Homogeneous K�hler manifolds andT-algebras inN=2 supergravity and superstrings

Abstract

Motivated by the problem of the moduli space of superconformal theories, we classify all the (normal) homogeneous Kahler spaces which are allowed in the coupling of vector multiplets toN=2 SUGRA. Such homogeneous spaces are in one-to-one correspondence with the homogeneous quaternionic spaces (≠ℍH n ) found by Alekseevskii. There are two infinite families of homogeneous non-symmetric spaces, each labelled by two integers. We construct explicitly the corresponding supergravity models. They are described by acubic functionF, as in flat-potential models. They are Kahler-Einstein if and only if they are symmetric. We describe in detail the geometry of the relevant manifolds. They are Siegel (bounded) domains of the first type. We discuss the physical relevance of this class of bounded domains for string theory and the moduli geometry. Finally, we introduce theT-algebraic formalism of Vinberg to describe in an efficient way the geometry of these manifolds. The homogeneous spaces allowed inN=2 SUGRA are associated to rank 3T-algebras in exactly the same way as the symmetric spaces are related to Jordan algebras. We characterize theT-algebras allowed inN=2 supergravity. They are those for which theungraded determinant is a polynomial in the matrix entries. The Kahler potential is simply minus the logarithm of this “naive” determinant.