Topic
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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01 Jan 2012
TL;DR: The Strominger-Yau-Zaslow conjecture has been extensively studied in the literature since its introduction in 1996 as discussed by the authors, including a survey of recent developments in this conjecture.
Abstract: This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert. After explaining the overall philosophy, I explain how recent results fit into this program.
28 citations
TL;DR: In this article, the authors established a linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve.
Abstract: We establish a \({\mathbb {Z}}[[t_1,\ldots , t_n]]\)-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to \(t_1= \cdots =t_n=0\) gives a \({\mathbb {Z}}\)-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Neron) n-gon. We prove that this equivalence extends to a \({\mathbb {Z}}\)-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for \(n=1\) were established in Lekili and Perutz (Arithmetic mirror symmetry for the 2-torus (preprint) arXiv:1211.4632, 2012).
28 citations
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TL;DR: In this paper, B-type tensor product branes in mirrors of two-parameter Calabi-Yau hypersurfaces, using the language of matrix factorizations, are discussed.
Abstract: We discuss B-type tensor product branes in mirrors of two-parameter Calabi-Yau hypersurfaces, using the language of matrix factorizations. We determine the open string moduli of the branes at the Gepner point. By turning on both bulk and boundary moduli we then deform the brane away from the Gepner point. Using the deformation theory of matrix factorizations we compute Massey products. These contain the information about higher order deformations and obstructions. The obstructions are encoded in the F-term equations, which we obtain from the Massey product algorithm. We show that the F-terms can be integrated to an effective superpotential. Our results provide an ingredient for open/closed mirror symmetry for these hypersurfaces.
28 citations
Posted Content•
TL;DR: In this paper, the universal covering of the moduli space of complex structures by non-commutative spaces is proposed, which are described by certain categories of sheaves with connections which are flat along foliations.
Abstract: We suggest to compactify the universal covering of the moduli space of complex structures by non-commutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of abelian varieties this approach gives quantum tori as a non-commutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.
28 citations
TL;DR: In this article, the authors studied complex algebraic K3 surfaces with an automorphism which acts trivially on the Neron-Severi group and showed that the order of the automomorphism is a 2-power and equals the rank of the transcendental lattice.
28 citations