Geometric Construction of the r-Map: From Affine Special Real to Special Kahler Manifolds

We give an intrinsic definition of ( affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N = TM carries a canonical structure of ( affine) special Kahler manifold. This gives an intrinsic description of the r-map as the map M -> N = TM. On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when M is any Hessian manifold.