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 = Gr(Nc, nf) is a circle compactification of the 3d N = 2 supersymmetric gauge theory with gauge group U(Nc)k, k+lNc and nf fundamental chiral multiplets, for any choice of the Chern-Simons levels (k, l) in the ‘geometric window’. For k=Nc−nf/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 QKT(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 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 QKT(X). We then use 3d A-model techniques to explicitly compute QKT(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.
Original language | English |
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Article number | 82 |
Number of pages | 55 |
Journal | Journal of High Energy Physics |
Volume | 2023 |
Issue number | 12 |
DOIs | |
Publication status | Published - 12 Dec 2023 |
Bibliographical note
Acknowledgments:We thank Mathew Bullimore, Stefano Cremonesi, Hans Jockers, Heeyeon Kim, Horia Magureanu, Qaasim Shafi, Leonardo Mihalcea, and Eric Sharpe for discussions and correspondence. CC is particularly indebted to Mathew Bullimore and Heeyeon Kim for seminal contributions and for collaboration on an earlier version of this project, and to Leonardo Mihalcea and Eric Sharpe for sharing their deep understanding of quantum K-theory. CC is a Royal Society University Research Fellow supported by the University Research Fellowship Renewal 2022 ‘Singularities, supersymmetry and SQFT invariants’. The work of OK is supported by the School of Mathematics at the University of Birmingham.
Keywords
- Supersymmetry and Duality
- Differential and Algebraic Geometry
- Field Theories in Lower Dimensions