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

TL;DR

This work extends Schubert line defects to partial flag manifolds $Fl(\\boldsymbol{k};n)$ within 3d $\\mathcal{N}=2$ GLSMs, realized by 1d $\\mathcal{N}=2$ quivers coupled to the bulk. The flavored Witten indices of these defects are shown to equal the recently introduced parabolic Whitney polynomials, which in turn generate parabolic quantum Grothendieck polynomials via the quantum K-theory relations; in the 2d limit these reduce to parabolic quantum Schubert polynomials, reproducing Schubert classes in quantum cohomology. The paper provides explicit constructions, dualities, and residue computations in numerous partial flag cases (notably Grassmannians), and introduces a complete framework for representing Schubert classes in equivariant quantum K-theory using physically realized line defects. A Mathematica notebook accompanies the work to compute the parabolic polynomials for general partial flags, enabling practical calculations of Schubert products and their quantum corrections. The results offer a physical realization and computational toolkit for parabolic polynomials that generalize known quantum Schubert and Grothendieck polynomials, with potential extensions to broader defect networks and algebraic structures in quantum K-theory.

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.

Figures (10)

• Figure 1: Limits connecting all the (quantum and classical) polynomials that we consider in this work (including part I). One can go from the parabolic Whitney polynomials to the parabolic quantum Grothendieck ones using the quantum K-theory ring relations. Similarly, one can go from the parabolic cohomological Whitney polynomials to their parabolic quantum Schubert counterparts via the quantum cohomology ring relations. Starting with the parabolic quantum double Grothendieck polynomial, one obtains all the other Toda polynomials in various limits. See table \ref{['tab:polys and refs']} for terminology and references. For the special case of the Grassmannian manifold (on the left of the figure), the partition $\lambda=\lambda_w$ is defined in \ref{['defn par k']}.
• Figure 2: The 2d quiver gauge theory of interest in this paper. The circle nodes denote vector multiplets for the gauge groups $U(k_\ell)$ and the square node stands for the flavor symmetry group $SU(n)$. The ranks of the gauge groups are such that $1\leq k_1<k_2<\cdots<k_s\leq n-1$. The arrows denote bifundamental chiral multiplets --- here the fields $\phi_{\ell}^{\ell+1}$ denote the corresponding bifundamental complex scalars. The classical Higgs branch of this theory is the partial flag manifold Fl$(k_1, \cdots, k_s;n)$.
• Figure 3: Quiver diagram of the 1d-3d coupled system realizing $X_w\subseteq {\rm Fl}({\boldsymbol{k}};n)$. The last column represents the 3d GLSM to the partial flag manifold Fl$({\boldsymbol{k}}; n)$. The square quiver in the blue box is the 1d quiver gauge theory defining the defect. The circle nodes represent 1d gauge groups of the indicated ranks ${\rm r}_{k_\ell,j}\equiv {\rm r}_{k_{\ell},j}^{w_0w}$. Horizontal and vertical arrows stand for 1d chiral matter multiplets in the bifundamental representation of the corresponding nodes. Moreover, diagonal arrows correspond to 1d Fermi multiplets in the bifundamental representation of the corresponding nodes. The gray squares in the last row represent fixed background data for the inclusion maps.
• Figure 4: Diagram of nested vector spaces in $\mathbb{C}^n$, with a fixed reference flag $E_\bullet$ shown in gray. The arrows indicate the inclusions of subspaces, while the circular arrows indicate that the squares commute. Here, $\dim(V_{\ell,j})\, =\, \text{r}_{k_\ell,j}$ and $\dim(F_{k_\ell}) \,=\,\ k_\ell$.
• Figure 5: The 1d-3d coupled system after integrating out the massive fermionic and bosonic degrees of freedom of the system given in figure \ref{['fig:GenProposal']}. In the last row, the red diagonal arrows now represent Fermi multiplets in the fundamental representation of the gauge group represented by the circle node and have a charge $-1$ under the $U(1)$ subgroup of the bulk flavor symmetry $SU(n)$ indexed by the square node.

Theorems & Definitions (10)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Example 6
• Example 7
• Example 8
• Example 9
• Example 10