(2,2) VACUUM CONFIGURATIONS FOR TYPE-IIA SUPERSTRINGS - N = 2 SUPERGRAVITY LAGRANGIANS AND ALGEBRAIC-GEOMETRY

The compactification of D = 10 type IIA supergravity (the point-field theory limit of the type IIA superstring) on a Calabi-Yau manifold ((2,2) vacuum) is performed. The resulting D = 4, N = 2 effective supergravity Lagrangian gives rise to an explicit construction of special Kahler manifolds for the (1, 1) moduli and dual quaternionic manifolds for the (2, 1) moduli. In both cases the scalar manifolds are characterized by a homogeneous holomorphic function of degree two. In the case of the (1, 1) moduli, the holomorphic function is given by a cubic polynomial, a result which is valid only in the classical large volume limit (compared to the string scale) of the internal manifold. In contrast, for the (2, 1) moduli the holomorphic function, which is related to the 'periods' of a holomorphic three form, is unrestricted and due to non-renormalization theorems, should coincide with the exact (tree-level) string result.