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

TL;DR

This paper investigates higher-order gravitational couplings in N=2, D=4 heterotic string compactifications, emphasizing the role of modular forms, non-holomorphic corrections, and symplectic covariance, with explicit calculations for a toroidal model.

Contribution

It establishes the form of higher-order couplings using duality constraints, symplectic covariance, and modular forms, connecting heterotic and type-II descriptions.

Findings

Derived explicit expressions for higher-order couplings in terms of modular forms.

Showed the equivalence of the symplectic anomaly equation with the holomorphic anomaly in certain limits.

Provided detailed calculations for a specific toroidal compactification with two moduli.

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 K\"ahler-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.