Special geometries

Abstract

In this chapter we will discuss the geometric concepts behind the special geometries. In terms of the supergravity theories, this means that we will restrict to the bosonic sector of these theories. In the first part of this chapter, we will examine the scalars of the vector multiplets, and we will see that the special Kähler geometry for the scalars is very much related to the symplectic transformations defined by the dualities of the vector fields. We will also discuss the attractor mechanism for black holes in this context. Then we will study the geometry of quaternionic-Kähler manifolds, related to the scalars of the hypermultiplets. Finally, we will present the relations between these manifolds and their versions in the different dimensions.