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Résumé |
In this talk, I will discuss the correspondence between 3d supersymmetric gauge theories living on R^2 x S^1 and quantum K-theory. This is a 3d uplift of the well-known relation between 2d gauged linear sigma models (GLSMs) with target variety X (a.k.a the Higgs branch) and the quantum cohomology of X. One way to state this correspondence is the following: half-BPS line operators (e.g. Wilson lines) wrapping the S^1 fibre are in one-to-one correspondence with coherent sheaves (e.g. vector bundles) on X. After reviewing these correspondences in general, I will focus on the case where X is a partial flag manifold. I will define a new set of half-BPS line operators, which I will call Schubert line defects, in terms of a supersymmetric quantum mechanical system coupled to the bulk 3d theory. In the infrared, I will argue that these lines flow to the Schubert classes of the quantum K-theory ring of X. The latter are defined in terms of the structure sheaves on the Schubert varieties in X, and they form a basis for the K-theory ring. Finally, using supersymmetric localization results, I will show in explicit examples how one can work out the quantum K-theory ring relations in terms of these line operators. |
