T-Duality and Homological Mirror Symmetry of Toric Varieties

By Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, Eric Zaslow. 2008

87 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c82adea2ad0315d436938d2e47830b8b7a19db24

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Citations in the corpus

On the homological mirror symmetry conjecture for pairs of pants (2010, background): “Complete or partial results in the Fano case are known for toric varieties [1, 2, 15], del Pezzo surfaces [8], and weighted projective planes [9].”
Homological mirror symmetry for punctured spheres (2011, background): “he case of Calabi-Yau manifolds. One direction of the mirror symmetry conjecture, in which the B-model consists of coherent sheaves on a Fano variety, has been established for toric Fano varieties in [10, 8, 44, 1, 14], 2010 Mathematics Subject Classiﬁcation. Primary 53D37, 14J33; Secondary 53D40, 5”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “Complete or partial results in the Fano case are known for toric varieties [13, 14, 15], del Pezzo surfaces [16], and weighted projective planes [17].”