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Mirror symmetry
About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.
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TL;DR: A relation between the number of rational curves of fixed degree on Calabi Yau threefolds and the Picard Fuchs equations, which was suggested as part of the study of mirror symmetry, is verified in the case of complete intersection of two cubics and lines.
Abstract: A relation between the number of rational curves of fixed degree on Calabi Yau threefolds and the Picard Fuchs equations, which was suggested as part of the study of mirror symmetry, is verified in the case of complete intersection of two cubics and lines.
85 citations
TL;DR: In this paper, it was shown that the wrapped Fukaya category of a punctured sphere is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture.
Abstract: We prove that the wrapped Fukaya category of a punctured sphere (S with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.
85 citations
TL;DR: In this paper, the topological string amplitudes for the FHSV model with Enriques fibres have been studied and closed expressions for the amplitudes on the total space in terms of modular forms, up to genus three.
Abstract: We study topological string amplitudes for the FHSV model using various techniques. This model has a type II realization involving a Calabi-Yau threefold with Enriques fibres, which we call the Enriques Calabi-Yau. By applying heterotic/type IIA duality, we compute the topological amplitudes in the fibre to all genera. It turns out that there are two different ways to do the computation that lead to topological couplings with different BPS content. One of them gives the standard D0-D2 counting amplitudes, and from the other one we obtain information about bound states of D0-D4-D2 branes on the Enriques fibre. We also study the model using mirror symmetry and the holomorphic anomaly equations. We verify in this way the heterotic results for the D0-D2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms, and up to genus three. This model turns out to be much simpler than the generic B-model and might be exactly solvable.
84 citations
TL;DR: Inspired by the homological mirror symmetry conjecture of Kontsevich [30], this paper constructed new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi-projective variety.
Abstract: Inspired by the homological mirror symmetry conjecture of Kontsevich [30], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi– projective variety. MSC (2000): 18E30; 14J32.
84 citations
TL;DR: In this paper, the deformed Hermitian-Yang-Mills equation on a holomorphic line bundle over a compact Kahler manifold X is studied, and it is shown that this equation is the Euler-Lagrange equation for a positive functional and that solutions are unique global minimizers.
Abstract: Let L be a holomorphic line bundle over a compact Kahler manifold X. Motivated by mirror symmetry, we study the deformed Hermitian–Yang–Mills equation on L, which is the line bundle analogue of the special Lagrangian equation in the case that X is Calabi–Yau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that X is a Kahler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when L is ample and X has non-negative orthogonal bisectional curvature.
84 citations