Special Kähler manifolds: a survey

This is a survey of recent contributions to the area of special Kähler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001. 1 Remarkable features of special Kähler manifolds A (pseudo-) Kähler manifold (M,J, g) is a differentiable manifold endowed with a complex structure J and a (pseudo-) Riemannian metric g such that (i) J is orthogonal with respect to the metric g, i.e. J∗g = g and (ii) J is parallel with respect to the Levi Civita connection D, i.e. DJ = 0. In the following we will always allow pseudo-Riemannian, i.e. possibly indefinite metrics. The prefix “pseudo” will be generally omitted. The following definition is by now standard, see [F]. Definition 1 A special Kähler manifold (M,J, g,∇) is a Kähler manifold (M,J, g) together with a flat torsionfree connection ∇ such that (i) ∇ω = 0, where ω = g(·, J ·) is the Kähler form and (ii) ∇J is symmetric, i.e. (∇XJ)Y = (∇Y J)X for all vector fields X and Y . More precisely, one should speak of affine special Kähler manifolds since there is also a projective variant of special Kähler manifolds. In fact, there is a class of (affine) special Kähler manifolds M , which are called conic special Kähler manifolds and which admit a certain C∗-action. The quotient of M by that action can be considered as projectivisation of M and is called a projective special Kähler manifold, see [ACD]. Originally [dWVP1], in the supergravity literature, by a special Kähler manifold one understood a projective special Kähler manifold. This terminology was mantained in the first mathematical papers on that subject [C1, C2, AC] and abandoned with the publication of [F]. Example 1: Let (M,J, g) be a flat Kähler manifold, i.e. the Levi Civita connection D is flat. Then (M,J, g,∇ = D) is a special Kähler manifold and ∇J = 0. Conversely, any special Kähler manifold (M,J, g,∇) such that ∇J = 0 satisfies ∇ = D = Levi Civita connection of the flat Kähler metric g. This is the trivial example of a special Kähler manifold. Before giving a general construction of special Kähler manifolds, which yields plenty of non-flat examples, I would like to offer some motivation for that concept. • The notion of special Kähler manifold was introduced by the physicists de Wit and Van Proeyen [dWVP1] and has its origin in certain supersymmetric field theories. More precisely, affine special Kähler manifolds are exactly the allowed targets for the scalars of the vector multiplets of field theories with N = 2 rigid supersymmetry on four-dimensional Minkowski spacetime. Projective special Kähler manifolds correspond to such theories with local supersymmetry, which describe N = 2 supergravity coupled to vector multiplets. N = 2 supergravity theories occur as low energy limits of type II superstrings and play a prominent role in the study of moduli spaces of certain two-dimensional superconformal field theories [CFG]. The structure of these