Gauge Theory and Invariants of Symplectic Manifolds

The first objective of this project is to study a class of equations known as generalized Seiberg-Witten equations. These little understood equations belong to the area of mathematics known as gauge theory, which originates from physics and describes the dynamics of gauge fields, such as the electromagnetic field and other fields carrying fundamental forces of nature. The importance of gauge fields in pure mathematics stems from their relationship to the geometry of shapes known as manifolds. This leads to the second objective of this project, which is to apply the theory of generalized Seiberg-Witten equations to find new ways of distinguishing different manifolds from one another. The Principal Investigator (PI) is specifically interested in applications lying at the intersection of different areas of research, such as algebraic geometry (which studies manifolds described by equations), symplectic geometry (which studies manifolds related to classical mechanics), topology (which studies properties of manifolds which remain unchanged under deformations), and string theory (a branch of modern theoretical physics). In addition, the project's goal is to expose a broad audience, including graduate and undergraduate students, as well as researchers in other areas, to some of the ideas and techniques of gauge theory and geometry. As part of the project, the PI will organize seminars and minicourses for students and write expository notes on gauge theory aimed at non-specialists.

This project will develop analytic foundations in the study of generalized Seiberg-Witten equations on three- and four-dimensional manifolds. Examples of such equations include the Vafa–Witten and Kapustin–Witten equations, which are expected to lead to new topological invariants of low-dimensional manifolds and knots, and the ADHM Seiberg–Witten equations which play an important role in defining conjectural invariants of higher-dimensional Riemannian manifolds with special holonomy. In recent years, there has been a lot of progress on the compactness problem for these equations, following groundbreaking work of Taubes. The first goal of the project is to solve the converse problem of describing the moduli spaces of solutions near the boundary. This will involve understanding singular solutions to the Fueter equation, a nonlinear generalization of the Dirac equation, which appears as the limit of rescallings of generalized Seiberg–Witten equations. In particular, little is at present known about deformation theory and gluing constructions for such singular sections, and making progress in this direction will involve developing new analytical tools for studying singular solutions of elliptic differential equations. The PI will apply this general theory to define a symplectic analog of the Pandharipande–Thomas invariants of Calabi–Yau threefolds. The invariant counts embedded pseudo-holomorphic curves with weights defined using moduli spaces of the ADHM Seiberg–Witten equations. A symplectic interpretation of the Pandharipande–Thomas invariant is likely to shed a new light on the Maulik–Nekrasov–Okounkov–Pandharipande conjecture, which remains one of the major open problems of enumerative geometry of Calabi–Yau threefolds.

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.