Special geometry and symplectic transformations

TL;DR

This paper explores the structure of special Kahler manifolds in N=2 supergravity and rigid supersymmetry, focusing on symplectic transformations, their role in isometries, and connections to other special geometries.

Contribution

It provides a detailed analysis of symplectic transformations in special Kahler geometry and their implications for supergravity and supersymmetric models.

Findings

Symplectic transformations induce isometries or reparametrizations.

The paper clarifies the relation between holomorphic functions and symplectic structures.

Connections to quaternionic and real special manifolds are discussed.

Abstract

Special Kahler manifolds are defined by coupling of vector multiplets to $N=2$ supergravity. The coupling in rigid supersymmetry exhibits similar features. These models contain $n$ vectors in rigid supersymmetry and $n+1$ in supergravity, and $n$ complex scalars. Apart from exceptional cases they are defined by a holomorphic function of the scalars. For supergravity this function is homogeneous of second degree in an $(n+1)$-dimensional projective space. Another formulation exists which does not start from this function, but from a symplectic $(2n)$- or $(2n+2)$-dimensional complex space. Symplectic transformations lead either to isometries on the manifold or to symplectic reparametrizations. Finally we touch on the connection with special quaternionic and very special real manifolds, and the classification of homogeneous special manifolds.