Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles

TL;DR

This paper establishes two presentations of the torus equivariant quantum K theory of Grassmannians, connecting physics-inspired Wilson line operators and Coulomb branch equations with algebraic structures, and computes related Gromov-Witten invariants.

Contribution

It introduces Whitney and Coulomb branch presentations for quantum K theory of Grassmannians, linking physics, algebra, and integrable systems, and computes related invariants.

Findings

Proved Whitney and Coulomb branch presentations for quantum K theory.

Connected presentations via transition matrices involving symmetric polynomials.

Calculated K-theoretic Gromov-Witten invariants of wedge powers of tautological bundles.

Abstract

We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch $\lambda_y$ classes of the tautological bundles. In physics, the $\lambda_y$ classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. Along the way, we calculate K-theoretic Gromov-Witten invariants of wedge powers of the tautological bundles on $\mathrm{Gr}(k;n)$, using the `quantum=classical' statement.