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Remodeling the B-model

Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti

TL;DR

The paper delivers a complete, non-perturbative B-model framework for computing open and closed topological string amplitudes on local Calabi–Yau geometries, including mirrors of toric manifolds, by adapting the Eynard–Orantin recursion to mirror curves in C^* × C^*. This formalism unifies brane framing, open/closed moduli, and phase transitions, enabling calculations at geometric and non-geometric points (e.g., orbifolds) and providing extensive cross-checks against the topological vertex and Chern–Simons dualities on lens spaces. It produces explicit disk and higher-genus open amplitudes across multiple examples (vertex, resolved conifold, local P^2, local F_n) and offers predictions for orbifold cases like C^3/Z_3, illustrating the method’s reach beyond large-radius expansions. The work further connects the B-model recursion to the chiral boson picture and holographically to matrix-model loop equations, laying groundwork for potential compact-CY generalizations and deepening the interplay between geometry, dualities, and non-perturbative topological string theory.

Abstract

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi-Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A_p fibrations, where the amplitudes compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C^3/Z_3.

Remodeling the B-model

TL;DR

The paper delivers a complete, non-perturbative B-model framework for computing open and closed topological string amplitudes on local Calabi–Yau geometries, including mirrors of toric manifolds, by adapting the Eynard–Orantin recursion to mirror curves in C^* × C^*. This formalism unifies brane framing, open/closed moduli, and phase transitions, enabling calculations at geometric and non-geometric points (e.g., orbifolds) and providing extensive cross-checks against the topological vertex and Chern–Simons dualities on lens spaces. It produces explicit disk and higher-genus open amplitudes across multiple examples (vertex, resolved conifold, local P^2, local F_n) and offers predictions for orbifold cases like C^3/Z_3, illustrating the method’s reach beyond large-radius expansions. The work further connects the B-model recursion to the chiral boson picture and holographically to matrix-model loop equations, laying groundwork for potential compact-CY generalizations and deepening the interplay between geometry, dualities, and non-perturbative topological string theory.

Abstract

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi-Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A_p fibrations, where the amplitudes compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C^3/Z_3.

Paper Structure

This paper contains 61 sections, 253 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Summary of the results
  4. Outline
  5. Toric Calabi-Yau threefolds with branes
  6. Mirror symmetry and topological strings on toric Calabi-Yau threefolds
  7. A-model geometry
  8. B-model mirror geometry
  9. Open string mirror symmetry
  10. Topological open string amplitudes
  11. Moduli spaces, periods and flat coordinates
  12. Moving in the moduli space
  13. Open string phase structure
  14. The open and closed mirror maps
  15. Closed flat coordinates
  16. ...and 46 more sections

Figures (9)

  • Figure 1: Open string phase structure.
  • Figure 2: Toric base of ${\cal O}(-3)\rightarrow \mathbb{P}^2$ with an outer brane and the mirror curve with the open cycle defining the mirror map.
  • Figure 3: A graphic representation of the recursion relation (\ref{['rec1']}).
  • Figure 4: The framed vertex and its mirror curve.
  • Figure 5: The framed vertex in two legs.
  • ...and 4 more figures