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...

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We prove a formula for certain cubic Z n -Hodge integrals in terms of loop Schur functions. We use this identity to prove the Gromov-Witten/Donaldson-Thomas correspondence for local Z n -gerbes over P 1 .14N35, 53D45; 05E05

The Remodeling Conjecture proposed by 13] relates all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifolds/3orbifolds X to the Eynard-Orantin invariants of the mirror curve of X . In this paper, we present a proof of the Remodeling Conjecture for open-closed orbifold Gromov-Witten invariants of an arbitrary affine toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa Lagrangian brane. This can be viewed as an all genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds.

In this paper, we establish equivariant mirror symmetry for the weighted projective line. This extends the results by B. Fang, C.C. Liu and Z. Zong, where the projective line was considered [Geometry & Topology 24:2049-2092, 2017. More precisely, we prove the equivalence of the R-matrices for A-model and B-model at large radius limit, and establish isomorphism for R-matrices for general radius. We further demonstrate that the graph sum of higher genus cases are the same for both models, hence establish equivariant mirror symmetry for the weighted projective line.