Quantum Langlands duality of representations of -algebras

Abstract

We prove duality isomorphisms of certain representations of Walgebras which play an essential role in the quantum geometric Langlands Program and some related results.1 A similar, but more subtle, equivalence is expected for rational values of κ as well.

Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks

“…(iii) More generally, if B is any boundary condition, then we expect that the vertex algebra 13 12 This has been proved in [5]. 13 This expectation is conditional on the existence of a physical junction D G 0,1 B which preserves the global G symmetry associated to the Dirichlet boundary condition.…”

“…where C(T, B 1 ) ∨ is the dual category to C(T, B 1 ). 5 Physically, the functor maps boundary lines to the spaces of local operators supported at points where the lines end at the junction. These naturally form a module for the vertex algebra V (T, B 1 B 2 ) of local operators supported at generic points of the junction.…”

“…It is natural to consider conformal blocks (more precisely, their dual spaces called spaces of coinvariants, see [28,12] and Section 6 below) of a junction vertex algebra V (T, B 1 B 2 ), possibly involving the V (T, B 1 B 2 )-modules associated to the corresponding ribbon categories. 5 In special situations, the bulk 4d theory may admit non-trivial category S(T) of bulk line defects, with functors to the categories of boundary lines. If that's the case, the product should taken over this category.…”

Universal two-parameter 𝒲∞-algebra and vertex algebras of type 𝒲(2, 3, …,N)

“…The principal -algebras have appeared prominently in several important problems in mathematics and physics including the Alday–Gaiotto–Tachikawa correspondence [AGT10, SV13, BFN16, MO19], and the quantum geometric Langlands program [Fre07, Gai16, CG20, FG20, AF19, Gai]. They are also closely related to the classical -algebras which arose in the context of integrable hierarchies of soliton equations in the work of Adler, Gelfand, Dickey, Drinfeld, and Sokolov [Adl78, GD78, DS84, Dic91].…”

Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

“…The construction of W-algebras was firstly introduced by Feigin and Frenkel [19] for f a principal nilpotent element, and was extended for general nilpotent elements by Kac, Roan and Wakimoto [26]. The theory of W-algebras is related with integrable systems [24], the two-dimensional conformal field theory, the geometric Langlands program [23,20,10], and the 4d/2d duality [8,12,13,33] in physics.…”