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Symmetry structure of special geometries

B. de Wit, F. Vanderseypen, A. Van Proeyen

TL;DR

The article analyzes how the isometry algebras of special geometries—real, Kähler, and quaternionic—are shaped by dimensional reduction via the r-map and c-map, revealing systematic enhancements and hidden symmetries. It develops a detailed, algebraic framework for the symmetry structure, including root-lattice decompositions, solvable subalgebras, and Iwasawa decompositions, and applies this to both generic and homogeneous spaces. A central outcome is that homogeneous d-spaces can be fully characterized as coset spaces G/H, with explicit constructions based on Clifford algebras and cubic data d_{ABC}, and with symmetry content that distinguishes symmetric from non-symmetric cases. These results illuminate how higher-dimensional origins constrain lower-dimensional moduli spaces and duality groups, advancing understanding of dualities and geometric structures in N=2 supergravity and string compactifications.

Abstract

Using techniques from supergravity and dimensional reduction, we study the full isometry algebra of Kähler 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.

Symmetry structure of special geometries

TL;DR

The article analyzes how the isometry algebras of special geometries—real, Kähler, and quaternionic—are shaped by dimensional reduction via the r-map and c-map, revealing systematic enhancements and hidden symmetries. It develops a detailed, algebraic framework for the symmetry structure, including root-lattice decompositions, solvable subalgebras, and Iwasawa decompositions, and applies this to both generic and homogeneous spaces. A central outcome is that homogeneous d-spaces can be fully characterized as coset spaces G/H, with explicit constructions based on Clifford algebras and cubic data d_{ABC}, and with symmetry content that distinguishes symmetric from non-symmetric cases. These results illuminate how higher-dimensional origins constrain lower-dimensional moduli spaces and duality groups, advancing understanding of dualities and geometric structures in N=2 supergravity and string compactifications.

Abstract

Using techniques from supergravity and dimensional reduction, we study the full isometry algebra of Kähler 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 . These include all homogeneous quaternionic spaces.

Paper Structure

This paper contains 19 sections, 179 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Special Kähler manifolds
  4. Special real manifolds
  5. Special quaternionic manifolds
  6. The symmetry algebra of special Kähler and quaternionic spaces
  7. Symmetries of $d$-spaces.
  8. Symmetries of the real space.
  9. Symmetries of the Kähler space.
  10. Symmetries of the quaternionic space.
  11. Summary
  12. Homogeneous $d$-spaces.
  13. Symmetries of the Clifford algebra.
  14. Symmetries of the real space.
  15. Symmetries of the Kähler space.
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Root lattice corresponding to the isometries of an $n=3$ special Kähler manifold discussed in the text.
  • Figure 2: Root lattice corresponding to the isometries of an $n=1$ special quaternionic manifold discussed in the text.
  • Figure 3: Roots of quaternionic $d$-spaces in the $\beta$-$\epsilon^0$ plane