Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Properties of Topological String Amplitudes and Orbifold Invariants

M. Alim, J. D. Laenge, P. Mayr

TL;DR

The paper develops a global polynomial representation for topological string amplitudes on local Calabi–Yau threefolds, leveraging BCOV holomorphic anomaly equations and special geometry. By carefully constructing propagators and absorbing non-holomorphic data, it fixes holomorphic ambiguities using boundary conditions at large complex structure, conifold points, and orbifold loci. The authors apply the framework to local P^2, F_0, and F_2 to compute higher-genus orbifold Gromov–Witten invariants, including new predictions for C^3/Z_4, and demonstrate how multiple moduli-space patches constrain the holomorphic ambiguity. This approach offers a practical, phase-spanning method to access topological-string data across moduli space and supports future extensions to compact geometries and D-brane configurations.

Abstract

We derive topological string amplitudes on local Calabi-Yau manifolds in terms of polynomials in finitely many generators of special functions. These objects are defined globally in the moduli space and lead to a description of mirror symmetry at any point in the moduli space. Holomorphic ambiguities of the anomaly equations are fixed by global information obtained from boundary conditions at few special divisors in the moduli space. As an illustration we compute higher genus orbifold Gromov-Witten invariants for C^3/Z_3 and C^3/Z_4.

Global Properties of Topological String Amplitudes and Orbifold Invariants

TL;DR

The paper develops a global polynomial representation for topological string amplitudes on local Calabi–Yau threefolds, leveraging BCOV holomorphic anomaly equations and special geometry. By carefully constructing propagators and absorbing non-holomorphic data, it fixes holomorphic ambiguities using boundary conditions at large complex structure, conifold points, and orbifold loci. The authors apply the framework to local P^2, F_0, and F_2 to compute higher-genus orbifold Gromov–Witten invariants, including new predictions for C^3/Z_4, and demonstrate how multiple moduli-space patches constrain the holomorphic ambiguity. This approach offers a practical, phase-spanning method to access topological-string data across moduli space and supports future extensions to compact geometries and D-brane configurations.

Abstract

We derive topological string amplitudes on local Calabi-Yau manifolds in terms of polynomials in finitely many generators of special functions. These objects are defined globally in the moduli space and lead to a description of mirror symmetry at any point in the moduli space. Holomorphic ambiguities of the anomaly equations are fixed by global information obtained from boundary conditions at few special divisors in the moduli space. As an illustration we compute higher genus orbifold Gromov-Witten invariants for C^3/Z_3 and C^3/Z_4.

Paper Structure

This paper contains 19 sections, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Polynomial Method
  3. Review of the Polynomial Structure of Topological Strings
  4. Constructing the Propagators
  5. Holomorphic ambiguity and boundary conditions
  6. Application to Local Mirror Symmetry
  7. Local $\mathbbm{P}^2$
  8. Local $\mathbbm{F}_0$
  9. Local $\mathbbm{F}_2$
  10. Conclusion
  11. The Ambiguities
  12. Local $\mathbbm{P}^2$
  13. Local ${\mathbbm{F}_0}$
  14. Local ${\mathbbm{F}_2}$
  15. Gopakumar-Vafa and Orbifold Gromov-Witten Invariants
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Fan and secondary fan for local $\mathbbm{P}^2$.
  • Figure 2: Fan and secondary fan for local $\mathbbm{F}_0$.
  • Figure 3: Fan and secondary fan for local $\mathbbm{F}_2$.