Geometry of Non-abelian Hodge Structures

托尼·潘特夫使用阿贝尔和非阿贝尔霍奇理论的技术来处理具体的几何问题以及宇宙学、数学物理学和弦理论中的问题。研究者和他的合作者研究了四个问题。 第一个描述了一个自然的霍奇理论准则，用于决定是否一个给定的同伦类型的基础上光滑的射影簇。第二个问题是两个球面上的四个流形的辛拓扑。第三个方案设计了Gerby K3曲面上层的显式构造，并讨论了它与场的非对易变形的物理概念的关系。第四个项目提出了利用二级Abel-Jacobi映射在Eucledian $D$-膜的模空间上构造特殊坐标的方法，对这些问题的理解对于统一代数几何、辛拓扑和数学物理中的各种线性化过程是必不可少的。它使我们更接近于理解代数簇的基本结构，并增强了建立弦理论真空的显式模型的技术的几何武器库。代数几何不仅是现代数学中研究最深入的部分之一，也是最近从弦论物理学中涌现出来的革命性数学思想的重要试验场。研究者的研究为我们目前对弦对偶性的理解所产生的困难的处理奠定了基础，并扩大了理解弦物理学的可用工具库。研究者的方法适用于天体物理学、量子引力和弦理论中的广泛具体问题。

Tony Pantev uses techniques from both the abelian and non-abelian Hodge theory to approach concrete geometric questions as wellas problems in cosmology, mathematical physics and string theory. The investigator and his collaborators study four problems. The first onedescribes a conjectural Hodge theoretic criterion for deciding whethera given homotopy type underlies a smooth projective variety. Thesecond concerns the symplectic topology of four manifolds, fiberedover the two sphere. The third projectdesigns an explicit construction of sheaves on a gerby K3 surfaceand discusses its relations to the physical notion of non-commutativedeformation of fields. The fourth project proposes an explicitconstruction of the special coordinates on the moduli space ofEucledian $D$-branes by means of secondary Abel-Jacobi maps.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology and mathematical physics. It brings us closer to understanding the basic structure of algebraic varieties and enhances the geometric arsenal of techniques for building explicitmodels of string theory vacua.This is a research in the field of algebraic geometry. Algebraicgeometry is not only one of the most intensively researched partsof modern mathematics, but also a crucial testing ground for therecent influx of revolutionary mathematical ideas coming from stringtheory physics. The investigator's research sets the stage for arigorous treatment of the hard conjectures generated by ourcurrent understanding of string dualities and enlarges the arsenal ofavailable tools for understanding the physics of strings. Theinvestigator's methods are applicable to a wide range of concreteproblems in astrophysics, quantum gravity and string theory.