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

点击翻译按钮获取中文摘要

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.