Problems on Higgs bundles and other gauge theoretic moduli spaces

About This Grant

The notion of a moduli space plays an important role in geometry and physics. It has also proved useful to certain applied fields such as robotics. 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. The current project seeks to extend the PIs 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 grant will further our understanding of the relationship between the geometric, analytic, and algebraic properties of moduli spaces. The specific goals of this project lie in several areas of complex geometry related to holomorphic bundles, gauge theory, and moduli problems. The first consists of problems stemming from previous work of the PI on moduli spaces of Higgs bundles on Riemann surfaces. These include the following subprojects: (1) Giving a gauge theoretic construction of a joint moduli space of Higgs bundles over varying Riemann surfaces; (2) further understanding the asymptotic structure of the moduli space and its topological properties. The latter is related to important conjectures concerning the geometry of the Hitchin moduli space, in part arising from supersymmetric gauge theories; (3) extending work on Chern-Simons line bundles to moduli spaces of parabolic Higgs bundles. In the second area of proposed research, the PI will study higher dimensional generalizations of the Yang-Mills equations and their relationship to the complex geometry of holomorphic bundles. This includes: (1) a study of the adjoint Seiberg-Witten equations on Kähler surfaces; (2) giving a gauge theoretic proof of the Bogomolov-Miyaoka-Yau inequality for projective surfaces of general type using monopoles; (3) studying the nature of Z/2 harmonic forms and spinors as they relate to asymptotics of coupled moduli spaces. 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.