T-Duality and Homological Mirror Symmetry of Toric Varieties

Let $X_{\mathrm{\Sigma }}$ be a complete toric variety. The coherent-constructible correspondence $\kappa$ of \cite{FLTZ} equates ${\text{Perf}}_T(X_{\mathrm{\Sigma }})$ with a subcategory $S{h}_{cc}(M_{\text{bR}};\text{LS})$ of constructible sheaves on a vector space $M_{\text{bR}}\mathrm{.}$ The microlocalization equivalence $\mu$ of \cite{NZ,N} relates these sheaves to a subcategory $Fuk(T^{\ast }{M}_{\text{bR}};\text{LS})$ of the Fukaya category of the cotangent $T^{\ast }{M}_{\text{bR}}$. When $X_{\text{Si}}$ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, $DCo{h}_T(X_{\text{Si}})\cong DFuk(T^{\ast }{M}_{\text{bR}};\text{LS})$, which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence $\bar{\kappa }$ of \cite{T} embeds $\text{Perf}(X_{\text{Si}})$ into a subcategory $S{h}_c(T_{\text{bR}}^{\vee };{\bar{\mathrm{\Lambda }}}_{\text{Si}})$ of constructible sheaves on a compact torus $T_{\text{bR}}^{\vee }$. When $X_{\text{Si}}$ is nonsingular, the composition of $\bar{\kappa }$ and microlocalization yields a version of homological mirror symmetry, $DCoh(X_{\mathrm{\Sigma }})\hookrightarrow DFuk(T^{\ast }{T}_{\text{bR}};{\bar{\mathrm{\Lambda }}}_{\text{Si}})$, which is a full embedding of triangulated tensor categories. When $X_{\text{Si}}$ is nonsingular and projective, the composition $\tau=\mu\circ \kappa$ is compatible with T-duality, in the following sense. An equivariant ample line bundle $\text{cL}$ 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 $\mathbb{L}$ on the universal cover $T^{\ast }{M}_{\text{bR}}$ of the dual real torus fibration. We prove $\mathbb L\cong \tau(\cL)$in$Fuk(T^*M_\bR;\LS).$ Thus, equivariant homological mirror symmetry is determined by T-duality.