A Representation-Theoretic Approach to $qq$-Characters

Symmetry Integrability and Geometry-Methods and Applications

article

Henry Liu¹ 

[1] Mathematical Institute, University of Oxford

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

DOI  :  10.3842/SIGMA.2022.090

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown