Abstract
We study general linear perturbations of a class of 4d real-dimensional hyperkähler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d + 1 variables, as opposed to the functions of d + 1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kähler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah–Hitchin manifold away from its negative mass Taub-NUT limit.
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Acknowledgments
We are very grateful to R. Ionas for correspondence on [29], M. Roček for informing us of his upcoming work [33], and to A. Neitzke for enlightening discussions. The research of S. Alexandrov is supported by CNRS and by the contract ANR-05-BLAN-0029-01. The research of B. Pioline is supported in part by ANR(CNRS-USAR) contract no. 05-BLAN-0079-01. F. Saueressig acknowledges financial support from the ANR grant BLAN06-3-137168. S. Vandoren 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 network MRTN-CT-2004-005104 “Constituents, Fundamental Forces and Symmetries of the Universe”.
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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 Hyperkähler Metrics. Lett Math Phys 87, 225–265 (2009). https://doi.org/10.1007/s11005-009-0305-8
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DOI: https://doi.org/10.1007/s11005-009-0305-8