6. Embedding D -term models in string theory Attempts were made to embed D -term models within an explicit SUSY SU (6) model [66]. This will be discussed with other embedding of inflation in SUSY GUT below at section IV G. More recent modifications of the D -term models were motivated by the possibility that the model is generically realized at large field VEV. This opens up the importance of non- renormalizable corrections to the K¨ ahler potential [74, 512, 664], or corrections which are motivated from string theory [665–667]. It was found that the dynamics of the model remains mostly unchanged when considering any non-renormalizable corrections of order $_{M}$− 2 P in K¨ ahler potential [74, 664]. It was observed in Ref. [74] that despite the new contributions to the potential, the non-observation of cosmic strings still requires a suppressed superpotential coupling constant λ < ∼ 10 − $^{4}$. The D -term inflation models arising from string theories have generated a lot of activity in the past (within weakly coupled string theories) [10, 653, 654, 668], and more recently within a brane description or with a more phenomenological approach studying the moduli corrections to the model. The original interest for D -term inflation was in part due to the observation that anomalous U (1), generically appear in weakly coupled string theories with ∑ q$_{i}$ < 0. There are many problems to circumvent destabilizing the model by vacuum shifting due to ∑ q$_{i}$ < 0, or to generate the right amplitude for the FI term and the string coupling, see for discussion in [10, 653]. The D -term inflation has also been found to be the low energy description of brane inflation [462, 647, 648]. The existence of branes in string theory allows to construct a new class of inflationary models (see Sec. VIII), where the inflaton becomes a modulus describing the distance between two branes. The most studied examples are the D 3 /D 7 and the D 3 − D 3 [647, 669–671] brane systems. It has been shown that such a system give rise to a D -term model of inflation, the inter-brane distance possessing a flat direction at tree level, and the open string degrees of freedom between the two branes playing the role $^{of the waterfall fields Φ}$±, as one of them becomes tachyonic below a certain inter-brane separation, i.e. near the string length scale. The formation of cosmic (super)strings occurs at the end, in the form of D 1-branes or fundamental ( F -)strings [460, 672, 673]. A more phenomenological approach is to study the modifications of D -term models when embedded in string inspired SUGRA [665–667]. These corrections arise from the coupling (at 125