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About
President
Governance
Partner Institutions
Visit
People
Management
Faculty
Postdocs
Visiting Scholars
Staff
Research
Research Groups
Courses
Seminars
Join Us
Faculty
Postdocs
Students
Events
Conferences
Workshops
Forum
Life @ BIMSA
Accommodation
Transportation
Facilities
Tour
News
News
Announcement
Downloads
Qiuzhen College, Tsinghua University
Yau Mathematical Sciences Center, Tsinghua University (YMSC)
Tsinghua Sanya International  Mathematics Forum (TSIMF)
Shanghai Institute for Mathematics and  Interdisciplinary Sciences (SIMIS)
BIMSA > BIMSA Lecture Vertex Operators and L-operators of Elliptic Quantum Toroidal Algebras
Vertex Operators and L-operators of Elliptic Quantum Toroidal Algebras
Organizers
Nicolai Reshetikhin , Takashi Takebe
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.
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