Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials

TL;DR

This paper constructs Schubert line defects in 3d supersymmetric GLSMs for partial flag manifolds, linking them to quantum K-theory and introducing new polynomials representing Schubert classes.

Contribution

It generalizes previous work to partial flag manifolds, connecting defect indices to parabolic Whitney and quantum Grothendieck polynomials.

Findings

Witten indices reproduce parabolic Whitney polynomials for Schubert classes.

Conversion of these polynomials yields new parabolic quantum Grothendieck polynomials.

In the 2d limit, the polynomials reduce to known quantum Schubert polynomials.

Abstract

We construct Schubert line defects in the 3d $\mathcal{N}=2$ supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifold $X={\rm Fl}({\boldsymbol{k}};n)$, generalizing our construction for complete flag manifolds given in a companion paper arXiv:2512.19802 (part I). In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d $\mathcal{N}=2$ supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varieties $X_w \subseteq X$ in the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of $[\mathcal{O}_w]$ -- more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes $[X_w]$ in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials.