T-Duality and Homological Mirror Symmetry of Toric Varieties

Abstract

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