Title: Asymptotic representations of shifted quantum affine algebras from critical K-theory
Description: In this thesis we explore the geometric representation theory of shifted quantum affine algebras 𝒜^𝜇, using the critical K-theory of certain moduli spaces of infinite flags of quiver representations resembling the moduli of quasimaps to Nakajima quiver varieties. These critical K-theories become 𝒜^𝜇-modules via the so-called critical R-matrix 𝑅, which generalizes the geometric R-matrix of Maulik, Okounkov, and Smirnov. In the asymptotic limit corresponding to taking infinite instead of finite flags, singularities appear in 𝑅 and are responsible for the shift in 𝒜^𝜇. The result is a geometric construction of interesting infinite-dimensional modules in the category 𝒪 of 𝒜^𝜇, including e.g. the pre-fundamental modules previously introduced and studied algebraically by Hernandez and Jimbo. Following Nekrasov, we provide a very natural geometric definition of qq-characters for our asymptotic modules compatible with the pre-existing definition of q-characters. When 𝒜^𝜇 is the shifted quantum toroidal gl₁ algebra, we construct asymptotic modules DT_𝜇 and PT_𝜇 whose combinatorics match those of (1-legged) vertices in Donaldson--Thomas and Pandharipande--Thomas theories.

Such vertices control enumerative invariants of curves in toric 3-folds, and finding relations between (equivariant, K-theoretic) DT and PT vertices with descendent insertions is a typical example of a wall-crossing problem. We prove a certain duality between our DT_𝜇 and PT_𝜇 modules which, upon taking q-/qq-characters, provides one such wall-crossing relation.