Journal of High Energy Physics (Aug 2020)

3d N $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

  • Kazushi Ueda,
  • Yutaka Yoshida

DOI
https://doi.org/10.1007/JHEP08(2020)157
Journal volume & issue
Vol. 2020, no. 8
pp. 1 – 44

Abstract

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Abstract We study a correspondence between 3d N $$ \mathcal{N} $$ = 2 topologically twisted Chern-Simons-matter theories on S 1 × Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S 1 × Σ g satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

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