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

A bstract We further investigate the 3d gauged linear sigma model (GLSM)/ quantum K-theory correspondence for partial flag manifolds X ≡ Fl( k ; n ). This is a 3d uplift of the 2d GLSM/quantum cohomology correspondence with the 3d theory compactified on R²× Sβ¹ ℝ 2 × S β 1 . 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, going beyond the existing results in the literature. 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.