Monodromy Theorems, Affine Quantum Groups, and Meromorphic Tensor Categories

Northeastern University, Boston MA

Investigators

Abstract

Quantum groups are deformations of the most basic symmetries in nature. They were discovered in the mid-eighties as symmetries of one- and two-dimensional statistical mechanical models describing, for example, thin layers of ice. Amazingly, quantum groups have recently arisen as the symmetries of four-dimensional gauge theories, which describe the interaction of elementary particles such as quarks, and their five- and six-dimensional generalizations, as well as the constraints of a class of counting problems in geometry. Braid groups are another pervasive class of symmetries that arise in the mathematical study of knots, the analysis of three-dimensional shapes, the statistics of elementary particles constrained to move in two dimensions, and quantum computing. Because the presence of symmetries greatly constrains these systems, the mathematical study of the intrinsic and extrinsic structures of their groups of symmetries is key in understanding their physical and mathematical properties. This research project will lead to a deeper understanding of the relationship between the quantum groups corresponding to the plane, the sphere, and the torus. These have been traditionally thought to sit on distinct rungs in a ladder of increasing complexity. Research in progress is uncovering the striking fact that these quantum groups are, in fact, equivalent. Such equivalences are interesting in themselves, but they also allow precise characterization of the appearance of braid groups in disparate mathematical and physical contexts. This research project centers on infinite-dimensional quantum groups associated to one-dimensional algebraic groups: Yangians, quantum loop algebras, and elliptic curves. The project builds on the equivalence of finite-dimensional representations of Yangians and quantum loop algebras. The PI and collaborators will promote this equivalence of finite-dimensional algebras to an equivalence between infinite-dimensional quantum loop algebras and elliptic quantum groups, thereby elucidating the structure of the latter, which is still little understood outside type A. These equivalences may be thought of as q-deformed, meromorphic versions of the Kazhdan-Lusztig equivalence between affine Lie algebras and finite-dimensional quantum groups. A key aspect of this project is the study of the meromorphic braided categories of representations of these affine quantum groups, which provides new examples of such categories. Using the newly derived equivalence, the monodromy of rational affine, trigonometric, and elliptic Casimir connections will be computed in terms of quantum Weyl group operators in a way reminiscent of the Kohno-Drinfeld quantum group. This research will have applications to Yangians, quantum loop algebras and elliptic quantum groups, quantum integrable systems, and questions in enumerative geometry.