Twist geometry of the c-map
type: Item
instance of: scholarly article
title: Twist geometry of the c-map
zbMATH Open document ID: 1327.53097
author: Óscar Maciá
author: Andrew Swann
published in: Communications in Mathematical Physics
publication date: 2015-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1404.0785
review text: 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 …
reviewed by: Hugues Auvray
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6424635
zbMATH Keywords: T-duality
zbMATH Keywords: twist construction
zbMATH Keywords: conic special Kähler
zbMATH Keywords: rotating circle symmetry
MaRDI profile type: Publication
OpenAlex ID: W3099277263
arXiv ID: 1404.0785
cites work: CLASSIFICATION OF QUATERNIONIC SPACES WITH A TRANSITIVE SOLVABLE GROUP OF MOTIONS
cites work: Quaternionic Kähler metrics associated with special Kähler manifolds
cites work: Conification of Kähler and hyper-Kähler manifolds
cites work: Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence
cites work: Linear perturbations of quaternionic metrics
cites work: Linear perturbations of hyperkähler metrics
cites work: Q3969490
cites work: Homogeneous quaternionic Kähler structures on eight-dimensional non-compact quaternion-Kähler symmetric spaces
cites work: GEOMETRY OF TYPE II SUPERSTRINGS AND THE MODULI OF SUPERCONFORMAL FIELD THEORIES
cites work: Alekseevskian spaces
cites work: Completeness in supergravity constructions
cites work: Modifying hyper-Kähler manifolds with circle symmetry
cites work: Special geometry, cubic polynomials and homogeneous quaternionic spaces
cites work: Hyperkähler metrics on cotangent bundles
cites work: Special Kähler manifolds
cites work: A note on manifolds whose holonomy group is a subgroup of Sp(n) Sp(1)
cites work: Hyper-Kähler and quaternionic Kähler manifolds with \(S^{1}\)-symmetries
cites work: Q3787091
cites work: Q3626795
cites work: The Hyperholomorphic Line Bundle
cites work: On the hyperkähler/quaternion Kähler correspondence
cites work: A canonical hyperkähler metric on the total space of a cotangent bundle
cites work: Q5748047
cites work: String loop corrected hypermultiplet moduli spaces
cites work: Q3810469
cites work: HyperKähler and quaternionic Kähler geometry
cites work: Q3171517
cites work: Twisting Hermitian and hypercomplex geometries
cites work: Twists versus modifications
DOI: 10.1007/S00220-015-2314-Z
description: scientific article