The Euler characteristic correction to the Kaehler potential - revisited

TL;DR

This paper confirms the leading $ ext{α'}^3$ correction to the Kähler potential in type IIB compactifications, providing explicit solutions for the corrected internal metric and analyzing the impact on the moduli space in 4D theories.

Contribution

It explicitly derives the $ ext{α'}^3$-corrected internal metric and kinetic terms, clarifying the correction's effect on the Kähler potential and moduli space in type IIB Calabi-Yau compactifications.

Findings

Confirmed the $ ext{α'}^3$ correction proportional to Euler characteristic.

Derived the explicit form of the corrected internal metric.

Analyzed the correction's impact on the Kähler moduli space in 4D theories.

Abstract

We confirm the leading $\alpha'^3$ correction to the 4d, $\mathcal N = 1$ K\"{a}hler potential of type IIB orientifold compactifications, proportional to the Euler characteristic of the Calabi-Yau threefold (BBHL correction). We present the explicit solution for the $\alpha'^3$-modified internal background metric in terms of the non-harmonic part of the third Chern form of the leading order Calabi-Yau manifold. The corrected internal manifold is almost Calabi-Yau and admits an $SU(3)$ structure with non-vanishing torsion. We also find that the full ten-dimensional Einstein frame background metric is multiplied by a non-trivial Weyl factor. Performing a Kaluza-Klein reduction on the modified background we derive the $\alpha'^3$-corrected kinetic terms for the dilaton and the K\"{a}hler deformations of the internal Calabi-Yau threefold for arbitrary $h^{1,1}$. We analyze these kinetic terms in the 4d, $\mathcal N = 2$ un-orientifolded theory, confirming the expected correction to the K\"ahler moduli space prepotential, as well as in the 4d, $\mathcal N = 1$ orientifolded theory, thus determining the corrections to the K\"ahler potential and K\"ahler coordinates.