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Direct Integration of the Topological String

Thomas W. Grimm, Albrecht Klemm, Marcos Marino, Marlene Weiss

TL;DR

This paper introduces a direct-integration approach to solve BCOV holomorphic anomaly equations for type B topological strings by exploiting the tight interplay between non-holomorphic dependence and modularity. It provides a generic formalism valid for any Calabi–Yau, and develops detailed implementations for two key examples: the Seiberg–Witten local Calabi–Yau and the Enriques Calabi–Yau, yielding closed expressions for $F^{(g)}$ at low genus and systematic boundary data to fix holomorphic ambiguities. The Enriques case yields a rich automorphic structure, a fiber all-genus product formula, and a clear field-theory limit revealing modular properties of $SU(2)$ with $N_f=4$; these results are then extended via a big-moduli-space framework to generic CYs. The work further derives a general recursive and Feynman-rule–based formalism on the big moduli space, enabling direct integration of higher-genus amplitudes with manifest modularity, and outlining practical directions for extracting holomorphic ambiguities using boundary conditions and dualities. Overall, the methods greatly enhance computational control over higher-genus topological strings on compact geometries and connect them to automorphic structures and field-theory limits.

Abstract

We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non-holomorphic dependence of the amplitudes, and relies on the interplay between non-holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi-Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its gravitational corrections. The second example is the Enriques Calabi-Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N=2 super Yang-Mills theory with four massless hypermultiplets.

Direct Integration of the Topological String

TL;DR

This paper introduces a direct-integration approach to solve BCOV holomorphic anomaly equations for type B topological strings by exploiting the tight interplay between non-holomorphic dependence and modularity. It provides a generic formalism valid for any Calabi–Yau, and develops detailed implementations for two key examples: the Seiberg–Witten local Calabi–Yau and the Enriques Calabi–Yau, yielding closed expressions for at low genus and systematic boundary data to fix holomorphic ambiguities. The Enriques case yields a rich automorphic structure, a fiber all-genus product formula, and a clear field-theory limit revealing modular properties of with ; these results are then extended via a big-moduli-space framework to generic CYs. The work further derives a general recursive and Feynman-rule–based formalism on the big moduli space, enabling direct integration of higher-genus amplitudes with manifest modularity, and outlining practical directions for extracting holomorphic ambiguities using boundary conditions and dualities. Overall, the methods greatly enhance computational control over higher-genus topological strings on compact geometries and connect them to automorphic structures and field-theory limits.

Abstract

We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non-holomorphic dependence of the amplitudes, and relies on the interplay between non-holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi-Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its gravitational corrections. The second example is the Enriques Calabi-Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N=2 super Yang-Mills theory with four massless hypermultiplets.

Paper Structure

This paper contains 36 sections, 391 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The holomorphic anomaly equations
  3. Solving Seiberg-Witten theory by direct integration
  4. The Seiberg-Witten geometry
  5. Direct integration
  6. Boundary conditions
  7. A first look at the Enriques Calabi-Yau
  8. Special geometry of the classical moduli space
  9. Genus one and the free energies on the Enriques fiber
  10. An all--genus product formula on the fiber
  11. Automorphic forms
  12. Direct Integration on the Enriques Calabi-Yau
  13. A simple direct integration and $F^{(g)}$ to genus six
  14. Propagators and homolomorphic automorphic forms
  15. Direct integration of the holomorphic anomaly
  16. ...and 21 more sections

Figures (3)

  • Figure 1: The singular loci in the moduli space ${\cal N}_8$, leading to two different gauge theories in the field theory limit.
  • Figure 2: A sample pair of Young diagrams $Y_\lambda,Y_{\tilde{\lambda}}$ contributing to \ref{['ZNek']}.
  • Figure 3: The Feynman graph expansion for $F^{(2)}$.