Abstract
We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifold \({\mathcal{M}}\) are in one-to-one correspondence with deformations of its 4d + 4-dimensional hyperkähler cone \({\mathcal{S}}\). The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space \({\mathcal{Z}_\mathcal{S}}\), with a suitable homogeneity condition that ensures that the hyperkähler cone property is preserved. Equivalently, we show that the deformations of \({\mathcal{M}}\) can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space \({\mathcal{Z}_\mathcal{M}}\) of \({\mathcal{M}}\), by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-Kähler metrics with d + 1 commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.
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Acknowledgements
We are grateful to A. Neitzke for discussions and former collaboration on related topics. The research of S.A. is supported by CNRS and by the contract ANR-05-BLAN-0029-01. The research of B.P. is supported in part by ANR(CNRS-USAR) contract no.05-BLAN-0079-01. F.S. acknowledges financial support from the ANR grant BLAN06-3-137168. S.V. thanks the Federation de Recherches “Interactions Fondamentales” and LPTHE at Jussieu for hospitality and financial support. Part of this work is also supported by the EU-RTN networkMRTN-CT-2004-005104 “Constituents, Fundamental Forces and Symmetries of the Universe”.
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Communicated by N.A. Nekrasov
Unité mixte de recherche du CNRS UMR 5207.
Unité mixte de recherche du CNRS UMR 7589.
Unité mixte de recherche du CNRS UMR 8549.
Unité de recherche associée au CNRS URA 2306.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Alexandrov, S., Pioline, B., Saueressig, F. et al. Linear Perturbations of Quaternionic Metrics. Commun. Math. Phys. 296, 353–403 (2010). https://doi.org/10.1007/s00220-010-1022-y
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DOI: https://doi.org/10.1007/s00220-010-1022-y