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.

该项目将研究在数学和物理中先验不同情况下出现的各种不变量的跨墙公式。在数学上，这些不变量通常被描述为某些模空间的虚拟欧拉特征。越墙现象与参数空间中存在实余维一“墙”有关，其中不变量会跳跃。对于 Donaldson-Thomas 不变量，墙位于适当 Calabi-Yau 类别上的 Bridgeland 稳定性条件的模空间中。类似的墙也出现在箭袋和簇代数的表示理论中。在镜像对称中，壁对应于由 SYZ 拉格朗日纤维的环面纤维界定的伪全纯圆盘数量的跳跃。在物理学的超对称规范理论中，BPS 状态的数量会跨越“边际稳定性之墙”。因此，Donaldson-Thomas 不变量的 Kontsevich-Soibelman 穿墙公式出现在有关 4 维超对称理论和超对称黑洞的向量多重组的模空间等主题的物理文献中。由于这些不同的跨墙公式看起来非常相似，因此人们可以要求一种共同的形式主义。 FRG 的目的是研究潜在的“跨墙结构”，并证明上述相似之处并非巧合，而是反映了深层的基础理论。数值量原则上取决于各种参数实际上对于一般参数值来说是恒定的（它们是“不变量”），但沿着参数空间中的某些“墙”跳跃。跨墙公式定量地描述了这些“跳跃”。由于与数学和物理的许多不同领域的相关性，穿墙这一主题最近成为一个非常活跃的主题。该项目的目的是严格发展“穿墙结构”的概念，并将其应用于出现穿墙公式的新旧问题。该项目产生的结果将受到数学界和物理学界的需求。 FRG 还将围绕这一协调努力建立一个研究社区，包括初级和高级研究人员的混合、研究生的培训机会以及组织多个研讨会。