Casimir connections, Yangians and quantum loop algebras

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Quantum groups were discovered in the mid--eighties as symmetries of 1 and 2-dimensional Statistical Mechanical models. After a period of intense development, they reemerged more recently as the symmetries of 4-dimensional supersymmetric quantum gauge theories, through the work of Nekrasov--Shatashvilii, and as the constraints governing the enumerative geometry of Nakajima quiver varieties, through the work of Maulik-Okounkov. The present proposal bears upon two such infinite-dimensional quantum groups which are associated to a complex semisimple Lie algebra: the Yangian and the quantum loop algebra. An important component of the proposal is joint between the PI and S. Gautam, and builds upon a precise link they recently discovered between these quantum groups. It endeavors on the one hand to promote the above link to an equivalence between categories of finite-dimensional representations and, on the other, to use this equivalence to compute the monodromy the trigonometric Casimir connection of the Yangian in terms of the quantum Weyl group operators of the quantum loop algebra, in a way reminiscent of the Kohno--Drinfeld theorem. The proposal has potential implications for the representation theory of Yangians and quantum loop algebras, quantum integrable systems and enumerative geometry.Quantum groups are deformations of the most basic symmetries of Nature. They were discovered in the mid-eighties as symmetries of 1 and 2-dimensional statistical mechanical models and, amazingly, reemerged recently as the symmetries of 4-dimensional quantum gauge theories, as well as the constraints of a class of enumerative problems in geometry. The mathematical study of their intrinsic and extrinsic structures is often key in understanding, and solving, the physical and mathematical systems they govern, since the presence of these symmetries greatly constrains these systems. The goal of this project is to better understand the relationship between two such classes of infinite-dimensional quantum groups, and to use this relation to describe the evolution of the systems governed by the first as some of its physical parameters (masses for example) are tuned, in terms of the second.