CAREER: Geometric Applications of Gauge Theory

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AbstractAward: DMS-1151693Principal Investigator: Andrew Neitzke The PI's research is focused on the interface between physics and geometry. In a long-running collaboration with Davide Gaiotto and Greg Moore, he has studied mathematical applications of N=2 supersymmetric quantum field theory. This work has recently led to quite unexpected connections between physics and mathematical subjects such as enumerative geometry, hyperkahler geometry, and cluster algebras, and hence also to purely mathematical connections between these subjects. The proposed research builds on and extends this recent work in several directions. First, in continuing collaboration with Gaiotto and Moore, the PI will investigate new examples of generalized Donaldson-Thomas invariants, which "count" certain networks of geodesic trajectories on Riemann surfaces. This counting problem corresponds to determining the numbers of stable particles in certain physical theories. Also in collaboration with Gaiotto and Moore, the PI will continue development of a new method for constructing hyperkahler metrics on total spaces of integrable systems. In this new method the generalized Donaldson-Thomas invariants play a key role; the eventual goal is to produce asymptotic or convergent series representations for K3 metrics. Finally, in solo work, the PI will explore new mathematical structures which appear when the N=2 supersymmetric field theory is studied taking spacetime to be Taub-NUT space. This construction is expected to lead to new insights into the science of nonabelian theta functions. The PI will also produce a library of video clips, explaining concepts from physics for mathematicians at a variety of levels (from undergraduate level to other researchers). These clips will be freely available over the Web. In addition the PI will continue his active program of teaching, expository writing and talks, aimed at disseminating his methods and results more broadly.The PI studies geometric problems using tools imported from particle physics. Recently, he and his collaborators have devised a new scheme for solving the "field equation" which governs the curvature of spacetime. This equation, first written down by Einstein, is notoriously intractable. In particular, it is known that this equation has a large family of solutions describing a closed four-dimensional universe (so-called "K3 metrics"), but nobody has ever been able to write down actual formulas representing these solutions. The PI and collaborators have found a surprising connection between Einstein's equation and particle physics, which relates the problem of finding K3 metrics (and other similar solutions) to the problem of understanding how subatomic particles can decay. This connection has led to insight into both problems, which the PI and collaborators are now actively developing. The PI will also produce a library of video clips, explaining concepts from physics for mathematicians at a variety of levels (from undergraduate level to other researchers). These clips will be freely available over the Web.