We study moduli stabilization in the context of M-theory on compact manifolds with G 2 holonomy, using superpotentials from flux and membrane instantons, and recent results for the Kähler potential of such models. The existence of minima with negative cosmological constant, stabilizing all moduli, is established. While most of these minima preserve supersymmetry, we also find examples with broken supersymmetry. Supersymmetric vacua with vanishing cosmological constant can also be obtained after a suitable tuning of parameters.
JHEP12(2004)018M-theory on (compact) manifolds X of G 2 holonomy leads to non-chiral [42] fourdimensional effective theories with N = 1 supersymmetry [43,44]. Non-abelian gauge groups and matter fields, which may account for the standard model particles, arise when the G 2 space develops singularities [45]-[53]. In this paper we will not consider such possible matter field sectors, but focus on the gravity/moduli part of the theory, which contains b 3 (X) chiral moduli multiplets Φ I . Their real parts parameterize the moduli space of the G 2 manifold, while their imaginary parts corresponds to axions. We are interested in stabilizing these moduli fields Φ I , combining the effects of flux and membrane instantons. The flux superpotential has been computed in refs. [27,44], while the result for the structure of membrane instanton contributions to the superpotential can be found in ref. [54]. The scalar potential of four-dimensional N = 1 supergravity also depends on the Kähler potential for which we rely on the results of ref.[55]. We will be focusing on compact G 2 manifolds constructed by blowing up the singularities of G 2 orbifolds [56]-[58] and [45]. For such G 2 manifolds, the moduli naturally split into two classes, (Φ I ) = (T a , U i ), namely the "bulk" moduli T a associated with the underlying torus and the "blow-up" moduli U i that arise from blowing up the singularities. We will address the stabilization of both types of moduli for realistic models.A practical problem is that there are no simple compact G 2 manifolds with, say, b 3 (X) = 1 available. In particular, the calculation of the Kähler potential in ref. [55] has been carried out for an example with b 3 (X) = 43. We will, therefore, have to deal with a large number of moduli. This task will be approached by starting with relatively simple, but characteristic, toy models and then working our way up to include the full complications of more realistic cases. After setting up the structure of the models in the next section, section 3 discusses the stabilization of the bulk moduli T a , first for a simple universal case, and then including all bulk moduli. Section 4 includes the blow-up moduli U i , again starting with a simple universal toy model and then moving on to include all moduli. Conclusions and further directions are presented in section 5.
General structure of four-dimensional effective theoriesIn this section, we will review the structure of four-dimensional N = 1 theories, which originate from M-theory on (compac...