Stable envelopes, introduced by Maulik and Okounkov, give a family of bases for the equivariant cohomology of symplectic resolutions. The theory of stable envelopes provides a fascinating interplay between geometry, combinatorics and integrable systems. This learning seminar is devoted to the cohomological stable envelopes and their role in the context of 3d mirror symmetry.
Topics:
Nakajima quiver varieties
Quantum difference equation
Definition of Stable envelopes and examples
"Geometrization" of quantum group action, computation of Yang's matrix in terms of stable envelopes
Vertex functions, KZ equation
Monodromy matrix in terms of stable envelopes
3d mirror symmetry
Equality of monodromies for mirror-dual vertex functions
References:
Davesh Maulik and Andrei Okounkov. Quantum Groups and Quantum Cohomology. Vol. 408. Ast ́erisque. Soci ́et ́e Math ́ematique de France, 2019
Andrei Okounkov. Lectures on K-theoretic computations in enumerative geometry, Geometry of Moduli Spaces and Representation Theory. AMS, 2017
Andrei Okounkov. Inductive construction of stable envelopes and applications, II. Non- abelian actions. Integral solutions and monodromy of quantum difference equations. 2020
Richa ́rd Rim ́anyi, Andrey Smirnov, Zijun Zhou, and Alexander Varchenko. “Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes”. In: International Mathematics Research Notices 2022 (Feb. 2021)
Alexander Braverman, Davesh Maulik, Andrei Okounkov,Quantum cohomology of the Springer resolution, Advances in Mathematics, Volume 227, Issue 1, 2011
Okounkov, A., Smirnov, A. Quantum difference equation for Nakajima varieties. Invent. math. 1203–1299 (2022)
Neil Chriss, Victor Ginzburg. Representation theory and complex geometry
Schedule:
Date: Speaker: Abstract: References: