Collaborative Research: Instantons, Monopoles, and Relations among their invariants

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Manifolds are shapes which locally resemble Euclidean space but may have complicated global structure. Four-dimensional manifolds (with three spatial and one temporal direction) are used in General Relativity as models for the Universe. Manifolds of other dimensions, such as two or six, are used by theoretical physicists in String Theory models which may lead to a unification of Quantum Field Theory and Gravity. Because the solution sets of many equations arising in theoretical physics and applied mathematics are manifolds, the ability to distinguish between manifolds is useful throughout the mathematical sciences, as well as in Topology, Geometry, and Theoretical Physics. The work undertaken in this project will lead to a rare mathematical proof of a prediction from supersymmetric quantum field theory. The discovery in 2012 at CERN of the Higgs boson confirms the Standard Model, but supersymmetry has not yet been detected by the Large Hadron Collider experiments. However, the Principal Investigators' research shows that at least some consequences of supersymmetry can be mathematically verified. The activities in this project should also lead to greater involvement of minorities, especially African-American and Hispanic students, and women in mathematics research, given the Principal Investigators' record and continuing desire to encourage women and minorities to pursue careers in mathematics and to provide mentorship and training. To communicate this work to a wider audience, the Principal Investigators will write a research monograph based on this project, and organize conferences at Rutgers University and Florida International University and special sessions at national American Mathematical Society meetings each year from 2015 through 2018. In 1994, using supersymmetric quantum field theory, Edward Witten derived his celebrated formula relating Donaldson and Seiberg-Witten invariants of a closed, oriented, smooth four-dimensional manifold with admissible topology and simple type. The first goal of this project is to complete the proof of Witten's formula, employing a mathematically rigorous method based on moduli spaces of non-Abelian monopoles; the Principal Investigators have completed all steps of this program except their work on the gluing theorem for non-Abelian monopoles. They will also use non-Abelian monopoles to define new invariants of four-dimensional manifolds, investigate higher- rank Donaldson invariants, and derive relations between the instanton and Seiberg-Witten Floer homologies of closed three-dimensional manifolds.