Finally, the presence of anomalous U (1) symmetries in weakly coupled string theo- ries [649–652] generated a lot of hope, though there are problems in fully embedding the model and the generation of the FI term [653, 654]. Also, the model tends to require large inflaton values (Planckian, even super-Planckian in part of the parameter space) unlike F -term inflation, which represents another challenge for the model, and thus necessitates D -term inflationary models to be studied in the context of SUGRA with a VEV below M$_{P}$ . where g$_{a}$ and T a are respectively the gauge coupling constants and the generators of each factors of the symmetry of the action, ′ a ′ running over all factors of the symmetry, and f ( φ ) is the gauge kinetic function. If this symmetry contains a factor U (1)$_{ξ}$, not originating from a larger non-abelian group, the most general action allows for the presence of an additional constant contribution ξ . Below we will assume that such an abelian symmetry is the only symmetry of the inflaton sector. The simplest realization of D -term inflation reproduces the hybrid potential with three chiral superfields, S , φ$_{+}$ , and $^{φ}$− with non-anomalous U (1)$_{ξ}$ (an abelian theory is said to be anomalous if the trace of the generator is non-vanishing ∑ q$_{n}$ = 0) charges q$_{n}$ = 0 , +1 , − 1 [645, 646]. The superpotential can be written as (where K m ≡ ∂K/∂ Φ$_{m}$) give rise to a scalar potential within N = 1 SUGRA: Let us consider both F -and D -terms contribute to the scalar potential. Given a K¨ ahler potential K (Φ$_{m}$ , Φ$_{n}$), the D -terms 1. Minimal hybrid inflation from D -terms In what follows, we assume the minimal structure for f (Φ$_{i}$) (i.e., f (Φ$_{i}$)=1) and take the 115 D a = − g$_{a}$ [ D$_{a}$ = φ$_{i}$ ( T$_{a}$ ) $^{i}$$_{j}$Kj + ξ$_{a}$ ] V $^{SUGRA}$( $_{φ, φ}$∗$_{) =}$ 1 2 [Re f ( φ $_{)]}$− 1 ∑ D $^{a}$D$_{a}$ + F − terms (246) W D = λSφ$_{+}$$^{φ}$− . (247)