ScalingStacks

Remark 9.8 . [03KA]

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Remark 9.8.

If we take each collection of wjw_{j} monopole points in 𝒩wj4​(−Tj−1,Tj+1)\mathcal{N}_{w_{j}}^{4}(-T_{j}-1,T_{j+1}) to have distances exactly proportional to β−1\beta^{-1} (in the flat metric on 𝕋2×ℝ\mathbb{T}^{2}\times\mathbb{R}) from each other, then the corresponding bubble limit will be a multi-Taub-NUT ALF-Awj−1A_{w_{j}-1} metric instead of having wjw_{j} Taub-NUT bubbles. It is also possible to obtain nontrivial bubble-trees. For example, if the distances of the monopole points in a collection of monopole points from each other is proportional to β−2\beta^{-2}, then there will be a first bubble which is a ALF orbifold with an orbifold point which is cyclic of order wjw_{j}, and the deepest bubble will then be an ALE-Awj−1A_{w_{j}-1} metric.

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