A representation-theoretic approach to $qq$-characters

Abstract

We raise the question of whether (a slightly generalized notion of) qq-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal gl 1 algebra, geometric engineering of adjoint matter produces an explicit vertex operator RR which computes certain qq-characters, namely Hirzebruch χ y -genera, completely analogously to how the R-matrix R computes q-characters. We prove the independence of preferred direction for the refined vertex in this and more general non-toric settings.

Higgsing $qq$-character and irreducibility

“…The qq-character (6.6) is interpreted to be associated with the infinite dimensional Fock module of the quantum toroidal algebra of gl 1 denoted by U q ( gl 1 ) [NPS18, FJMM15, FJMM17] and its elliptic uplift [KO21]. See also [Liu22] for a related geometric representation theoretical perspective of the qq-character of A 0 quiver. Similarly to finite-type quivers, we can reduce the module by tuning the parameters.…”