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

TL;DR

This paper explores the duality between ordered and disordered phases in 2d supersymmetric theories, linking D-brane categories, mirror symmetry, and geometric objects through categorification and interface defects.

Contribution

It introduces a novel perspective on Higgs-Coulomb duality, connecting D-brane categories with homological mirror symmetry and categorifying hypergeometric series analytic continuation.

Findings

D-brane categories on IR vacua are equivalent under duality.

Interface defects correspond to D-brane parallel transport functors.

Categorification of hypergeometric series via Atiyah flop.

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 $\mathcal{N}=(2,2)$ theories. We consider these examples of QFTs in the Schr\"odinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated 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.