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Topological open strings on orbifolds

Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti

Abstract

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C^3/Z_3 orbifold point of local P^2, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov-Witten invariants of C^3/Z_3, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

Topological open strings on orbifolds

Abstract

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C^3/Z_3 orbifold point of local P^2, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov-Witten invariants of C^3/Z_3, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

Paper Structure

This paper contains 30 sections, 2 theorems, 118 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Open B-model on mirrors of toric Calabi-Yau threefolds
  3. The geometry
  4. Disk amplitude
  5. General formalism
  6. Modularity
  7. Picard-Fuchs equations, monodromy and modularity of the closed amplitudes
  8. Modularity of the open amplitudes
  9. Modular forms and functionals
  10. The annulus amplitude
  11. Higher amplitudes
  12. Open orbifold Gromov-Witten invariants of ${\mathbb C}^3 / {\mathbb Z}_3$
  13. Geometry
  14. Open and closed mirror maps
  15. Closed mirror map
  16. ...and 15 more sections

Key Result

Proposition 2.11

Let be the elementary symmetric polynomials in the four branch points, and let be the holomorphic differential. The annulus amplitude can be written as where $\tau$ is the modular parameter, $E_2(\tau)$ is the second Eisenstein series, and the rational function $f_0^{(0,2)}(x_1,x_2)$ readsRecall from remark rem:dep that we do not write explicitly the dependence on $z$ for simplicity.

Theorems & Definitions (16)

  • Example 2.1
  • Remark 2.2
  • Conjecture 2.3: AKVAV
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Conjecture 2.9
  • Remark 2.10
  • ...and 6 more