3d N=2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians

Abstract

We study a correspondence between 3d $\mathcal{N}=2$ topologically twisted Chern-Simons-matter theories on $S^1 \times \Sigma_g$ and quantum $K$-theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter $\beta$ associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with $\beta=-1$ is isomorphic to the (equivariant) small quantum $K$-ring of the Grassmannian, and the Frobenius algebra with $\beta=0$ 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 $\beta=-1$. 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 $S^1 \times \Sigma_g$ satisfy the axiom of 2d TQFT. For $\beta=0$, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for $\beta=0$, 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.