One-loop Einstein-Hilbert term in minimally supersymmetric type IIB orientifolds

TL;DR

The paper computes one-loop string corrections to the Einstein-Hilbert term in toroidal, minimally supersymmetric type IIB orientifolds with D-branes, highlighting their relevance for quantum corrections to the Kahler metric of moduli. It develops a general graviton 1-loop framework across all world-sheet surfaces and splits the results into ${\mathcal N}=1$ and ${\mathcal N}=2$ sectors, with new technical integrals arising in the ${\mathcal N}=1$ analysis for ${\mathbb Z}_6'$. In explicit models, the ${\mathcal N}=1$ contributions yield a universal finite piece proportional to the Clausen function ${\mathrm Cl}_2(\pi/3)$, while ${\mathcal N}=2$ sectors introduce moduli-dependent corrections via non-holomorphic Eisenstein series $E_2(U)$, including a large-volume winding contribution in ${\mathcal Z}_6'$. The Z$_3$ and Z$_6'$ examples illustrate how these corrections combine to give finite, scheme-dependent results that inform the quantum-corrected Kahler potential and field redefinitions, with implications for low-energy effective actions in string phenomenology.

Abstract

We evaluate string one-loop contributions to the Einstein-Hilbert term in toroidal minimally supersymmetric type IIB orientifolds with D-branes. These have potential applications to the determination of quantum corrections to the moduli Kahler metric in these models.