A Representation-Theoretic Approach to $qq$-Characters

Henry Liu

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

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 $\mathrm{R}$-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.

Détail

Keywords: $qq$-characters, geometric engineering, vertex operators, Pandharipande–Thomas theory.
Mots-clés : $\mathrm{R}$-matrices

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