c--Map,very Special Quaternionic Geometry and Dual Kahler Spaces

TL;DR

The paper establishes a four-dimensional construction of a dual Kähler geometry for any very special geometry via a reduced c-map, introducing a dual metric $g^{ab}=V^{-2}(G^{-1})^{ab}$ and a dual Kähler potential $\\hat{K}=-2\ln V$. Through an $N=1$ truncation of the $N=2$ very special sigma-model, the authors derive a consistent reduction whose complex coordinates are $oldsymbol{\eta}_a=t_a+2\sqrt{2}\,i\tilde{\zeta}_a$, yielding a Kähler sector dual to the original very special geometry. They analyze the isometry structure, showing at least $2n+4$ symmetries in the dual quaternionic space and a residual $(n+4)$-dimensional subgroup after truncation, with an ${ m SL}(2,\mathbb{R})$ factor appearing ubiquitously; the dual and original manifolds generally differ, coinciding only for homogeneous-symmetric cases. The results connect to Calabi–Yau orientifolds in Type II strings, where consistent orientifold projections yield a distinct Kähler sector and no-scale properties, and they discuss potential ${\rm N}=4$ analogues and broader 4D applicability.

Abstract

We show that for all very special quaternionic manifolds a different N=1 reduction exists, defining a Kaehler Geometry which is ``dual'' to the original very special Kaehler geometry with metric G_{a\bar{b}}= - \partial_a \partial_b \ln V (V={1/6}d_{abc}λ^a λ^b λ^c). The dual metric g^{ab}=V^{-2} (G^{-1})^{ab} is Kaehler and it also defines a flat potential as the original metric. Such geometries and some of their extensions find applications in Type IIB compactifications on Calabi--Yau orientifolds.