Towards matter inflation in heterotic string theory

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