content_image/373557.jpg content_image/373558.jpg
The phase of the conformon, Λ$_{Y}$ , is free and should be fixed to break the invariance of the theory under the K¨ ahler transformations that leave the Lagrangian invariant. Λ$_{Y}$ can be fixed by imposing that the superconformal superpotential is real, W = W ∗ . This leads to the regular description of N = 1 SUGRA, where the Lagrangian is fully described by the function G ≡ K/M 2 $_{P}$+ln( | W | $^{2}$/M6 $_{P}$). But during D -term inflation, this choice is meaningless as the superpotential is vanishing [462]. Alternatively, another choice to fix the gauge is to assume that Y is real [658], Y = Y ∗ (see [658] for the full description of the Lagrangian with this choice). 

The transformation of Y under U (1)$_{ξ}$ can be written as an imaginary constant, 

<img src='content_image/373557.jpg'>

where ξ is the FI term. From Eq. (257), it is clear that imposing the invariance of the conformal superpotential, δ$_{α}$ W = 0 = 3 δ$_{α}$Y + δ$_{α}$W , implies that the existence of the FI term requires the superpotential W not to be invariant, δ$_{α}$W = ig$_{ξ}$ξ/M 2 $_{P}$. 

For W = S Φ$_{+}$$^{Φ}$−, δ$_{α}$W = ig$_{ξ}$ ∑ q$_{i}$ , and the presence of a FI term imposes the anomaly (non-vanishing sum of charges) of the U (1)$_{ξ}$ symmetry, which is given by ξ/M 2 $_{P}$. This can be accommodated by modifying the charges from 0 , +1 , − 1, for example to 0, (1 − ξ/ 2 M 2 $_{P}$) and ( − 1 − ξ/ 2 M 2 $_{P}$) for S , Φ$_{+}$ $^{and Φ}$−, respectively. This leads to a tree level potential of the form: 

<img src='content_image/373558.jpg'>

This affects only the D -terms of the potential, which modify the effective masses of the $^{components of Φ}$± involving the expression for the one-loop contribution to V ( S ), but note that the amplitude, ξ/M 2 $_{P}$, is very small [74, 462], since √$_{ξ}$ is found at least three orders of magnitude below the Planck mass, from the normalization of the CMB anisotropies and the non-observation of cosmic strings. It is worth noting that this construction makes the D -term inflation more robust against non-renormalizable corrections to W , since the new charge assignment prevents, by symmetry, all terms of the form S (Φ$_{+}$$^{Φ}$−) $^{n}$/M 2 n − 2 P [74, 462]. 

## 4. D -term inflation without cosmic strings 

Several modifications of the original D -term model have also been proposed to avoid the formation of cosmic strings and therefore lift the constraint on the coupling constant, λ given 

