Abstract：

The quantum difference equation (qde) is the $q$-difference equation which is proposed by Okounkov and Smirnov to encode the $K$-theoretic twisted quasimap counting for the Nakajima quiver varieties. Its construction relies on the K-theoretic stable envelope and it is conjectured that the construction is related to the quantum affine algebra of the corresponding quiver type.

In this talk, I will focus on the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mf{sl}}}_{n})$ and give an analog of the qde. We explicitly construct the qde and give the explicit formula for the case of instanton moduli space and Hilbert scheme of A_n-singularities. We also discuss its connection to the Dubrovin connection of the quantum cohomology with the example of the instanton moduli space.