Line bundles on spectral curves and the generalised Legendre transform construction of hyperkähler metrics

Bielawski, Roger

doi:10.1016/j.geomphys.2008.11.010

Cited by 5 publications

(6 citation statements)

References 30 publications

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“…, S r−1 . It has been shown in [5] that conditions such as (3.1) and (3.2) on spectral curves correspond to a particular choice of the function G and the contour c in formula (3.4). In fact, one can replace the usually multi-valued function G with a single-valued function on a branched cover of P 1 .…”

Section: Generalised Legendre Transformmentioning

confidence: 99%

Deformations of Instanton Metrics

Bielawski¹,

Yannic²,

Cherkis³

2023

SIGMA

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Section: Generalised Legendre Transformmentioning

confidence: 99%

Deformations of Instanton Metrics

Bielawski¹,

Yannic²,

Cherkis³

2023

SIGMA

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“…Since x is real, so is the scale factor ρ by (25). Without loss of generalities we may assume that the scale factor ρ is positive.…”

Section: The Function F For the Atiyah-hitchin Manifoldmentioning

confidence: 99%

Revisiting Atiyah-Hitchin manifold in the generalized Legendre transform

Arai¹,

Baba²,

Ionas³

2022

Preprint

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We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This is originally studied by Ivanov and Rocek and is subsequently investigated more by Ionas, in the latter of which the explicit forms of the Kähler potential and the Kähler metric are calculated. There is a difference between the former and the latter. In the generalized Legendre transform approach, a Kähler potential is constructed from the contour integration of one function with holomorphic coordinates. The choice of the contour in the latter is different from the former's one, whose difference may yield a discrepancy in the Kähler potential and eventually in the Kähler metric. We show that the former only gives the real Kähler potential, which is consistent with its definition, while the latter yields the complex one. We derive the Kähler potential and the Kähler metric for the Atiyah-Hitchin manifold for the former's choice and show that they are consistent with the ones evaluated in the latter. * arai(at)sci.kj.yamagata-u.ac.jp † kurando.baba(at)rs.tus.ac.jp ‡ radu.ionas(at)stonybrook.edu

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“…where we have used the Monge-Ampère equation [31], which holds for any 4-dimensional hyperkähler manifold,…”

Section: Brief Review Of the Generalized Legendre Transformmentioning

confidence: 99%

“…Here, we have used F z z + F xx = 0 in (31). We then obtain the Kähler metric of the Taub-NUT manifold as…”

Section: Taub-nut Manifoldmentioning

confidence: 99%

Special Lagrangian Submanifolds and Cohomogeneity One Actions on the Complex Projective Space

Arai¹,

Baba²

2019

Tokyo J. Math.

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We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.

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Line bundles on spectral curves and the generalised Legendre transform construction of hyperkähler metrics

Cited by 5 publications

References 30 publications

Deformations of Instanton Metrics

Deformations of Instanton Metrics

Revisiting Atiyah-Hitchin manifold in the generalized Legendre transform

Special Lagrangian Submanifolds and Cohomogeneity One Actions on the Complex Projective Space

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