The SYZ mirror symmetry and the BKMP remodeling conjecture

The Remodeling Conjecture proposed by Bouchard-Klemm-Mari\~{n}o-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (open and closed Gromov-Witten invariants) of a symplectic toric Calabi-Yau 3-fold to Eynard-Orantin invariants of its mirror curve. The Remodeling Conjecture can be viewed as a version of all genus open-closed mirror symmetry. The SYZ conjecture explains mirror symmetry as $T$-duality. After a brief review on SYZ mirror symmetry and mirrors of symplectic toric Calabi-Yau 3-orbifolds, we give a non-technical exposition of our results on the Remodeling Conjecture for symplectic toric Calabi-Yau 3-orbifolds. In the end, we apply SYZ mirror symmetry to obtain the descendent version of the all genus mirror symmetry for toric Calabi-Yau 3-orbifolds.Comment: 18 pages. Exposition of arXiv:1604.0712

Open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial generic stabilizers and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten invariants are defined by torus localization and depend on the choice of a framing which is an integer. We also provide another definition of all genus open-closed Gromov-Witten invariants based on algebraic relative orbifold Gromov-Witten theory; this generalizes the definition in Li-Liu-Liu-Zhou [arXiv:math/0408426] for smooth toric Calabi-Yau 3-folds. When the toric DM stack a toric Calabi-Yau 3-orbifold (i.e. when the generic stabilizer is trivial), we define generating functions of open-closed Gromov-Witten invariants or arbitrary genus $g$ and number $h$ of boundary circles; it takes values in the Chen-Ruan orbifold cohomology of the classifying space of a finite cyclic group of order $m$. We prove an open mirror theorem which relates the generating function of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of the toric Calabi-Yau 3-orbifold. This generalizes a conjecture by Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] (proved in full generality by the first and the second authors in [arXiv:1103.0693]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.Comment: 42 pages, 7 figure

T-Duality and Homological Mirror Symmetry of Toric Varieties

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.Comment: 34 pages, 2 figures. The previous version of this paper has now been broken into two parts. The other part is available at arXiv:1007.005