FRG: Collaborative Research: Wall-crossings in Geometry and Physics

FRG：合作研究：几何和物理的跨越

• 批准号: 1265230
• 负责人: Ludmil Katzarkov
• 金额: $ 20.02万
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265230&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Wall; crossings

This project will investigate wall-crossing formulas for a wide class of invariants which appear in a priori different situations in mathematics and physics. Mathematically, those invariants are typically described as virtual Euler characteristics of some moduli spaces. The wall-crossing phenomenon is related to the presence of real codimension one "walls" in the space of parameters, where the invariants jump. In the case of Donaldson-Thomas invariants, the walls live in the moduli space of Bridgeland stability conditions on the ppropriate Calabi-Yau categories. Similar walls also occur in the theory of representations of quivers and cluster algebras. In mirror symmetry, walls correspond to jumps in the number of pseudo-holomorphic discs bounded by the torus fibers of an SYZ Lagrangian fibration. In supersymmetric gauge theories in physics, the number of BPS states jumps across "walls of marginal stability". The Kontsevich-Soibelman wall-crossing formulas for Donaldson-Thomas invariants thus occur in the physics literature on topics such as moduli spaces of vector ultiplets of 4-dimensional supersymmetric theories and supersymmetric black holes. Since these various wall-crossing formulas look so similar, one can ask for a common formalism. The aim of the FRG is to study the underlying "wall-crossing structures" and demonstrate hat the above-mentioned similarities are not coincidental, but rather reflect a deep underlying theory.It is a frequently encountered situation in mathematics and physics that numerical quantities which in principle depend on various parameters actually are constant for general parameter values (they are "invariants"), but jump along certain "walls" in the parameter space. Wall-crossing formulas describe these "jumps" quantitatively. The subject of wall-crossing has recently become a very active one due to its relevance to a number of different areas of mathematics and physics. The aim of this project is to develop the concept of "wall-crossing structure" rigorously and apply it to problems both old and new in which wall-crossing formulas appear. The results arising from this project will be in demand by both the mathematics and physics communities. The FRG will also build a research community around this coordinated effort, involving a mix of junior and senior researchers, training opportunities for graduate students, and the rganization of several workshops.

该项目将调查在数学和物理学领域的先验情况下出现的各种不变性的壁挂公式。从数学上讲，这些不变性通常被描述为某些模量空间的虚拟欧拉特征。墙壁交叉现象与在不变的参数空间中存在真实的condimension“壁”的存在有关。对于唐纳森 - 托马斯不变的，墙壁生活在布里奇兰稳定性条件的模量空间中，这是对帕拉比野的类别的。在震颤和集群代数的表示理论中也出现了类似的壁。在镜像对称性中，墙壁对应于由Syz Lagrangian振动的圆环纤维界定的伪型圆光盘数量的跳跃。在物理学上的超对称仪表中，BPS状态的数量跳过了“边际稳定墙”。因此，Kontsevich-Soibelman的唐纳森 - 托马斯不变式的kontsevich-soibelman隔离公式出现在有关主题的物理学文献中，例如4维超对称理论的矢量末端和超对比的黑洞的模量空间。由于这些各种墙壁交叉的公式看起来如此相似，因此可以要求共同的形式主义。 FRG的目的是研究潜在的“墙壁结构”，并证明上述相似性不是偶然的，而是反映了一个深层的基本理论。原则上，取决于各种参数实际上是常规参数值（它们是“不变”），但在参数空间中沿某些“壁”跳跃。隔离式公式定量地描述了这些“跳跃”。墙面交叉的主题由于与许多不同领域的数学和物理学领域相关，因此最近变得非常活跃。该项目的目的是严格地开发“隔离结构”的概念，并将其应用于出现墙壁交叉公式的新老问题。该项目产生的结果将由数学和物理社区的需求。 FRG还将围绕这项协调的工作建立一个研究社区，其中包括初级和高级研究人员，研究生培训机会以及几个研讨会的培训机会。