Aspects of Special Kähler Geometry and Moduli Space Theory in String Compactifications and (2,2) Superconformal Models

In this thesis we are concerned with some aspects of special Kahler geometry arising in string theory and superconformal models. In the first chapter we present a general discussion on moduli and moduli spaces, with the purpose of clarifying some topics sketched in this introduction. In the second one, after a suitable definition of special Kahler geometry, we consider orbifolds of homogeneous special Kahler manifolds, namely varieties of the type M = M'/T where M' is a special Kahler coset manifold G / H and T C G is a discrete subgroup of its isometry group. Varieties of this type appear as moduli spaces in orbifold compactification of superstrings, where r plays the role of target space modular group. We show that the construction of the homogeneous function F(X), encoding the special geometry of M', can be systematically derived from the symplectic embedding of the isometry group G into Sp(2n + 2, R), n being the complex dimension of M' [43]. This is actually related to the Gaillard-Zumino [39] construction of Lagrangians with duality symmetries. Different embeddings yield different F(X). For the case of M 1 = SUU((li )l ) X so(S0(2 n) h th t 0 t . "bl t bt . 1 t• t• 2)xSO(n), we s ow a 1 1s poss1 e o o am a new symp ec ic sec 10n n = (X, i8F(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 r into Sp(2n + 2, Z). This embedding yields the explicit rule to give a formal definition of a PS L(2, Z) x 80(2, n, Z) automorphic superpotential for any n. As a second application we consider the duality group r = SU(3, 3, Z) [44] for the Narain lattice (45] of the T 6 / Z3 orbifold and its action on the corresponding moduli space M 3.3/r, where M 3 .3 = SU(J)~~i(~?xu(i)" A symplectic embedding of the momenta and winding numbers allows to connect the orbifold lattice to the special geometry of M 3.3 . A formal expression for an automorphic function, which is a candidate for a non-perturbative superpotential, is given. In the last chapter we consider the realization of (2, 2) superconformal models in terms of free first-order (b, c, ,B 1'Y )-systems [46, 4 7), and show that an arbitrary Landau-Ginzburg interaction with quasi-homogeneous potential can be introduced without spoiling the (2, 2) superconformal invariance (48]. We discuss the topological twisting and the renormalization group properties of these theories, and compare them to the conventional topological Landau-Ginzburg models (49). By deforming the theory with relevant and marginal operators it is possible to define perturbed correlation functions, and a suitable metric in the coupling constant space. After a proper bosonization of the first-order systems, explicit calculations of perturbed topological correlation functions are performed by using standard Coulomb gas techniques [50, 51, 52]. In the coupling constant space of topological field theories there is a preferred coordinate frame in which the metric is constant. These "flat coordinates" are strongly connected with special coordinate systems in special geometry. We show that in our formulation the parameters multiplying deformation terms in the potential are flat coordinates. We retrieve known results for minimal models and for the c = 3 cubic torus. The extension of the techniques presented here to a general c = 3d theory should give more information on the special geometry structure of the moduli spaces. At the end of the chapter we elaborate, for a particular example, on the relation between Picard-Fuchs equations and topological Landau-Ginzburg models Finally in the appendices we deserve some technical remarks on the topics treated in the thesis.