Quantum K-theory in Physics and Schubert Line Defects

Abstract

In this thesis, we study the interplay between supersymmetric gauge theories and quantum K-theory. We begin by reviewing 2d mathcalN=(2,2) gauged linear sigma models and their connection to quantum cohomology. We then consider their K-theoretic uplift to 3d mathcalN=2 Chern--Simons matter theories, which provide a physical framework for quantum K-theory. We then discuss the computation of the quantum K-theory ring of partial flag manifolds from a physical perspective, analyzing the Bethe ansatz equations arising from the effective twisted superpotential on the Coulomb branch. We further introduce suitable symmetrizations of these Bethe ansatz equations that lead to a new presentation of the quantum K-theory ring. Finally, we introduce new half-BPS line defects, which we call Schubert line defects, in three-dimensional theories associated with a class of homogeneous spaces. These defects are proposed to realize Schubert basis elements in the quantum K-theory ring. In their presence, the 3d Higgs branch is expected to localize to the corresponding Schubert varieties. We construct these line defects by coupling one-dimensional mathcalN=2 supersymmetric quantum mechanics to the three-dimensional bulk theory, and show that the resulting flavored Witten index reproduces the polynomials associated with Schubert classes.