Exponential Periods, Bispectrality and Affine Quantum Groups

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The study of symmetry is a basic paradigm in science, with fundamental consequences for crystallography, particle physics, general relativity, quantum computing and signal processing. The study of the (bilateral, translation, rotational, scaling and other) symmetries of a given chemical, physical or mathematical system is often key to its understanding, and ultimate solution, since their presence greatly constrains the system. Quantum groups are deformations of the most basic symmetries of Nature. They were discovered in the 1980s as symmetries of one and two-dimensional statistical mechanical models that describe, for example, thin layers of ice. Quantum groups have re-emerged very recently as the symmetries of four-dimensional gauge theories that describe the interaction of elementary particles such as quarks. Quantum groups form a hierarchy that depends on the strength of the deformation and their dependence on a spectral parameter: constant, rational, trigonometric or elliptic. As the hierarchical level rises, an interesting trade-off occurs: some of the structure becomes far more intricate, while some simplifies radically. This project will further uncover relations that exist between different members of the hierarchy by building conceptual bridges between them. These conceptual bridges will have fundamental consequences for the quantum groups they link. On the one hand, they will give a radically simpler description of the multivaluedness of the differential or difference equations for one quantum group in terms of its hierarchical superior. On the other, they will give a self-contained description of the latter in terms of the former which, in addition to clarifying its structure, has potentially far reaching arithmetic consequences. More precisely, the present project will explore the relations between affine quantum groups. One main theme of the project is to extend the description of the monodromy of the rational Casimir connection of a symmetrisable Kac-Moody algebra, in terms of quantum Weyl group operators, to numerical values of the deformation parameter, and to the difference analogue of the connection. The second main theme of the project is to use the meromorphic tensor equivalence of finite dimensional representations of Yangians and quantum loop algebras, and of quantum loop algebras and elliptic quantum groups, to compute the monodromy of the trigonomeric Casimir connection and of the rational and trigonometric qKZ equations, thus proving the difference analog of the Drinfeld-Kohno theorem conjectured by Frenkel and Reshetikhin.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.