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

这个项目将研究一大类不变量的跨墙公式，这些不变量在数学和物理中出现在先验的不同情况下。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。穿墙现象与参数空间中实余维一“墙”的存在有关，不变量在参数空间中跳跃。在Donaldson-Thomas不变量的情况下，壁生活在适当的Calabi-Yau范畴上的Bridgeland稳定性条件的模空间中。类似的墙也出现在箭图和簇代数的表示理论中。在镜像对称中，壁对应于被Syz Lagrangian纤维的环面纤维包围的伪全纯圆盘数量的跳跃。在物理学中的超对称规范理论中，BPS态的数量跨越了“边缘稳定墙”。因此，Kontsevich-Soibelman关于Donaldson-Thomas不变量的穿墙公式出现在物理文献中，主题包括四维超对称理论的向量多重模空间和超对称黑洞。由于这些不同的跨墙公式看起来如此相似，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“穿墙结构”，并证明上述相似之处不是巧合的，而是反映了一个深层次的理论。在数学和物理中经常遇到的情况是，原则上依赖于各种参数的数值对于一般参数值来说实际上是恒定的(它们是“不变量”)，但在参数空间中沿着某些“墙”跳跃。穿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相关，越墙问题最近成为一个非常活跃的主题。本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。这一项目产生的结果将受到数学界和物理界的需求。联邦德国政府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，为研究生提供培训机会，并组织几个讲习班。

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.