On the remodeling conjecture for toric Calabi-Yau 3-orbifolds

Abstract

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...

On explicit formulae of LMOV invariants

“…The open string partition function Z (X,D) str (g s , ω, x) can be computed by the method of topological vertex [4,41] and the method of topological recursion developed by Eynard and Orantin [15]. The second approach was first proposed by Mariño [58], and studied further by Bouchard, Klemm, Mariño and Pasquetti [9], the equivalence of the two methods was proved in [16,20] In the following, we only need to consider the case of L=1. It is also useful to introduce the generating function of K µ,g,Q in the fixed genus g as follow:…”

“…Furthermore, the open Gromov-Witten invariants of higher genus with more holes can be computed by using Eynard-Orantin topological recursions [15]. This approach named as BKMP conjecture, was proposed by Bouchard, Klemm, Mariño and Pasquetti [9], and was fully proved in [16,20] for any toric Calabi-Yau 3-fold with AV-brane, so one can also use the BKMP method to compute the LMOV invariants for (X, D τ ). To determine the mirror curve of (X, D τ ), there are standard methods in toric geometry.…”

Graph sums in the remodeling conjecture

“…In fact, there is an explicit construction of a flat family of toric surfaces over a neighborhood of q = 0 in [45]. Each generic fiber is a toric surface isomorphic to S P , while the central fiber is ∪ σ∈Σ(3) S Pσ , a normal crossing union of several toric surfaces -each corresponds to the polytope P σ in the triangulation of the defining polytope P .…”

“…In this section we survey the proof from [45] on how to match the graph sums. The key idea is that graph sum ingredients are genus 0 information, and the genus 0 open-closed mirror theorem can be used to match them.…”

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The BKMP Remodeling Conjecture [81,13,14] preditcs all genus open-closed Gromov-Witten invariants for a toric Calabi-Yau 3-orbifold by Eynard-Orantin's topological recursion [35] on its mirror curve. The proof of the Remodeling Conjecture by the authors [42,45] relies on comparing two Feynman-type graph sums in both A and Bmodels. In this paper, we will survey these graph sum formulae and discuss their roles in the proof of the conjecture.

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