Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles
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| Format: | Preprint |
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2022
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| _version_ | 1866914020561256448 |
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| author | Gu, Wei Mihalcea, Leonardo C. Sharpe, Eric Zou, Hao |
| author_facet | Gu, Wei Mihalcea, Leonardo C. Sharpe, Eric Zou, Hao |
| contents | 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 $λ_y$ classes of the tautological bundles. In physics, the $λ_y$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_01091 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles Gu, Wei Mihalcea, Leonardo C. Sharpe, Eric Zou, Hao Algebraic Geometry High Energy Physics - Theory Combinatorics 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 $λ_y$ classes of the tautological bundles. In physics, the $λ_y$ 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. |
| title | Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles |
| topic | Algebraic Geometry High Energy Physics - Theory Combinatorics |
| url | https://arxiv.org/abs/2208.01091 |