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

Abstract

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 $$ {\mathbb{R}}^2\times {S}_{\beta}^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>β</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> . 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.