量子场论和弦理论的几何应用

Mathematical foundations of quantum field theory and string theory have been developed quickly in the last decades. One breakthrough is about the rigorous renormalization/quantization method and its deep connection with geometry and topology. In the PhD thesis, the applicant uses the rigorous quantum field theory method to study the quantum gauge theory on complex manifolds, and develops a general mathematical foundation of quantization over the complex structures of Calabi-Yau manifolds. It leads to the first established example of higher genus mirror symmetry on compact Calabi-Yau manifolds. Based on this work, the applicant develops a homological renormalization method for a class of quantum gauge theories, and completely classifies their intrinsic geometric structures in low dimensions. These include the relations between one dimensional homological renormalization and index theory, as well as two dimensional homological renormalization and vertex algebras. In recent years, the applicant has been frequently invited to international conferences to present related works, including the annual meetings String-Math 2015, String 2016, String-Math 2017, String-Math 2018.

量子场论与弦理论的数学基础在近年来得到了飞速的发展,其中一个重要的突破为重整量子化方法的严格化及其与几何拓扑的深刻联系。申请人在博士论文中提出并建立了Calabi-Yau复空间上量子化的一般性数学理论,在领域中首次实现了关于紧致Calabi - Yau空间上的高亏格镜像对称。在此基础上,申请人发展了一类规范场同调量子化方法，完整刻画了低维空间上同调量子化的几何结构，其中包括一维同调量子化和指标定理，以及二维同调量子化与顶点算子代数的联系。近年来申请人多次被邀请在国际会议上做大会报告介绍相关工作，包括领域中的年度大会String-Math 2015, String 2016, String-Math 2017, String-Math 2018。

量子场论与弦理论的数学基础在近年来得到了飞速的发展,其中一个重要的突破为重 整量子化方法的严格化及其与几何拓扑的深刻联系。项目主要结果包括：.1. 构造了orbifold奇点的形变理论并得到了其量子不变量的计算方法.2. 严格构造了拓扑量子力学和指标定理的联系.3. 构造了二维场论中的几何重整化方法-regularized积分技术.4. 构造了镜像对称中B模型Kodaira-Spencer引力的可积系统..研究成果多次发表在Commun. Math. Phys. 、Adv. Math. 等国际一流杂志。

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