Landau-Ginzburg模型上的稳定条件

在这个项目中，申请人将研究一类特殊的三角范畴，这种范畴具有[2]=[0]的特殊性质。[2]=[0]这种现象在数学领域中普遍存在，我们可以在诸如数学物理（矩阵分解），几何表示论（Treumann引进的Smith理论的层化），K理论（Bott周期律，周期循环上同调），和数论（循环群的Tate上同调）等分支中发现此类现象。..申请人将主要研究矩阵分解构成的范畴。矩阵分解最初被Eisenbud发现，之后物理学家根据Kontsevich的提议用它们来描述Landau-Ginzburg模型上的D膜。申请人将试着应用Bridgeland的想法，在此类范畴上给出一个S^1分次的概念。这会给出Landau-Ginzburg模型上稳定条件的定义。之后申请人会研究不同稳定条件下的半稳定对象的模空间的性质。

In this project, the applicant will study a special kind of triangulated categories, which has the miraculous property [2]=[0]. This phenomenon of [2]=[0] is ubiquitous in mathematics, we can find such ones in mathematical physics (matrix factorizations), geometric representation theory (the sheafification of Smith theory introduced by Treumann), K theory (Bott's periodicity, periodic cyclic cohomology), and number theory (Tate cohomology of cyclic groups) etc...The applicant will mainly focus on categories of matrix factorizations. Matrix factorizations were firstly discovered by Eisenbud, then physicists used them to describe the D-branes in Landau-Ginzburg models by following a proposal of Kontsevich. The applicant will try to adapt Bridgeland’s idea to define S^1 gradings on such categories. This will give us the notion of stability conditions on Landau-Ginzburg models. Then the applicant will try to investigate the properties of moduli spaces of semi-stable objects under various stability conditions.