How to remedy the eta-problem of SUSY GUT hybrid inflation via vector
  backreaction
  It is shown that, in supergravity models of inflation where the gauge kinetic
function of a gauge field is modulated by the inflaton, we can obtain a new
inflationary attractor solution, in which the roll-over of the inflaton suffers
additional impedance due to the vector field backreaction. As a result,
directions of the scalar potential which, due to strong Kaehler corrections,
become too steep and curved to normally support slow-roll inflation can now
naturally do so. This solves the infamous eta-problem of inflation in
supergravity and also keeps the spectral index of the curvature perturbation
mildly red despite eta of order unity. This mechanism is applied to a model of
hybrid inflation in supergravity with a generic Kaehler potential. The spectral
index of the curvature perturbation is found to be 0.97 - 0.98, in excellent
agreement with data. The gauge field can act as vector curvaton generating
statistical anisotropy in the curvature perturbation. However, this anisotropy
could be possibly observable only if the gauge coupling constant is unnaturally
small.
