Symmetry structure of special geometries

TL;DR

This paper investigates the symmetry structures of special geometries like Kähler and quaternionic manifolds, revealing how dimensional reduction and supergravity techniques uncover their isometry algebras and hidden symmetries.

Contribution

It provides a detailed analysis of the isometry algebra of special geometries, clarifies the emergence of extra and hidden symmetries, and characterizes homogeneous spaces as coset spaces.

Findings

Identification of full isometry algebras for special geometries

Conditions for the existence of hidden symmetries

Classification of homogeneous special manifolds as coset spaces

Abstract

Using techniques from supergravity and dimensional reduction, we study the full isometry algebra of K\"ahler and quaternionic manifolds with special geometry. These two varieties are related by the so-called c-map, which can be understood from dimensional reduction of supergravity theories or by changing chirality assignments in the underlying superstring theory. An important subclass, studied in detail, consists of the spaces that follow from real special spaces using the so-called r-map. We generally clarify the presence of `extra' symmetries emerging from dimensional reduction and give the conditions for the existence of `hidden' symmetries. These symmetries play a major role in our analysis. We specify the structure of the homogeneous special manifolds as coset spaces $G/H$. These include all homogeneous quaternionic spaces.