Threshold corrections, generalised prepotentials and Eichler integrals

TL;DR

This paper analyzes one-loop integrals in $ =2$ heterotic string vacua, expressing them via Fourier series linked to Niebur-Poincaré series, revealing their connection to holomorphic prepotentials and Eichler integrals, and exploring their modular properties and quantum monodromies.

Contribution

It introduces a novel framework connecting one-loop integrals to holomorphic prepotentials and Eichler integrals, extending the analysis to non-factorisable tori and orbifolds with explicit formulas.

Findings

Fourier series representation of integrals in terms of Niebur-Poincaré series

Holomorphic prepotentials transform as Eichler integrals under T-duality

Explicit formulas for integrals involving ${ m extGamma}_0(N)$ Hauptmodul

Abstract

We continue our study of one-loop integrals associated to BPS-saturated amplitudes in $\mathcal{N}=2$ heterotic vacua. We compute their large-volume behaviour, and express them as Fourier series in the complexified volume, with Fourier coefficients given in terms of Niebur-Poincar\'e series in the complex structure modulus. The closure of Niebur-Poincar\'e series under modular derivatives implies that such integrals derive from holomorphic prepotentials $f_n$, generalising the familiar prepotential of $\mathcal{N}=2$ supergravity. These holomorphic prepotentials transform anomalously under T-duality, in a way characteristic of Eichler integrals. We use this observation to compute their quantum monodromies under the duality group. We extend the analysis to modular integrals with respect to Hecke congruence subgroups, which naturally arise in compactifications on non-factorisable tori and freely-acting orbifolds. In this case, we derive new explicit results including closed-form expressions for integrals involving the ${\varGamma}_0(N)$ Hauptmodul, a full characterisation of holomorphic prepotentials including their quantum monodromies, as well as concrete formulae for holomorphic Yukawa couplings.