  The main problem of inflation in string theory is finding the models with a
flat potential, consistent with stabilization of the volume of the compactified
space. This can be achieved in the theories where the potential has (an
approximate) shift symmetry in the inflaton direction. We will identify a class
of models where the shift symmetry uniquely follows from the underlying
mathematical structure of the theory. It is related to the symmetry properties
of the corresponding coset space and the period matrix of special geometry,
which shows how the gauge coupling depends on the volume and the position of
the branes. In particular, for type IIB string theory on K3xT^2/Z with D3 or D7
moduli belonging to vector multiplets, the shift symmetry is a part of
SO(2,2+n) symmetry of the coset space [SU(1,1)/ U(1)]x[SO(2,2+n)/(SO(2)x
SO(2+n)]. The absence of a prepotential, specific for the stringy version of
supergravity, plays a prominent role in this construction, which may provide a
viable mechanism for the accelerated expansion and inflation in the early
universe.
