh(X)）。这一类允许三角形结构。相干束的派生范畴有一个由完美配合物构成的三角子范畴Perf(X)。如果品种X是光滑的，则任何相干束都具有有限型局部自由束的有限分辨率，并且完美配合物的子范畴与整个D（coh(X)）重合。但对于奇异变种，这个性质就不满足了。我们引入了奇异点的三角化范畴DSg(X)的概念，作为由完美复形的完全三角化子范畴Perf(X)的三角化范畴D（coh(X)）的商。范畴DSg(X)反映了X的奇点性质，并且“不依赖于X的全部”。例如，我们证明了它对于Zariski拓扑中的局部化是不变的（命题1.14）。当X为Gorenstein时，DSg(X)类具有良好的性质。在这种情况下，如果奇点轨迹是完备的，那么所有物体之间的Hom轨迹都是有限维向量空间（推论1.24）。这类范畴的研究受到了同调镜像对称猜想（[21]）的启发。拓扑弦理论的研究工作主要关注具有Calabi-Yau目标空间的N=2超共形σ模型的情况。在这种情况下，场论有两个拓扑扭曲的版本：模型和b模型。相应的d膜称为a膜和b膜。镜像对称应该是这两类d膜的交换。从数学的观点来看，Calabi-Yau上的b膜范畴是其上相干束（[21],[7]）的派生范畴。作为Calabi-Yau流形上a -膜的候选范畴，所谓的Fukaya范畴已被提出。它的对象
In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models It seems that two different types of categories can be associated with singularities (or singularities of maps). Categories of the first type are connected with vanishing cycles and closely related to the categories which were introduced in [24] for symplectic Picard-Lefschetz pencils. Categories of the second type are purely algebraic and come from derived categories of coherent sheaves. Categories of this type will be central in this work. An important notion here is the concept of a perfect complex, which was introduced in [3]. A perfect complex is a complex of sheaves which locally is quasi-isomorphic to a bounded complex of locally free sheaves of finite type (a good reference is [25]). To any algebraic variety X one can attach the bounded derived category of coherent sheaves D(coh(X)) . This category admits a triangulated structure. The derived category of coherent sheaves has a triangulated subcategory Perf(X) formed by perfect complexes. If the variety X is smooth then any coherent sheaf has a finite resolution of locally free sheaves of finite type and the subcategory of perfect complexes coincides with the whole of D(coh(X)). But for singular varieties this property is not fulfilled. We introduce a notion of triangulated category of singularities DSg(X) as the quotient of the triangulated category D(coh(X)) by the full triangulated subcategory of perfect complexes Perf(X) . The category DSg(X) reflects the properties of the singularities of X and ”does not depend on all of X ”. For example we prove that it is invariant with respect to a localization in Zariski topology (Proposition 1.14). The category DSg(X) has good properties when X is Gorenstein. In this case, if the locus of singularities is complete then all Hom’s between objects are finite-dimensional vector spaces (Corollary 1.24). The investigation of such categories is inspired by the Homological Mirror Symmetry Conjecture ([21]). Works on topological string theory are mainly concerned with the case of N=2 superconformal sigmamodels with a Calabi-Yau target space. In this case the field theory has two topologically twisted versions: Amodels and B-models. The corresponding D-branes are called A-branes and B-branes. The mirror symmetry should interchange these two classes of D-branes. From the mathematical point of view the category of B-branes on a Calabi-Yau is the derived category of coherent sheaves on it ([21],[7]). As a candidate for a category of A-branes on Calabi-Yau manifolds so-called Fukaya category has been proposed. Its objects