29761 2606.15261 10.1007/JHEP08(2026)103 103 103 On the Schubert calculus of the quantum K-theory for partial flag manifolds: a 3d A-model perspective 01 Mathematical Sciences 0101 Pure Mathematics Regular Article - Theoretical Physics 1 58 2026 8 13 2026 6 24 2026 6 27 2026 8 13 2026 8 13 Swiss National Science Foundation TMPFP2_234009 National Natural Science Foundation of China 12405083 National Natural Science Foundation of China 12475005 Shanghai Pujiang Program 24PJA119 The Author(s) 2026 Open Access. This article is distributed under the terms of the Creative Commons Attribution License ( CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 13130 2026 2026 8 8 Zhihao Duan xduanz@gmail.com Osama Khlaif osama.khlaif@phys.ens.fr Hao Zou haozou@fudan.edu.cn https://ror.org/01swzsf04 grid.8591.5 0000 0001 2175 2154 Section de Mathématiques Université de Genève 1211 Genève 4 Switzerland https://ror.org/01ggx4157 grid.9132.9 0000 0001 2156 142X Theoretical Physics Department, CERN 1211 Geneva 23 Switzerland https://ror.org/013cjyk83 grid.440907.e 0000 0004 1784 3645 Philippe Meyer Institute, Physics Department, École Normale Supérieure (ENS) Université PSL 24 rue Lhomond F-75231 Paris France https://ror.org/013q1eq08 grid.8547.e 0000 0001 0125 2443 Center for Mathematics and Interdisciplinary Sciences Fudan University Shanghai 200433 China https://ror.org/0489cd473 Shanghai Institute for Mathematics and Interdisciplinary Sciences Shanghai 200433 China Abstract 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 ℝ 2 × S β 1 $$ {\mathbb{R}}^2\times {S}_{\beta}^1 $$ . 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. Keywords Supersymmetric Gauge Theory Wilson ’t Hooft and Polyakov loops ArXiv ePrint: 2606.15261