On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry

Abstract

We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d N = (2, 2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated Abelian categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.

Interfaces and Quantum Algebras, I: Stable Envelopes

“…This naturally leads to a special class of Janus interfaces that can be called phase interfaces, or phase walls. They interpolate between distinct phases in different spacetime regions, and understanding their properties allows to establish mapping of various objects across the phases (e.g., study the brane transport, see [155][156][157][158], especially the recent work [159]). The interfaces we study in this paper will be of such type: they will interpolate between Higgs, Coulomb, mixed, as well as the CFT phases of the given quiver gauge theory.…”