  In this paper we obtain a model of Natural Inflation from string theory with
a Planckian decay constant. We investigate D-brane dynamics in the background
of the warped resolved conifold (WRC) throat approximation of Type IIB string
compactifications on Calabi-Yau manifolds. When we glue the throat to a compact
bulk Calabi-Yau, we generate a D-brane potential which is a solution to the
Laplace equation on the resolved conifold. We can exactly solve this equation,
including dependence on the angular coordinates. The solutions are valid down
to the tip of the resolved conifold, which is not the case for the more
commonly used deformed conifold. This allows us to exploit the effect of the
warping, which is strongest at the tip. We inflate near the tip using an
angular coordinate of a D5-brane in the WRC which has a discrete shift
symmetry, and feels a cosine potential, giving us a model of Natural Inflation,
from which it is possible to get a Planckian decay constant whilst maintaining
control over the backreaction. This is because the decay constant for a wrapped
brane contains powers of the warp factor, and so can be made large, while the
wrapping parameter can be kept small enough so that backreaction is under
control.
