On the Schubert calculus of the quantum K-theory for partial flag manifolds: a 3d A-model perspective

Abstract

We further investigate the 3d gauged linear sigma model (GLSM)/quantum K-theory correspondence for partial flag manifolds $X \equiv {\rm Fl}(\boldsymbol{k};n)$. This is a 3d uplift of the 2d GLSM/quantum cohomology correspondence with the 3d theory compactified on $\mathbb{R}^2\times S^1_β$. Recently, a set of half-BPS line operators, called Schubert line defects, were constructed that correspond to the Schubert classes in the K-theory ring of $X$. Utilizing algebro-geometric algorithms, we compute $2$-point and $3$-point correlation functions of these line operators in the 3d A-model regime of the theory. These are interpreted as genus-$0$ K-theoretic Gromov--Witten invariants, and they produce the K-theoretic Littlewood--Richardson coefficients of the quantum K-theory ring of $X$. We show how this works explicitly in examples. Taking the small $β$ limit, we apply these techniques to the resulting 2d GLSM. We explicitly compute the quantum cohomology ring relations of $X$ for some cases and match with existing results in the literature in examples.