A Representation-Theoretic Approach to $qq$-Characters

TL;DR

This paper explores a representation-theoretic construction of $qq$-characters using quantum toroidal algebras, providing geometric proofs of their properties and connecting them to Hirzebruch $ ext{chi}_y$-genera.

Contribution

It introduces a new geometric approach to constructing $qq$-characters via vertex operators in quantum toroidal algebras, extending previous algebraic methods.

Findings

Explicit vertex operator $ extsf{RR}$ computes $qq$-characters.

Proves independence of preferred direction in refined vertex.

Connects $qq$-characters to Hirzebruch $ ext{chi}_y$-genera.

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 $\mathfrak{gl}_1$ algebra, geometric engineering of adjoint matter produces an explicit vertex operator $\mathsf{RR}$ which computes certain $qq$-characters, namely Hirzebruch $\chi_y$-genera, completely analogously to how the R-matrix $\mathsf{R}$ computes $q$-characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.