Vertex Operators and L-operators of Elliptic Quantum Toroidal Algebras
Organizers
Speaker
Hitoshi Konno
Time
Thursday, September 12, 2024 3:30 PM - 5:00 PM
Venue
A3-4-301
Online
Zoom 518 868 7656
(BIMSA)
Abstract
We start from a review of the elliptic quantum group U_{q,p}(\widehat{\mathfrak{sl}}_N) and its correspondences to the elliptic cohomology of the cotangent bundle to the partial flag variety and to the deformed W-algebras. We emphasize the different roles of the two vertex operators defined by the two different co-algebra structures associated with the standard comultiplication \Delta and the Drinfeld comultiplication $\Delta^D$. Then we discuss the elliptic quantum toroidal algebra U_{t_1,t_2,p}({\mathfrak{gl}}_{1,tor}) and construct the two vertex operators associated with \Delta^D and \Delta. By using them, we show the same correspondence of U_{t_1,t_2,p}({\mathfrak{gl}}_{1,tor}) to the Jordan quiver W-algebras (an operator version of the qq-character) and to the elliptic cohomology of the instanton moduli spaces ${\cal M}(n,r)$. The former further yields the instanton calculus for the 5d and 6d lifts of the 4d {\cal N}=2^* SUSY gauge theory, whereas the latter yields the shuffle product formula for the elliptic stable envelopes, the K-theoretic vertex functions and $L$-operators satisfying the RLL=LLR^* relation with R and R^* being the elliptic dynamical instanton R-matrices. If time allows, we also discuss their higher rank extensions, the elliptic quantum toroidal algebra U_{t_1,t_2,p}({\mathfrak{gl}}_{N,tor}) and its connections to the A^{(1)}_{N-1} and A_\infty quiver varieties. This talk is based on the works done with Kazuyuki Oshima and Andrey Smirnov.