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

arXiv comment39 pages; changes in this version: removed section about filtration of the QK theory ring (no longer needed); changed proofs of Thm. 8.2 and Thm. 11.12 so they now use an argument based on finiteness of certain completions (this fixed a gap in the previous version); added a reference to a finiteness criterion in Remark A.4; fixed few typos

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Cited by in corpus (2)

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