Gauge Theory on Manifolds with Special Holonomy

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Gauge theory is a subject that originated at the interface of mathematics and high energy physics. In special cases, gauge theory is believed to underlie the mathematical description of the universe; for example, the standard model for particle physics is a gauge theory in four dimensions. The subject also has deep links with many areas of mathematics, including partial differential equations, representation theory, algebraic geometry, differential geometry and topology. However, most research in gauge theory and almost all applications so far have focused on dimensions two, three and four. In this project, the PI intends to push the boundary of our understanding of gauge theory in higher dimensions. The PI's results are expected to have wide implications for geometric analysis, because the fundamental system of equations that define gauge theory (the Yang-Mills equations) poses additional challenges, in terms of controlling the solutions, that are not present in lower dimensions. The PI's work will also have impact through integrated research and training, as the PI plans to involve undergraduates in research opportunities related to his project.The PI's research project consists of three parts. The first part of the project will extend this dictionary between gauge theory and complex algebraic geometry to the context of Hermitian-Yang Mills connections with singularities, and specifically understand the fine structure at the singularities in terms of the complex algebraic geometry of the corresponding reflexive sheaves. In the second part of this project, the PI will develop a deformation theory for a certain class of Yang-Mills connections called G2-instantons with one-dimensional singular set, and use this to construct concrete examples of singular G2-instantons from algebro-geometric input. The third part of the PI's research project will further explore the role that generalized Seiberg-Witten equations (specifically the ones discovered in joint work of Andriy Haydys and the PI) play in gauge theory in higher dimensions. As alluded to above, the main obstacle to progress in gauge theory in higher dimension is our lack of understanding of the non-compactness issues arising from the fact that the Yang-Mills equations become super-critical starting in dimension five and leading to the formation of non-removable singularities and bubbling phenomena.