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We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold X with boundary condition specified by a framed Aganagic-Vafa outer brane (L , f) and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold X ˜ , proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps: • First, a correspondence between the disk invariants of (X , L , f) and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold (Y , D) , where Y is a toric partial compactification of X by adding a smooth toric divisor D. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of (X , L , f) are identified with the formal relative Gromov-Witten invariants of the formal completion of (Y , D) along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry. • Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of (Y , D) and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold X ˜ = O Y (− D). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting. [ABSTRACT FROM AUTHOR]

Abstract: Let be a complete toric variety. The coherent-constructible correspondence κ of Fang et al. (2011) equates with a subcategory of constructible sheaves on a vector space . The microlocalization equivalence μ of Nadler and Zaslow (2009) and Nadler (2009) relates these sheaves to a subcategory of the Fukaya category of the cotangent . When is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, , which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence of Treumann (preprint) embeds into a subcategory of constructible sheaves on a compact torus . When is nonsingular, the composition of and microlocalization yields a version of homological mirror symmetry, , which is a full embedding of triangulated tensor categories. When is nonsingular and projective, the composition is compatible with T-duality, in the following sense. An equivariant ample line bundle has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane on the universal cover of the dual real torus fibration. We prove in . Thus, equivariant homological mirror symmetry is determined by T-duality. [Copyright &y& Elsevier]