Symplectic Embeddings and Special Kahler Geometry of CP(n-1,1)

TL;DR

This paper explores the embedding of isometry groups into symplectic groups for certain coset spaces, revealing implications for special geometry in N=2 supergravity and duality symmetries in heterotic string compactifications.

Contribution

It demonstrates the embedding of isometry groups into symplectic groups and analyzes the existence of holomorphic prepotentials in these geometries, impacting supergravity and string theory dualities.

Findings

Certain embeddings lack a homogeneous prepotential.

Gauge kinetic terms depend on scalar fields via these embeddings.

Results inform duality symmetries in heterotic orbifold compactifications.

Abstract

The embedding of the isometry group of the coset spaces SU(1,n)/ U(1)xSU(n) in Sp(2n+2,R) is discussed. The knowledge of such embedding provides a tool for the determination of the holomorphic prepotential characterizing the special geometry of these manifolds and necessary in the superconformal tensor calculus of N=2 supergravity. It is demonstrated that there exists certain embeddings for which the homogeneous prepotential does not exist. Whether a holomorphic function exists or not, the dependence of the gauge kinetic terms on the scalars characterizing these coset in N=2 supergravity theory can be determined from the knowledge of the corresponding embedding, \`a la Gaillard and Zumino. Our results are used to study some of the duality symmetries of heterotic compactifications of orbifolds with Wilson lines.