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Higher-Order Gravitational Couplings and Modular Forms in $N=2,D=4$ Heterotic String Compactifications

Bernard de Wit, Gabriel Lopes Cardoso, Dieter Lüst, Thomas Mohaupt, Soo-Jong Rey

TL;DR

De Wit, Lopes Cardoso, Lust, Mohaupt, and Rey develop a symmetry-guided framework to constrain higher-genus gravitational couplings ${\cal F}^{(g)}$ in $N=2$, $D=4$ heterotic strings by enforcing target-space duality through symplectic covariance and holomorphic anomaly analysis. They show that non-holomorphic corrections are essential to form symplectic functions and that, in models with a type-II dual, the heterotic symplectic anomaly aligns with the BCOV holomorphic anomaly in a large Kähler-class limit. Focusing on the $S$-$T$-$U$ heterotic model, they solve the holomorphic anomaly equations for ${\cal F}^{(2)\rm\ cov}$ and ${\cal F}^{(3)\rm\ cov}$, expressing the results entirely in terms of modular forms (e.g., ${\hat G}_2$) and logarithmic modular objects like $\log(j(T)-j(U))$, thereby achieving duality-covariant, unambiguous covariant sections. They further argue that ${\cal F}^{(2)\rm\ cov}$ is fixed without holomorphic ambiguities by duality and leading singularities, enabling potential extraction of genus-2 instanton data on the dual Calabi–Yau, and strengthening the link between heterotic and type-II descriptions. The work provides a concrete, symmetry-driven method to compute higher-genus couplings in heterotic vacua and clarifies their modular structure and duality relations.

Abstract

The restrictions of target--space duality are imposed at the perturbative level on the holomorphic Wilsonian couplings that encode certain higher-order gravitational interactions in $N=2, D=4$ heterotic string compactifications. A crucial role is played by non-holomorphic corrections. The requirement of symplectic covariance and an associated symplectic anomaly equation play an important role in determining their form. For models which also admit a type-II description, this equation coincides with the holomorphic anomaly equation for type-II compactifications in the limit that a specific Kähler-class modulus grows large. We explicitly evaluate some of the higher-order couplings for a toroidal compactification with two moduli $T$ and $U$, and we express them in terms of modular forms.

Higher-Order Gravitational Couplings and Modular Forms in $N=2,D=4$ Heterotic String Compactifications

TL;DR

De Wit, Lopes Cardoso, Lust, Mohaupt, and Rey develop a symmetry-guided framework to constrain higher-genus gravitational couplings in , heterotic strings by enforcing target-space duality through symplectic covariance and holomorphic anomaly analysis. They show that non-holomorphic corrections are essential to form symplectic functions and that, in models with a type-II dual, the heterotic symplectic anomaly aligns with the BCOV holomorphic anomaly in a large Kähler-class limit. Focusing on the -- heterotic model, they solve the holomorphic anomaly equations for and , expressing the results entirely in terms of modular forms (e.g., ) and logarithmic modular objects like , thereby achieving duality-covariant, unambiguous covariant sections. They further argue that is fixed without holomorphic ambiguities by duality and leading singularities, enabling potential extraction of genus-2 instanton data on the dual Calabi–Yau, and strengthening the link between heterotic and type-II descriptions. The work provides a concrete, symmetry-driven method to compute higher-genus couplings in heterotic vacua and clarifies their modular structure and duality relations.

Abstract

The restrictions of target--space duality are imposed at the perturbative level on the holomorphic Wilsonian couplings that encode certain higher-order gravitational interactions in heterotic string compactifications. A crucial role is played by non-holomorphic corrections. The requirement of symplectic covariance and an associated symplectic anomaly equation play an important role in determining their form. For models which also admit a type-II description, this equation coincides with the holomorphic anomaly equation for type-II compactifications in the limit that a specific Kähler-class modulus grows large. We explicitly evaluate some of the higher-order couplings for a toroidal compactification with two moduli and , and we express them in terms of modular forms.

Paper Structure

This paper contains 13 sections, 119 equations.

Table of Contents

  1. Introduction
  2. The holomorphic sections ${\cal F}^{({g})}(z)$
  3. Symplectic transformations of the ${\cal F}^{({g})}(z)$ and holomorphic anomalies
  4. The holomorphic anomaly equations and their solutions for the heterotic string
  5. The weak-coupling limit in the heterotic string and target--space duality
  6. The anomaly equation for the heterotic $S$-$T$-$U$ model
  7. Leading holomorphic singularities in heterotic models
  8. The modular covariant section ${\cal F}^{({2})\,{\rm cov}}$ in the $S$-$T$-$U$ model
  9. The modular covariant section ${\cal F}^{({3})\,{\rm cov}}$ in the $S$-$T$-$U$ model
  10. Conclusions
  11. Some asymptotic results for ${S+\bar{S}}\to\infty$
  12. A class of non-holomorphic corrections to ${\cal F}^{({2})\,{\rm cov}}$ and ${\cal F}^{({3})\,{\rm cov}}$ in the $S$-$T$-$U$ model
  13. Power-series expansion of ${\cal F}^{({2})\,{\rm cov}}$