content_image/333233.jpg content_image/333234.jpg content_image/333235.jpg content_image/333236.jpg
It was noticed in [482] that the potential for N = 2 SUSY gauge theory at ξ$_{+}$ = $^{ξ}$− = 0, ξ$_{3}$ = | ξ | are similar to the case of D -term inflation studied before in [645] with W = λS Φ$_{+}$$^{Φ}$− and D = | Φ$_{+}$ | 2 − | $^{Φ}$− | 2 − ξ , for which the potential is given by: 

with an assumption, λ = √$_{2}$$_{g}$$_{. If instead,}$ ξ$_{+}$ = $^{ξ}$− = 2 M 2 = ξ , one recovers a potential of an F -term inflation model with, W = λS (Φ ′$_{+}$ Φ ′ − − M $^{2}$), and, D = | Φ $^{′$_{+}$}$| 2 − | Φ ′ − | $^{2}$, with a potential: 

The first term of the potential now coincides with the F -term inflation potential in N = 1 theory, proposed by [62], under the assumption that the gauge coupling g = $_{λ/}$√$_{2.}$ 

<img src='content_image/333233.jpg'>

<img src='content_image/333234.jpg'>

The P -term model could be coupled to N = 1 SUGRA and the inflationary potential is given by [462, 482] 

The inflating trajectory takes place at Φ$_{+}$ $^{= Φ}$− = 0. After adding the 1-loop correction from gauge fields, one obtains [482] 

<img src='content_image/333235.jpg'>

<img src='content_image/333236.jpg'>

where f = ξ 2 $_{1}$+ ξ 2 2 ξ 2 , 0 ≤ f ≤ 1 and . . . stands for terms | S | $^{6}$/ 2 M 6 p and higher order gravitational corrections. Special case f = 0 corresponds to D-term inflation, and f = 1 corresponds to F-term inflation. A general P-term inflation model has an arbitrary 0 ≤ f ≤ 1. The running of the spectral index from blue to red and the amplitude of the perturbations are similar to what has been studied in [513]. 

## G. Embedding inflation in SUSY GUTs 

Embedding inflation within GUT has a long history. We give here an overview of some of the old attempts and the current status of some of these models, and then turn to more recent developments. 

