Level correspondence of $K$-theoretic $I$-function in Grassmann duality

Abstract

In this paper, we prove a class of nontrivial q-Pochhammer symbol identities with extra parameters by iterated residue method. Then we use these identities to find relations of the quasi-map $K$-theoretical $I$-functions with level structure between Grassmannian and its dual Grassmannian. Here we find an interval of levels within which two $I$-functions are the same, and on the boundary of that interval, two $I$-functions are intertwining with each other. We call this phenomenon level correspondence in Grassmann duality.

In this version, we fix typos and add a reference: arXiv:1912.03792 in which Y. Yoshida and K. Ueda established the correspondence between 3d gauge theory and the quantum $K$-ring and $I$-function of $Gr(r,n)$ independently of ref [14] in our paper