Trigonometric weight functions asK-theoretic stable envelope maps for the cotangent bundle of a flag variety

Defects and quantum Seiberg-Witten geometry

“…In this case, it will not be possible to provide an argument that lands directly on the Lax matrix formulation of the complex trigonometric RS model. Instead, we attempt to verify the mirror equations I⊂{1,...,N } |I|=r i∈I j / ∈I 33) related to those above by τ j ↔ µ j and η ↔ η −1 . Let us first consider the first independent Hamiltonian with r = 1.…”

“…Recently we became aware of reference [33] where quantum K-rings of flag varieties were discussed. The conjecture the authors make about the quantum K-ring is in agreement with Proposition 2.1.…”

“…Yet, the expressions are symmetric if one interchanges z = τ 1 /τ 2 with Qτ 2 /τ 1 . Starting from (4.39) and taking the limit described after (5.1) we arrive to the above expressions provided that Q is also rescaled as follows 33) as → 0. In addition to the above scaling one can redefine the FI parameters as 34) then the elliptic RS eigenfunctions take the form of (5.32).…”

Elliptic stable envelope for Hilbert scheme of points in the plane

“…In particular, all known explicit examples of elliptic stable envelopes correspond to this case. These are hypertoric varieties and cotangent bundles to partial flag varieties of A n -type [33,15,14]. In fact, for cotangent bundles of flag varieties they were known for more than 20 years under the name of elliptic weight functions for solutions of qKZ equations for gl n .…”

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

“…This new approach was enhanced by a discovery of a connection to the weight functions appearing in the hypergeometric integral solutions to the difference KZ equations. Gorbounov, Rimányi, Tarasov and Varchenko found an identification of rational weight functions with stable envelopes for torus-equivariant cohomology of the partial flag variety T * F λ [21] and extended this to the trigonometric ones for the equivariant K-theory [44]. Furthermore they succeeded to construct a geometric representation of the Yangian Y (gl N ) [21] and the quantum affine algebra U q ( gl N ) [44] on the equivariant cohomology and the equivariant K-theory, respectively.…”