A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds ofG₂Holonomy

Abstract

We consider the breaking of N = 1 supersymmetry by non-zero G-flux when Mtheory is compactified on a smooth manifold X of G 2 holonomy. Gukov has proposed a superpotential W to describe this breaking in the low-energy effective theory. We check this proposal by comparing the bosonic potential implied by W with the corresponding potential deduced from the eleven-dimensional supergravity action. One interesting aspect of this check is that, though W depends explicitly only on G-flux supported on X, W also describes the breaking of supersymmetry by G-flux transverse to X.

More dual fluxes and moduli fixing

“…In addition we consider the presence of seven 4-form backgrounds g IJKL corresponding to fluxes of the M-theory 3-form. The presence of these two types of fluxes gives rise to a superpotential [41,42,39] …”

New potentials from Scherk-Schwarz reductions

“…The corresponding Kähler potential reads 17) where due to the symmetries of the theory we can set X 0 = 1. The expressions for the period matrix N ΛΣ are given in (B.12), and the field strengths appearing in (3.10) are written as…”

Local Geometry of the G 2 Moduli Space

“…The metric (4.62) on M C arises naturally from the Kaluza-Klein reduction of the M -theory action. As shown in [19], it turns out that this metric is in fact Kähler, with the Kähler potential K given by…”

“…Let us now use our Kaluza-Klein ansatz to reduce the 11-dimensional action (4.45) to 4 dimensions. Here we follow [19], [20] and [14]. The term which interests us is the kinetic term for the z N .…”

“…Here we follow [19], but in [20] and [9], in particular, a different convention is used. Let us show that K is indeed the Kähler potential for G MN .…”