SYMPLECTIC EMBEDDINGS, SPECIAL KAHLER GEOMETRY AND AUTOMORPHIC-FUNCTIONS - THE CASE OF SK(N+1)=SU(1,1)/U(1)CIRCLE-TIMES-SO(2,N)/SO(2)CIRCLE-TIMES-SO(N)

In this paper we consider orbifolds of homogeneous special Kahler manifolds, namely varieties of the type L = L'/GAMMA where L' is a special Kahler coset manifold G/H and GAMMA subset-of G is a discrete subgroup of its isometry group. Varieties of this type appear as moduli spaces in orbifold compactification of superstrings, where GAMMA plays the role of target space modular group. Special varieties of this type may also be relevant in connection with topological field theories. We show that the construction of the homogeneous function F(X), encoding the special geometry of L', can be systematically derived from the symplectic embedding of the isometry group G into Sp(2n + 2, R), n being the complex dimension of L'. This is actually related to the Gaillard-Zumino construction of lagrangians with duality symmetries. Different embeddings yield different F(X). For the case defined in the title we obtain a new symplectic section OMEGA = (X, i partial derivative F(X)), generating a new set of special coordinates. They transform linearly under SO(n), differently from the old special coordinates that transform linearly only under SO(n-1). This solves an apparent paradox in superstring compactifications. From the embedding of G into Sp(2n + 2, R) one retrieves the embedding of GAMMA into Sp(2n + 2, Z). Recently a general formula has been proposed by Ferrara et al. [Nucl. Phys. B365 (1991) 431] to construct GAMMA-automorphic functions as infinite sums over a restricted set of integers. Our embedding yields the explicit rule to parametrize the restricted integers in terms of integers describing modular orbits. In particular, via this procedure we can give the formal definition of a PSL(2, Z) x SO(2, n, Z) automorphic function for any n.