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Integrability of the holomorphic anomaly equations

Babak Haghighat, Albrecht Klemm, Marco Rauch

Abstract

We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the $\frac{1}{N^2}$ expansion of matrix models with more then one cut.

Integrability of the holomorphic anomaly equations

Abstract

We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the expansion of matrix models with more then one cut.

Paper Structure

This paper contains 30 sections, 161 equations, 3 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Local Mirror Symmetry
  3. The local A-model
  4. The local B-model
  5. Integrability of the holomorphic anomaly equation
  6. Direct Integration in local Calabi-Yau geometries
  7. Fixing the ambiguity
  8. $\mathds{K}_{\mathds{P}^2}=\mathcal{O}(-3)\rightarrow\mathds{P}^2$
  9. Global properties of the moduli space of $\Sigma(z)$
  10. Periods and genus zero and one amplitudes in all patches
  11. Expansion near the large radius point
  12. Expansion near the conifold
  13. Coordinates and amplitudes at the Orbifold
  14. Direct integration for $\mathds{K}_{\mathds{P}^2}$
  15. Modular expressions for the $F_g$ on $\mathds{K}_{\mathds{P}^2}$
  16. ...and 15 more sections

Figures (3)

  • Figure 1: Definition of the monodromies in ${\cal M}(\Sigma(z))=\mathds{P}^1\setminus \{z=0,z=-\frac{1}{27},\frac{1}{z}=0\}$.
  • Figure 2: Resolved Moduli Space of $\mathds{F}_0$
  • Figure 3: Conifold coordinates