Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 090, 20 pages      arXiv:2203.07072      https://doi.org/10.3842/SIGMA.2022.090

A Representation-Theoretic Approach to $qq$-Characters

Henry Liu
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX26GG, UK

Received May 15, 2022, in final form November 17, 2022; Published online November 24, 2022

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.

Key words: $qq$-characters; geometric engineering; vertex operators; R-matrices; Pandharipande-Thomas theory.

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