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

Abstract

We construct new half-BPS line defects in 3d $\mathcal{N}=2$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds $X=\mathrm{Fl}(n)$. 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 $X$.

36 pages plus appendix. v2: fixed typos and references

References in corpus (30)

• Phases of $N=2$ Theories In Two Dimensions
• Supersymmetric vacua and Bethe ansatz
• Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
• A topologically twisted index for three-dimensional supersymmetric theories
• Witten Index and Wall Crossing
• Comments on twisted indices in 3d supersymmetric gauge theories
• Aspects of 3d N=2 Chern-Simons-Matter Theories
• Defects and Quantum Seiberg-Witten Geometry
• The equivariant A-twist and gauged linear sigma models on the two-sphere
• GLSM's for partial flag manifolds
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Quadrality for Supersymmetric Matrix Models
• Quantum K theory of symplectic Grassmannians
• Higher Cluster Categories and QFT Dualities
• Quantum K-theory of Quiver Varieties and Many-Body Systems
• B-branes and supersymmetric quivers in 2d
• Quantum K-Theory of Calabi-Yau Manifolds
• Wilson loop algebras and quantum K-theory for Grassmannians
• Twisted indices, Bethe ideals and 3d $\mathcal{N}=2$ infrared dualities
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• On Phases of 3d $\mathcal{N}=2$ Chern-Simons-Matter Theories
• Residue mirror symmetry for Grassmannians
• On the Witten index of 3d $\mathcal{N}=2$ unitary SQCD with general CS levels
• Quantum K theory rings of partial flag manifolds
• Decomposition squared
• Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles
• Quantum cohomology from mixed Higgs-Coulomb branches
• Schubert defects in Lagrangian Grassmannians
• Vacuum Structures Revisited
• Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials