Towards Matter Inflation in Heterotic String Theory
  Recently, a class of inflation models in supergravity with gauge non-singlet
matter fields as the inflaton has been proposed. It is based on a `tribrid'
structure in the superpotential and on a Heisenberg symmetry for solving the
eta-problem. We suggest that a generalization of this model class may be
suitable for realising inflation in heterotic orbifold compactifications, where
the Heisenberg symmetry is a property of the tree-level K"ahler potential of
untwisted matter fields. We discuss moduli stabilization in this setup and
propose a mechanism to stabilize the modulus associated to the inflaton, which
respects the symmetry in the large radius limit. Inflation ends via a waterfall
phase transition, as in hybrid inflation. We give conditions which have to be
satisfied for realising inflation along these lines in the matter sector of
heterotic orbifolds.
