How to get masses from extra dimensions

Curvatures and potential of M–theory inD= 4 with fluxes and twist

“…The other six A i vectors become massive because of the twisting of the torus. We conclude that in this theory there are always eight massless scalars and four massless vectors, in agreement with [16]. The effect of turning on g andg is not of giving extra masses, but of shifting the v.e.v.…”

“…(2.10),(2.11) and (2.12) can be checked by further applying to them the D operator. As explained in reference [32], the set of curvatures (F I , F IJ ) do not form an ordinary Lie algebra since the "structure constants" f ΛΣ Γ entering the quadratic part of: 16) do not satisfy the Jacobi identities of a Lie algebra:…”

“…The scalar potential in (4.13) and (4.14) has the property that V K ≥ 0, V C−S ≥ 0 while V E has no definite sign [16]. Therefore in general we may have vacua with different signs of the cosmological constant.…”

Supersymmetric completion of M-theory 4D-gauge algebra from twisted tori and fluxes

“…A popular example of this class of theories corresponds to the compactification on twisted tori with form-fluxes turned on [10,11,12,13,14,15,16,17]. These models [18,19,20,21,22,23,24], depending on the choice of fluxes, can give rise to no-scale supergravities [25] with partially broken supersymmetry or other type of vacua.…”

“…Twisted tori can be described as seven dimensional group manifolds whose isometries are part of the gauge group of the theory [10]. This seven dimensional gauge algebra, which is always spontaneously broken for flat groups [10], enlarges to a bigger symmetry when the vectors (or their dual) coming from the 3-form are also included, thus realizing a non-trivial 28-dimensional subalgebra [28] of the maximal rigid symmetry E 7(7) [29].…”