Twist geometry of the c-map
The c-map construction, producing quaternionic Kähler manifolds (i.e., Riemannian manifolds with ``special'' holonomy, lying inside \(\mathrm{SP}(n)\cdot \mathrm{SP}(1)\)) out of certain Kähler manifolds, takes its origin in physics [\textit{S. Cecotti} et al., Int. J. Mod. Phys. A 4, No. 10, 2475--2529 (1989; Zbl 0681.58044)]; it can more specifically be related to some string theoretic duality. In the same vein, another type of string theoretic duality, the \textit{T-duality}, finds a mathematical avatar in the so-called \textit{twist construction}. This twist construction, analysed by the second author in previous works [in: Proceedings of the XV international workshop on geometry and physics, 2006. Madrid: Real Sociedad Matemática Española. 83--94 (2007; Zbl 1229.53074); Duke Math. J. 155, No. 2, 403--431 (2010; Zbl 1214.53060)], gives a method of producing hypercomplex structures. The ``central purpose'' of the authors of the present article is precisely to adapt the twist construction so that it produces quaternionic Kähler manifolds as well; combined with a uniqueness result, they moreover get a twist description of the c-map via this adaptation. It is worth mentioning that, for the sake of generality, the authors essentially work with metrics of arbitrary signature. The c-map can be decomposed into two steps; the first one, known as the \textit{rigid c-map} [\textit{D. S. Freed}, Commun. Math. Phys. 203, No. 1, 31--52 (1999; Zbl 0940.53040)], ends up with a hyper-Kähler manifold on the whole cotangent bundle. This is reminded and reformulated in the appropriate language -- that of structure bundles, allowing the twist interpretation all along the constructions -- in Section 2 of the article; a stress is also put on the \textit{special} condition for the Kähler manifolds on which the rigid c-map applies, and on its naturality, regarding the aim of constructing non-local hyper-Kähler metrics. Add a \textit{conic} condition to the special Kähler condition; the resulting \textit{conic special Kähler} manifolds -- or possibly their Kähler quotients, up to some extra data -- then become \textit{hyper-Kähler manifolds with a rotating circle symmetry} under the rigid c-map (Section 3). The main result (Theorem 4.1), concerning the second step of the c-map, then follows, that specifies the only possible 1-parameter family of adapted twists producing quaternionic Kähler manifolds from hyper-Kähler manifolds with a rotating symmetry. These twists imply conformal change of the metric and conformal scalings in the quaternionic directions of the symmetry. The two steps of the c-map are joined together in Section 5, where several geometric features of these twists applied to conic special Kähler manifolds are described. Finally, Section 6 deals with two examples (flat and non-flat) based on the real hyperbolic plane, and Section 7 addresses some modifications, enlightened by the twist picture, one may perform to the c-map construction, preserving the quaternionic Kähler property of the manifolds it produces.
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