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We concentrate on a treatment of a Higgs-Coulomb duality as an absence ofmanifest phase transition between ordered and disordered phases of 2d$\mathcal{N}=(2,2)$ theories. We consider these examples of QFTs in theSchr\"odinger picture and identify Hilbert spaces of BPS states with morphismsin triangulated categories of D-brane boundary conditions. As a result ofHiggs-Coulomb duality D-brane categories on IR vacuum moduli spaces areequivalent, this resembles an analog of homological mirror symmetry. Followingconstruction ideas behind the Gaiotto-Moore-Witten algebra of the infrared oneis able to introduce interface defects in these theories and associate them toD-brane parallel transport functors. We concentrate on surveying simpleexamples, analytic when possible calculations, numerical estimates and simplephysical picture behind curtains of geometric objects. Categorification ofhypergeometric series analytic continuation is derived as an Atiyah flop of theconifold. Finally we arrive to an interpretation of the braid group action onthe derived category of coherent sheaves on cotangent bundles to flag varietiesas a categorification of Berry connection on the Fayet-Illiopolous parameterspace of a sigma-model with a quiver variety target space.

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