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Register a new userOn Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
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Authors
Dmitry Galakhov
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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 Schrodinger 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.
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This is an expository article on the A-side of Kontsevichs Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_infty$-categories and their homological algebra, and later restrict to Fukaya categories, with particular emphasis on the basics of the underlying Floer theory, and the geometric features therein. Finally, we place the theory in the context of mirror symmetry, building towards its main predictions.
We describe supersymmetric A-branes and B-branes in open N=(2,2) dynamically gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold target spaces. For a subset of toric manifolds, these equivariant branes have a mirror description as branes in gauged Landau-Ginzburg models with neutral matter. We then study correlation functions in the topological A-twisted version of the GNLSM, and identify their values with open Hamiltonian Gromov-Witten invariants. Supersymmetry breaking can occur in the A-twisted GNLSM due to nonperturbative open symplectic vortices, and we canonically BRST quantize the mirror theory to analyze this phenomenon.
We prove homological mirror symmetry for the universal centralizer $J_G$ (a.k.a Toda space), associated to any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center). This is the first and the main part of a two-part series, dealing with $G$ of adjoint type. The general case will be proved in the forthcoming second part [Jin2].
This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painleve I equation, including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov-Witten invariants.
Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_infty$ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Floer cohomology approaches to mirror symmetry. More generally, our result continues to hold if the Lagrangian has cohomology the free graded algebra on a graded vector space $V$ concentrated in odd degree, and the anti-symplectic involution acts on the cohomology of the Lagrangian by the induced map of negative the identity on $V.$ It suffices for the Maslov class to vanish modulo $4.$
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