The log Kazhdan--Lusztig correspondence

Abstract: A landmark discovery of the 1980s, due to many mathematicians and physicists (Drinfeld, Kohno, Witten, etc.), was the close relationship between quantum groups and affine Lie algebras. Kazhdan–Lusztig established a sharp form of this in representation theory via an equivalence of braided tensor categories of modules. The subtlest cases of their result occur when the quantum parameter q is a root of unity, where one has to pick the right form of the quantum group (the so-called Lusztig, or divided-powers form) in order for the equivalence to hold. In the mid-2000s, Feigin–Gainutdinov–Semikhatov–Tipunin conjectured a similar ‘log Kazhdan–Lusztig correspondence’ between representations of another version of the quantum group, the small quantum group, and a vertex algebra known as the triplet, at certain roots of unity. After providing a survey of these influential works for nonspecialists, we will propose a conjecture extending that of Feigin et. al. to all roots of unity. Time permitting, we will indicate a way to prove it conditional on some foundational conjectures in quantum geometric Langlands.

algebraic geometrydifferential geometryquantum algebrasymplectic geometry

Audience: researchers in the topic

Boston University Geometry/Physics Seminar

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