Grothendieck lines in 3d $\mathcal{N}=2$ SQCD and the quantum K-theory of the Grassmannian

TL;DR

This paper introduces Grothendieck lines in 3d $ N=2$ SQCD, providing a physical realization of the quantum K-theory ring of the Grassmannian and connecting line defects to Schubert classes and enumerative invariants.

Contribution

It identifies new half-BPS line defects called Grothendieck lines in 3d $ N=2$ GLSM, linking them to Schubert varieties and quantum K-theory, and computes related invariants.

Findings

Grothendieck lines flow to structure sheaves of Schubert varieties.

The 1d Witten index reproduces Schubert class Chern characters.

Explicit computation of quantum K-theory and enumerative invariants.

Abstract

We revisit the 3d GLSM computation of the equivariant quantum K-theory ring of the complex Grassmannian from the perspective of line defects. The 3d GLSM onto $X={\rm Gr}(N_c, n_f)$ is a circle compactification of the 3d $\mathcal{N}=2$ supersymmetric gauge theory with gauge group $U(N_c)_{k, k+l N_c}$ and $n_f$ fundamental chiral multiplets, for any choice of the Chern-Simons levels $(k,l)$ in the `geometric window'. For $k=N_c-\frac{n_f}{2}$ and $l=-1$, the twisted chiral ring generated by the half-BPS lines wrapping the circle has been previously identified with the quantum K-theory ring QK$_T(X)$. We identify new half-BPS line defects in the UV gauge theory, dubbed Grothendieck lines, which flow to the structure sheaves of the (equivariant) Schubert varieties of $X$. They are defined by coupling $\mathcal{N}=2$ supersymmetric gauged quantum mechanics of quiver type to the 3d GLSM. We explicitly show that the 1d Witten index of the defect worldline reproduces the Chern characters for the Schubert classes, which are written in terms of double Grothendieck polynomials. This gives us a physical realisation of the Schubert-class basis for QK$_T(X)$. We then use 3d $A$-model techniques to explicitly compute QK$_T(X)$ as well as other K-theoretic enumerative invariants such as the topological metric. We also consider the 2d/0d limit of our 3d/1d construction, which gives us local defects in the 2d GLSM, the Schubert defects, that realise equivariant quantum cohomology classes.