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

Emmy Noether项目的目标是发展对物理理论构建块的数学和物理理解。重点是物理理论，其内容是由卡-丘（CY）流形的变形家庭，如超对称场论和拓扑弦理论。拓扑弦理论将CY流形族与一个配分函数联系起来，该配分函数被定义为弦耦合中的微扰渐近展开。这个配分函数编码了CY流形的所有格罗莫夫-威滕不变量。这个配分函数的数学定义是通过模空间上的几何量子化问题得到的。与此同时，一个非微扰完成的拓扑弦配分函数预计编码的唐纳森-托马斯不变量的基础家庭，物理上对应的不变量表征BPS状态。后者受到稳定性条件的影响，并表现出跨壁现象。Emmy Noether项目的目标是实现对几何量子化和BPS几何的构建块以及两者之间的精确关系的数学理解。理解和解决这个问题的一般性是一个具有挑战性的，但非常有益的任务，无论是物理和数学。

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.