NUCLEAR Nuclear Physics B 371 (1992) 659—679 P H VS I CS B North-Holland ________________ Symplectic embeddings, special Kähler geometry and automorphic functions *: The case of SK(n + 1) = SU(1, 1)/U(1) ® SO(2, n)/SO(2) ® SO(n) Pietro Fré and Paolo Soriani SISSA, International Schoolfor Advanced Studies, VIa Beirut 2, 1-34100 Trieste, and INFN, sezione di Trieste, Italy Received 10 July 1991 Accepted for publication 3 October 1991 In this paper we consider orbifolds of homogeneous special Kähler manifolds, namely varieties of the type .9’ 5”/F where .9” is a special Kähler coset manifold G/H and F c G is a discrete subgroup of its isometry group. Varieties of this type appear as moduli spaces in orbifold compactification of superstrings, where F 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 5”, can be systematically derived from the symplectic embedding of the isometry group G into Sp(2n + 2, II), n being the complex dimension of .9”. 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 £2 = (X, iaF(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, Il) one retrieves the embedding of F into Sp(2n + 2, Z). Recently a general formula has been proposed by Ferrara et al. [NucI. Phys. B365 (1991) 431) to construct F-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 PSU2, Z) ® SO(2, n, 7L) automorphic function for any n. 1. Introduction In recent times the geometry of special Kahler manifolds ~ [1—8] has emerged as a crucial structure in dealing with several problems related to the moduli of N = 1 superstring compactifications on Calabi—Yau manifolds [9—13], N = 2 super- * Work supported in part by Ministero dell’Universita’e della Ricerca Scientifica e Tecnologica. 0550-3213/92/$05.00 © 1992 — Elsevier Science Publishers B.V. All rights reserved