Schubert line defects in 3d GLSM, part I: Complete flag manifolds and quantum Grothendieck polynomials

TL;DR

This work establishes a physical realization of Schubert classes in the quantum (K-theoretic) geometry of complete flag manifolds via 3d N=2 GLSMs. By coupling 1d N=2 SQM defect quivers to the 3d GLSM, the authors construct Schubert line defects that flow to objects supported on Schubert varieties $X_w$, with the 1d sector providing Bott–Samelson resolutions of $X_w$. The flavored Witten indices of these defects reproduce the (equivariant) Chern characters of the structure sheaves $\mathcal{O}_w$, equating to double quantum Grothendieck polynomials ${\mathfrak G}_w^{(q)}(x,y)$, and in the small-circle limit yield Schubert classes in quantum cohomology via a 0d–2d coupled system. The paper thus gives a direct 3d GLSM/quantum K-theory correspondence for complete flags and outlines paths to generalize to partial flags and to compute ring structure constants using these defect bases.

Abstract

We construct new half-BPS line defects in 3d $\mathcal{N}=2$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds $X = {\rm Fl}(n)$. Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target space $X$ and the line defects flow to objects supported on Schubert varieties $X_w \subseteq X$. These Schubert line defects form an important basis of the quantum K-theory of $X$. They are realized as $\mathcal{N}=2$ supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert variety $X_w$, with the 1d degrees of freedom physically realizing a Bott--Samelson resolution of $X_w$. Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf $\mathcal{O}_{X_w}$ as a (double) quantum Grothendieck polynomial, generalizing previous results for $X$ a Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes $[X_w]$ in the quantum cohomology ring of $X$.

Figures (7)

• Figure 1: A simple example: the 3d quiver for $X={\rm Fl}(2)=\mathbb{P}^1$ coupled to a 1d $\mathcal{N}=2$ SQM engineering the skyscraper sheaf $\mathcal{O}_p$. The 1d quiver is indicated by the dashed blue rectangle. Note that the Fermi multiplet $\Gamma$ is not in the antifundamental of the $SU(2)$ global symmetry; instead, we have a single $\Gamma$ fields which breaks the flavor symmetry explicitly through the $E$-term \ref{['Egamma expl']}.
• Figure 2: The 2d quiver gauge theory of interest. The circle nodes are gauge groups $U(i)$, and the square node denotes the flavor symmetry group $SU(n)$. The fields $\phi_{i}^{i+1}$ are bifundamental scalars in chiral multiplets. The Higgs branch of this theory is the complete flag variety Fl$(n)$.
• Figure 3: Quiver diagram of the 1d-3d coupled system realizing $X_w\subseteq {\rm Fl}(n)$. The last column represents the 3d GLSM to the complete flag manifold Fl$(n)$. 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 --- for compactness, we write ${\rm r}_{i,j}\equiv {\rm r}_{i,j}^{w_0w}$, and ${\rm r}_i \equiv {\rm r}_{i,i}$. 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 defined in \ref{['eqn:background field']}.
• 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, $\dim(V_{i,j}) = \text{r}_{i,j}$ and $\dim(F_i) = i$.
• Figure 5: The 1d-3d coupled system after integrating out the massive 1d fermionic and bosonic degrees of freedom from the defect given in figure \ref{['fig:general proposal']}. 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.