T-duality and homological mirror symmetry for toric varieties

Let XΣ be a complete toric variety. The coherent-constructible correspondence κ of Fang et al. (2011) [14], equates PerfT(XΣ) with a subcategory Shcc(MR;ΛΣ) of constructible sheaves on a vector space MR. The microlocalization equivalence μ of Nadler and Zaslow (2009) [27], and Nadler (2009) [25] relates these sheaves to a subcategory Fuk(T⁎MR;ΛΣ) of the Fukaya category of the cotangent T⁎MR. When XΣ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCohT(XΣ)≅DFuk(T⁎MR;ΛΣ), which is an equivalence of triangulated tensor categories.The nonequivariant coherent-constructible correspondence of Treumann (preprint) [33] embeds Perf(XΣ) into a subcategory of constructible sheaves on a compact torus . When XΣ is nonsingular, the composition of and microlocalization yields a version of homological mirror symmetry, , which is a full embedding of triangulated tensor categories.When XΣ 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⁎MR of the dual real torus fibration. We prove L≅τ(L) in Fuk(T⁎MR;ΛΣ). Thus, equivariant homological mirror symmetry is determined by T-duality.