Moduli Spaces and String Theory

这一建议将在Calabi-Yau流形上传播的超弦的背景下分析模空间的结构，即参数空间的形式。将考虑三个模空间：N=(2，2)超共形场理论空间，它与Calabi Yau三重的模空间直接相关；N=(0，2)超共形场理论空间，它将这一概念扩展到更接近真实物理的东西，并与向量丛的模空间相关联；以及固定Calabi Yau流形上的D-膜，它与派生范畴中稳定对象的模空间相关联。超弦理论是一种基本物理模型，有望产生对基本粒子物理和时空结构的统一描述。虽然这一理论尚未与实验物理接触，但它在代数几何等数学领域的影响和成功是毋庸置疑的。这项拟议的研究打算既建立在理论物理方法和几何方法之间已建立的联系上，又寻求尚未探索的联系。这个提案中探讨的主要概念是一套所有可能的数学上一致的理论的想法。这个概念既可以用理论物理的语言来表达，也可以用数学的语言表达。然后，这些语言之间的比较往往会产生新的想法和结果。

This proposal will analyze the structure of moduli spaces, i.e., versions of parameter spaces, in the context of superstrings propagating on Calabi-Yau manifolds. Three moduli spaces will be considered: the space of N=(2,2) superconformal field theories, which relates directly to the moduli space of the Calabi Yau threefolds; the space of N=(0,2) superconformal field theories, which extends this notion to something closer to real physics and is associated with the moduli spaces of vector bundles; and D-branes on a fixed Calabi Yau manifold which is associated to the moduli space of stable objects in the derived category. Superstring theory is a model of fundamental physics expected to yield a unified description of fundamental particle physics and the structure of spacetime. While this theory has yet to make contact with experimental physics, its impact and success in field of mathematics in areas such as algebraic geometry is beyond doubt. The proposed research intends to both build on established links between methods of theoretical physics and geometry, and pursue as yet unexplored connections. The main notion explored in this proposal is the idea of a set of all possible mathematically consistent theories. This notion can be pursued in the language of either theoretical physics or mathematics. Comparisons between these languages then often yields new ideas and results.