Building Blocks of Physical Theories from the Geometry of Quantization and BPS States

艾美诺特项目的目标是发展对物理理论基础的数学和物理理解。重点是物理理论，其内容由Calabi-Yau(CY)流形的变形族决定，如超对称场理论和拓扑弦理论。拓扑弦理论将配分函数与一族CY流形联系起来，定义为弦耦合中的微扰渐近展开。这个配分函数编码了CY流形的所有亏格Gromov-Witten不变量。这种配分函数的数学定义是通过模空间上的一个几何量子化问题得到的。同时，拓扑弦配分函数的非微扰补全有望编码基础族的Donaldson-Thomas不变量，物理上对应于表征BPS状态的不变量。后者受稳定条件的制约，表现出穿墙现象。艾美诺特项目的目标是实现对几何量化和BPS几何的构建块以及两者之间的精确关系的数学理解。从总体上理解和解决这个问题是一项具有挑战性的任务，但在物理和数学上都是非常有意义的。

The goal of the Emmy Noether project is to develop a mathematical and physical understanding of the building blocks of physical theories. The focus is on physical theories, whose content is governed by deformation families of Calabi-Yau (CY) manifolds, such as supersymmetric field theories and topological string theory. Topological string theory associates to a family of CY manifolds a partition function, which is defined as a perturbative asymptotic expansion in the string coupling. This partition function encodes all genus Gromov-Witten invariants of the underlying CY manifold. The mathematical definition of this partition function is obtained through a geometric quantization problem on the moduli space. At the same time, a non-perturbative completion of the topological string partition function is expected to encode the Donaldson-Thomas invariants of the underlying family, physically corresponding to invariants characterizing BPS states. The latter are subject to stability conditions and exhibit wall crossing phenomena. The goal of the Emmy Noether project is to achieve a mathematical understanding of the building blocks of both geometric quantization and BPS geometries and the precise relation between the two. Understanding and solving this problem in generality is a challenging yet very rewarding task both physically and mathematically.