We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrallyconstructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety [M]. Specifically, let X be a proper toric variety of dimension n and let M R = Lie(T ∨ R ) ∼ = R n be the Lie algebra of the compact dual (real) torus T ∨ R ∼ = U (1) n . Then there is a corresponding conical Lagrangian Λ ⊂ T * M R and an equivalence of triangulated dg categories Perf T (X) ∼ = Shcc(M R ; Λ), where Perf T (X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Shcc(M R ; Λ) is the triangulated dg category of complex of sheaves on M R with compactly supported, constructible cohomology whose singular support lies in Λ. This equivalence is monoidal-it intertwines the tensor product of coherent sheaves on X with the convolution product of constructible sheaves on M R .

Mathematical treatments of mirror symmetry can take different forms. We will focus on the one most suited to our purpose, the homological mirror symmetry between a projective toric variety and its Hori-Vafa Landau-Ginzburg mirror.1 Several possible versions of this Fukaya category arise in the literature -see Section 2.2.3, 6.3 and References [45, 2, 26] -though we mainly work with the one defined in [42]. Road, Evanston, IL 60208

We extend our previous work [19] on coherent-constructible correspondence for toric varieties to include toric Deligne-Mumford (DM) stacks. Following Borisov-Chen-Smith [9], a toric DM stack X Σ is described by a "stacky fan" Σ = (N, Σ, β), where N is a finitely generated abelian group and Σ is a simplicial fan in N R = N ⊗ Z R. From Σ we define a conical Lagrangian Λ Σ inside the cotangent T * M R of the dual vector space M R of N R , such that torus-equivariant, coherent sheaves on X Σ are equivalent to constructible sheaves on M R with singular support in Λ Σ .The microlocalization theorem of Nadler and the last author [40,38] relates constructible sheaves on M R to a Fukaya category on the cotangent T * M R , giving a version of homological mirror symmetry for toric DM stacks.

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem for toric Calabi-Yau 3-orbifolds, which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This generalizes a conjecture by Aganagic-Vafa and Aganagic-Klemm-Vafa, proved by the first and the second authors, on disk invariants of smooth toric Calabi-Yau 3-folds. Similar results for compact Lagrangian tori.There are other open Gromov-Witten invariants relative to different types of Lagrangian submanifolds. C.-H. Cho [13] and J. Solomon [64] define disk invariants of a compact symplectic manifold in real dimension four and six relative to a Lagrangian submanifold which is the fixed locus of an anti-symplectic involution. The mirror theorem for disk invariants for the quintic 3-fold relative to the real quintic is conjectured in [66] and proved in [58]. It has been generalized to compact Calabi-Yau 3-folds which are projective complete intersections [59], where a mirror theorem for genus one open Gromov-Witten invariants (annulus invariants) is also proved.Open orbifold Gromov-Witten invariants of compact toric orbifolds with respect to Lagrangian torus fibers of the moment map are defined in [22], which generalizes the work of [33] on compact toric manifolds. The third author and collaborators prove mirror theorems on disk invariants in this context [18,16]. The third author and collaborators also prove mirror theorems on disk invariants of toric Calabi-Yau manifolds/orbifolds (which must be non-compact) with respect to Lagrangian torus fibers of the Gross fibration [19,17].1.5. Applications. The main theorem (Theorem 1.1) of this paper has several applications. Here we mention two important applications.As mentioned in Remark 1.2 above, Theorem 1.1 has been applied to prove an open version of Ruan's Crepant Resolution Conjecture for disk invariants of toric Calabi-Yau 3-orbifold relative to an effective Aganagic-Vafa brane [45]. The effective condition can be removed using the main result of this paper. This generalizes the Open Crepant Resolution Conjecture (OCRC) for disk invariants of [C 2 /Z n ] × C relative to Aganagic-Vafa branes proved in [11].Recently, the first two authors and Zong prove the BKMP Remodeling Conjecture for all semi-projective toric Calabi-Yau 3-orbifolds [32]. Theorem 1.1 is one of the key ingredients of this proof. 1.6. Overview of the paper. The rest of the paper is organized as follows. In Section 2 we review the necessary materials concerning toric DM stacks. In Section 3 we apply localization to relate open-closed Gromov-Witten invariants and descendant Gromov-Witten invariants of 3-dimensional Calabi-Yau toric DM stacks. In Section 4 we prove a mirror theorem for orbifold disk invariants.

The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds.In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera. 4.5. The equivariant small quantum cohomology 34 4.6. Dimensional reduction of the equivariant Landau-Ginzburg model 35 4.7. Action by the stacky Picard group 39 5. Geometry of the Mirror Curve 40 5.1. Riemann surfaces 40 5.2. The Liouville form 41 5.3. Differentials of the first kind and the third kind 41 5.4. Toric degeneration 42 5.5. Degeneration of 1-forms 44 5.6. The action of the stacky group on on the central fiber 45 5.7. The Gauss-Manin connection and flat sections 46 5.8. Vanishing cycles and loops around punctures 47 5.9. B-model flat coordinates 48 5.10. Differentials of the second kind 51 6. B-model Topological Strings 51 6.1. Canonical basis in the B-model: θ 0 σ and [Vσ(τ )]. 51 6.2. Oscillating integrals and the B-model R-matrix 52 6.3. The Eynard-Orantin topological recursion and the B-model graph sum 55 6.4. B-model open potentials 57 6.5. B-model free energies 58 7. All Genus Mirror Symmetry 58 7.1. Identification of A-model and B-model R-matrices 58 7.2. Identification of graph sums 59 7.3. BKMP Remodeling Conjecture: the open string sector 60 7.4. BKMP Remodeling Conjecture: the free energies 61 References 65On the B-model side, by the result in [37], the Eynard-Orantin recursion is equivalent to a graph sum formula. So the B-model potentialF g,n can also be expressed as a graph sum:The all genus open-closed Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds. The Crepant Transformation Conjecture, proposed by Ruan [82,83] and later generalized by others in various situations, relates GW theories of K-equivalent smooth varieties, orbifolds, or Deligne-Mumford stacks. To establish this equivalence, one may need to do change of variables, analytic continuation, and symplectic transformation for the GW potential. In general, the higher genus Crepant Transformation Conjecture is difficult to formulate and prove. Coates-Iritani introduced the Fock sheaf formalism and proved all genus Crepant Transformation Conjecture for compact toric orbifolds [29]. The Remodeling Conjecture leads to simple formulation and proof of all genus Crepant Transformation Conjecture for semi-projective toric CY 3-orbifolds (which are always non-compact). The key point here is that our higher genus B-model, defined in terms of Eynard-Orantin invariants of the mirror curve, is global and analyti...