Kahler Potential for M-theory on a G_2 Manifold

TL;DR

This paper presents a concrete calculation of the four-dimensional moduli Kahler potential and gauge-kinetic functions for M-theory compactified on a compact $G_2$ manifold. It employs Joyce's explicit compact example with small torsion, constructed by blowing up 12 co-dimension four fixed points of $T^7/\mathbb{Z}_2^3$ via Eguchi-Hanson spaces, and expresses the Kahler potential in terms of 43 real moduli and axions through controlled period/volume computations. The authors verify parts of the Kahler metric by direct integrals over (approximate) harmonic forms and derive the type-dependent gauge-kinetic functions, including the behavior in singular limits that enhance the gauge group. The results provide explicit, tree-level data for the 4D effective theory of M-theory on $G_2$ manifolds, enabling studies of moduli stabilization, SUSY breaking, and phenomenology in GUT-like setups with geometric origin of gauge sectors.

Abstract

We compute the moduli Kahler potential for M-theory on a compact manifold of G_2 holonomy in a large radius approximation. Our method relies on an explicit G_2 structure with small torsion, its periods and the calculation of the approximate volume of the manifold. As a verification of our result, some of the components of the Kahler metric are computed directly by integration over harmonic forms. We also discuss the modification of our result in the presence of co-dimension four singularities and derive the gauge-kinetic functions for the massless gauge fields that arise in this case.