We point out that the ability of some models of inflation, such as the Higgs inflation and the universal attractor models at large values of the coupling ξ, in reproducing the available data is due to their relation to the Starobinsky model of inflation. For large field values, where the inflationary phase takes place, all of these classes of models are indeed identical to the Starobinsky model. Nevertheless, the inflation is just an auxiliary field in the Jordan frame of the Starobinsky model, and this leads to two important consequences: first, the inflationary predictions of the Starobinsky model and its descendants are slightly different (albeit not measurably); second, the theories have different small-field behavior, leading to different ultraviolet cutoff scales. In particular, one interesting descendant of the Starobinsky model is the nonminimally coupled quadratic chaotic inflation. Although the standard quadratic chaotic inflation is ruled out by the recent Planck data, its nonminimally coupled version is in agreement with observational data and valid up to Planckian scales.

We discuss a D3-D7 system in type IIB string theory. The near-horizon geometry is described by AdS 5 × X 5 where X 5 is a U (1) bundle over a Kähler-Einstein complex surface S with positive first Chern class c 1 > 0. The surface S can either be P 1 × P 1 , P 2 or P n 1 ,...,n k , a blow up of P 2 at k points with 3 ≤ k ≤ 8. The P 2 corresponds to the maximally supersymmetric AdS 5 × S 5 vacuum while the other cases lead to vacua with less supersymmetries. In the F-theory context they can be viewed as compactifications on elliptically fibered almost Fano 3-folds. *

We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N=4 SU(N) super-Yang-Mills theory at large N. The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark-antiquark potential

We systematically examine corrections to the gravitational inverse square law, which are due to compactified extra dimensions. We find the induced Yukawa-type potentials for which we calculate the strength α and range. In general the range of the Yukawa correction is given by the wavelength of the lightest Kaluza-Klein state and its strength, relative to the standard gravitational potential, by the corresponding degeneracy. In particular, when n extra dimensions are compactified on an n-torus, we find that the strength of the potential is α = 2n, whereas the compactification on an n-sphere gives α = n + 1. For Calabi-Yau compactifications the strength can be at most α = 20.