content_image/357310.jpg content_image/357311.jpg content_image/357312.jpg content_image/357313.jpg
and y$_{i}$ = 0 are invariant. For a especial choice 

The N = 1 SUGRA potential reads [416, 424] 

<img src='content_image/357310.jpg'>

<img src='content_image/357311.jpg'>

Note that there is no cross term in the potential such as | φ ∗ $_{i}$W | $^{2}$. As a consequence any tree level flat direction remains flat even during inflation [416] (in fact it is the Heisenberg symmetry which protects the flat directions from obtaining Hubble induced masses [423]). A particular choice of K¨ ahler potential, i.e. 

<img src='content_image/357312.jpg'>

arises quite naturally from string compactifications [425, 426]. For a constant superpotential, W$_{0}$ , the F-term of the potential yields; 

and for the above choice of the K¨ ahler potential, V$_{F}$ = 0 for all ϕ , because the K¨ ahler potential satisfies ( K − $^{1}$) j $_{i}$K$^{i}$K$_{j}$ = 3, this is a property of no-scale model. The symmetry is broken by gauge interactions or by coupling in the renormalizable part of the K¨ ahler potential. Then the mass of the flat direction condensate arises from the running of the gauge couplings. 

## F. (SUSY) Grand Unified Theories 

<img src='content_image/357313.jpg'>

The observation that within SUSY the value of three gauge couplings nearly meet at ∼ 2 × 10 16 GeV has led to the idea 37 that the three gauge groups emerge from a single (Grand Unified) group G$_{GUT}$ with a single gauge coupling g$_{GUT}$ $^{38}$. Another motivation for 

