Moduli Spaces of Higgs Bundles, Gauge Theory, and Related Topics

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Moduli are parameters describing the variation of 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. This notion plays an important role in geometry and physics. It has also proved useful to certain applied fields such as robotics. This project seeks to extend the PI’s previous work on certain moduli spaces that arise naturally from the gauge theory of elementary particles. The research supported by this award will further our understanding of the relationship between the geometric, analytic, and algebraic properties of moduli spaces. The PI will continue mentoring undergraduate and graduate students, as well as other early career researchers. He will co-organize workshops and conferences to introduce students to the latest developments in the field and to give a platform to recent Ph.D. recipients to announce the results of their research. The specific goals of this project 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. These include three subprojects: (1) conformal limits for parabolic lambda-connections; (2) an analytic construction of the universal moduli space of Higgs bundles over varying Riemann surfaces; and (3) the asymptotic structure of the moduli space and its topological properties. In the second area of proposed research, the PI will prove further results surrounding his work on complex Chern-Simons bundles. He will investigate the link with hyperholomorphic bundles on moduli spaces, and also clarify the relationship of his constructions to important previous work on Atiyah algebras of determinants of cohomology. The third project is the study of higher dimensional generalizations of the Yang-Mills equations and their relationship to the complex geometry of holomorphic bundles. 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.