A Representation-Theoretic Approach to $qq$-Characters

DOI10.3842/SIGMA.2022.090arXiv2203.07072MaRDI QIDQ6393631

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Publication date: 14 March 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 $mathsfRR$ which computes certain $qq$-characters, namely Hirzebruch $ch{i}_y$-genera, completely analogously to how the R-matrix $mathsfR$ 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.

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras (17B67) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) (14N35)

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