Isometries of special manifolds

TL;DR

The paper systematically classifies and analyzes the isometries of special manifolds—special Kähler, very special real, and very special quaternionic—and their homogeneous counterparts, using dimensional-reduction maps $r$ and $c$ to connect real, Kähler, and quaternionic sectors. It develops both concrete geometric formalisms (Kähler, quaternionic, and real manifolds) and an algebraic framework based on real Clifford algebras to characterize continuous isometries, culminating in a one-to-one correspondence between homogeneous real, Kähler, and quaternionic manifolds via the $L(q,P)$ and $L(4m,P, ilde P)$ classifications. The authors provide explicit root-space decompositions and isotropy groups for the isometry algebras across the three sectors, including both symmetric and non-symmetric cases, and they demonstrate that all homogeneous non-symmetric Alekseevskii spaces arise as very special quaternionic manifolds through the $c$ map. The results illuminate the symmetry structure of vector multiplet moduli spaces in supergravity and string compactifications, with concrete implications for dualities, moduli stabilization, and the geometry of Calabi–Yau and related compactifications.

Abstract

We describe special Kahler geometry, special quaternionic manifolds, and very special real manifolds and analyze the structure of their isometries. The classification of the homogeneous manifolds of these types is presented.