Field Theory, Link Invariants, and Higher Moduli

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The PI studies problems of geometry and topology using methods imported from particle physics. This project is divided into two major parts. The first part concerns "knot invariants": these are ways of determining, given two pictures of knotted loops of string, whether it is possible to turn one into the other without cutting. The PI and his collaborators are developing a new method for calculating knot invariants using new tools from particle physics developed over the last decade. The second part concerns new ways of deforming quantum theories, which the PI expects to have many applications in mathematics, including to the theory of differential equations. 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 project will also contribute to the training of graduate students in both fields.The first part of the project concerns "q-nonabelianization". This is a new scheme for defining invariants of links in R^3 or more generally in 3-manifolds. It is a q-deformation of the method of "nonabelianization" using spectral networks, introduced earlier by the PI and collaborators for studying moduli spaces of flat GL(N)-connections. The PI and collaborators aim to construct q-nonabelianization on general 3-manifolds and for general values of N, beginning with the GL(2) case; in that case q-nonabelianization is closely related to constructions which have been introduced earlier by Bonahon-Wong. The second part of the project concerns a new approach to higher Teichmuller theory. The first key idea is to identify the "higher Teichmuller space" (also known as "Hitchin component") with a space of marginal and irrelevant deformations of a supersymmetric quantum field theory of class S; the second key idea is to develop the corresponding picture for the moduli spaces of marginal and irrelevant deformations of surface defects in the class S theory, with applications to the theory of Higgs bundles over Riemann surfaces.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.