Moduli Spaces of Higgs Bundles, Hermitian-Yang-Mills Connections, and Related Topics

模空间的概念在几何学和物理学中占有越来越重要的地位。事实证明，它在机器人等某些应用领域也很有用。模数是描述特定几何或代数结构中变化的参数。模空间的构造带来了对哪些几何结构在族中表现良好的更深层次的理解，而对模空间本身的几何分析揭示了它们所参数化的对象的不变性质。目前的项目试图扩展PI以前在某些模空间上的工作，这些空间自然地产生于基本粒子的规范理论。例如，杨-米尔斯方程是数学和理论物理的一个主要交汇点。利用Higgs丛的模空间研究了表面群表示为复李群及其非紧实形式的空间。它们出现在超对称规范理论中，在几何朗兰兹问题中也很重要。这一奖项所涵盖的研究项目将进一步加深我们对模空间的几何、解析和代数性质之间关系的理解。该奖项还支持研究生。具体目标在于与全纯丛、规范理论和模问题相关的复杂几何的三个领域。第一个继续PI关于Riemann曲面上Higgs丛的模空间的工作。重点研究了模空间的渐近结构及其拓扑性质。这与有关Hitchin模空间几何的重要猜想有关，部分源于超对称规范理论。PI将推广以前关于远离Fuchsian轨迹的Hitchin分量的压力度量的结果。他还将探索镜像对称性计算的含义，这些计算与希格斯和德罗姆模空间的莫尔斯分层的新性质有关，这是他最近工作的后续成果。第二个项目继续研究高维流形上的Hermitian-Yang-Mills联络。一个目标是更好地理解自然规范理论的紧凑化。PI还试图将射影流形的结果推广到Kaehler情形。一个相关的问题将研究广义Yang-Mills方程的解的跨墙性质。第三个项目建立在PI之前通过Deligne配对方法对解析扭转的全纯扩张的调查基础上。这将对复杂的Chern-Simons理论产生影响。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The notion of a moduli space occupies an ever increasingly important role in geometry and physics. It has also proved useful to certain applied fields such as robotics. Moduli are parameters describing the variation in a particular geometric or algebraic structure. The construction of a moduli space brings with it a deeper understanding of which geometric structures behave well in families, and the geometric analysis of the moduli space itself reveals invariant properties of the objects they parametrize. The current project seeks to extend the PI's previous work on certain moduli spaces that arise naturally from the gauge theory of elementary particles. The Yang-Mills equations, for example, are a major point of intersection between mathematics and theoretical physics. Moduli spaces of Higgs bundles have been used to study the space of representations of surface groups into complex Lie groups and their noncompact real forms. They appear in supersymmetric gauge theories and are also important in the Geometric Langlands problem. The research projects covered by this award will further our understanding of the relationship between the geometric, analytic, and algebraic properties of moduli spaces. The award also supports graduate students. The specific goals lie in three areas of complex geometry related to holomorphic bundles, gauge theory, and moduli problems. The first continues work of the PI on moduli spaces of Higgs bundles on Riemann surfaces. A special focus is given to understanding the asymptotic structure of the moduli space and its topological properties. This is related to important conjectures concerning the geometry of the Hitchin moduli space, in part arising from supersymmetric gauge theories. The PI will generalize previous results about the pressure metric on Hitchin components away from the Fuchsian locus. He will also explore implications for mirror symmetry calculations related to new properties of the Morse stratification of the Higgs and de Rham moduli spaces that follow from his recent work. The second project continues work on Hermitian-Yang-Mills connections on higher dimensional manifolds. One goal is to obtain a better understanding of natural gauge theoretic compactifications. The PI also seeks to extend results for projective manifolds to the Kaehler case. A related problem will study wall-crossing properties of solutions to generalized Yang-Mills equations. The third project builds on the PIs previous investigation of holomorphic extensions of analytic torsion via the approach of Deligne pairings.This will have implications for complex Chern-Simons theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.