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

DOI10.1007/JHEP10(2022)076arXiv2105.07602MaRDI QIDQ6367792

D. Galakhov

Publication date: 17 May 2021

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 $mathcalN=(2,2)$ 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.

Mathematics Subject Classification ID

Supersymmetric field theories in quantum mechanics (81T60) Yang-Mills and other gauge theories in quantum field theory (81T13) Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, (W)-algebras and other current algebras and their representations (81R10) String and superstring theories; other extended objects (e.g., branes) in quantum field theory (81T30) Topological field theories in quantum mechanics (81T45)

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