T-duality and homological mirror symmetry for toric varieties

Abstract

Let X Σ be a complete toric variety. The coherent-constructible correspondence κ of Fang et al. (2011) [14] equates P erf T ( X Σ ) with a subcategory Sh c c ( M R ; Λ Σ ) of constructible sheaves on a vector space M R . The microlocalization equivalence μ of Nadler and Zaslow (2009) [27] and Nadler (2009) [25] relates these sheaves to a subcategory Fuk ( T ⁎ M R ; Λ Σ ) of the Fukaya category of the cotangent T ⁎ M R . When X Σ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCoh T ( X Σ ) ≅ DFuk ( T ⁎ M R ; Λ Σ ) , which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence κ ¯ of Treumann (preprint) [33] embeds P erf ( X Σ ) into a subcategory S h c ( T R ∨ ; Λ ¯ Σ ) of constructible sheaves on a compact torus T R ∨ . When X Σ is nonsingular, the composition of κ ¯ and microlocalization yields a version of homological mirror symmetry, DCoh ( X Σ ) ↪ DFuk ( T ⁎ T R ; Λ ¯ Σ ) , which is a full embedding of triangulated tensor categories. When X Σ is nonsingular and projective, the composition τ = μ ∘ κ is compatible with T-duality, in the following sense. An equivariant ample line bundle L 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 L on the universal cover T ⁎ M R of the dual real torus fibration. We prove L ≅ τ ( L ) in Fuk ( T ⁎ M R ; Λ Σ ) . Thus, equivariant homological mirror symmetry is determined by T-duality.