FRG: Collaborative Research: Topological Invariants and Matrix Models

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Abstract Award: DMS 0244412 Principal Investigator: Sheldon Katz String theory has had a spectacular impact on many areas of modern mathematics, including algebraic geometry, differential geometry, topology, representation theory, analysis and combinatorial geometry. In particular, string dualities suggest unexpected relations between these diverse areas, many of which have been proven mathematically. Recent advances in matrix model techniques suggest further relations in mathematics. It can now be expected that number theory will become related as well. It is proposed to investigate unifying themes in duality symmetries in search of a deeper understanding of these symmetries. It is also proposed to work on a range of mathematical problems which these dualities inspire. Particular dualities include mirror symmetry, originally elucidated in the context of Gromov-Witten theory, and S-dualities, which are duality symmetries of supersymmetric gauge theories and string theories. More recently both have been related to Matrix integrals. In the case of supersymmetric gauge theories this gives a relation between perturbative Feynman diagrams on the one hand and certain computations on moduli spaces of instantons on the other, which the PIs propose to elucidate further. In the particular case of Gromov-Witten theory, connections have been found with knot theory. This leads to a complete computation of the corresponding invariants for non-compact toric Calabi-Yau threefolds. The PIs propose to extend these ideas to the compact case. The PIs will be exploiting existing connections and forging new ones between physics and mathematics to address a wide range of cutting-edge open problems in both fields. These ideas are expected to create new links between diverse areas of mathematics, as certain ideas in physics are equivalent to important unsolved problems in mathematics. The techniques introduced into mathematics are expected to have a revolutionary influence on core areas of mathematics as related techniques have in the past. This project occurs at the same time as an ongoing effort by the string theory community and will help set future directions in that field. String theory seeks to unify the force of gravity with the electromagnetic and nuclear forces; this is the problem that eluded Einstein. It is anticipated that numerous diverse areas of mathematics will become related to each other in unexpected ways and that this will have profound consequences for mathematics and physics. Due to the broad scope of the project, a multi-disciplinary and multi-institutional approach is indicated. The PIs will be bringing together their networks of collaborators, postdocs, and graduate students and directing an intense collaborative effort in these areas. An interdisciplinary math/physics curriculum will be created to train future leaders in areas at the interface of mathematics and physics. The PIs will expand their use of existing internet videoconferencing technologies for collaborative purposes, and will organize workshops on the topics of this project. This is a joint award of the Division of Mathematical Sciences programs in Geometric Analysis and Algebra, Number Theory, & Combinatorics, and the Physics Division program in Mathematical Physics.