Abelianization, Higgs Bundles and Conformal Blocks

The PI studies problems of geometry using methods imported from particle physics. This project is divided into two major parts. In the first part, the PI aims to prove new formulas for solutions of an equation first studied by Einstein, describing the geomery and curvature of space-time. The second part concerns the physics of two-dimensional systems which have "scale invariance," meaning they look the same both at short and long distances. The PI aims to study these systems by reducing them to simpler ones which can be solved exactly. The results of this work will be disseminated broadly both in the mathematics and high-energy physics communities, helping to bring these two areas closer together. The PI will continue outreach through expository lectures and articles. The project will also contribute to the training of graduate students in mathematics. In joint work with Davide Gaiotto and Greg Moore, the PI introduced a conjectural picture of the hyperkahler geometry of moduli spaces of Higgs bundles. Parts of the conjecture have been verified over the last several years, in the work of various authors, including the PI. Building on this recent progress, the PI will make a direct attack on proving the conjecture, as well as a detailed numerical study of the hyperkahler metric in a particular example. The PI will also employ a new approach to the construction of conformal blocks for the Virasoro vertex algebra and more generally the W(gl(N)) vertex algebras. Conformal blocks are much-studied functions arising in two-dimensional conformal field theory, which are notoriously difficult to describe in explicit terms. The new approach the PI will use involves a new technique of abelianization, which relates complicated vertex algebras to simpler ones. 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.