T-duality and homological mirror symmetry for toric varieties

Let X _σ be a complete toric variety. The coherent-constructible correspondence κ of Fang et al. (2011) [14] equates PerfT(Xσ) with a subcategory Shcc(M _{R{double-struck}}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*M _{R{double-struck}};Λσ) of the Fukaya category of the cotangent T*M _{R{double-struck}}. When X _σ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCoh _T(Xσ)*DFuk(T*M _{R{double-struck}}Λσ), which is an equivalence of triangulated tensor categories.The nonequivariant coherent-constructible correspondence κΛ of Treumann (preprint) [33] embeds Perf(Xσ) into a subcategory Shc(T _{R{double-struck}}VΛ̄σ) of constructible sheaves on a compact torus TRV. When X _σ is nonsingular, the composition of κΛ and microlocalization yields a version of homological mirror symmetry, DCoh(Xσ)=DFuk(T*T _{R{double-struck}}Λσ), which is a full embedding of triangulated tensor categories.When X _σ is nonsingular and projective, the composition τ=μ{ring operator}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{double-struck} on the universal cover T*M _{R{double-struck}} of the dual real torus fibration. We prove L*τ(L{double-struck}) in Fuk(T*MR{double-struck}Λσ). Thus, equivariant homological mirror symmetry is determined by T-duality.