nLab ALE space




An asymptotically locally Euclidean space or ALE space for short is a solution to the Euclidean Einstein equations which is a blow-up of an ADE-orbifold singularity 2/Γ\mathbb{C}^2/\Gamma for finite subgroup ΓSU(2)\Gamma \hookrightarrow SU(2).

geometry transverse to KK-monopolesRiemannian metricremarks
Taub-NUT space:
geometry transverse to
N+1N+1 distinct KK-monopoles
at r i 3i{1,,N+1}\vec r_i \in \mathbb{R}^3 \;\; i \in \{1, \cdots, N+1\}
ds TaubNUT 2U 1(dx 4+ωdr) 2+U(dr) 2, r 3,x 4/(2πR) U1+i=1N+1U i,AAωi=1N+1ω i U iR/2|rr i|,AA×ω=U i\array{d s^2_{TaubNUT} \coloneqq U^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U \coloneqq 1 + \underoverset{i = 1}{N+1}{\sum} U_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}(e.g. Sen 97b, Sect. 2)
ALE space
Taub-NUT close to NN close-by KK-monopoles
e.g. close to r=0\vec r = 0: |r i|R/2,|r|R/21\frac{{\vert \vec r_i\vert}}{R/2}, \frac{{\vert \vec r\vert}}{R/2} \ll 1
ds ALE 2U 1(dx 4+ωdr) 2+U(dr) 2, r 3,x 4/(2πR) Ui=1N+1U i,AAωi=1N+1ω i U iR/2|rr i|,AA×ω=U i\array{d s^2_{ALE} \coloneqq U'^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U' (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U' \coloneqq \underoverset{i = 1}{N+1}{\sum} U'_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U'_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}e.g. via Euler angles: ω=(N+1)R/2(cos(θ)1)dψ\vec \omega = (N+1)R/2(\cos(\theta)-1) d\psi
(e.g. Asano 00, Sect. 2)
A NA_N-type ADE singularity:
ALE space in the limit
where all N+1N+1 KK-monopoles coincide at vecr i=0vec r_i = 0
ds A NSing 2|r|(N+1)R/2(dx 4+ωdr) 2+(N+1)R/2|r|(dr) 2, r 3,x 4/(2πR)\array{d s^2_{A_N Sing} \coloneqq \frac{\vert\vec r\vert }{(N+1)R/2}(d x^4 + \vec \omega \cdot d \vec r)^2 + \frac{ (N+1)R/2}{\vert \vec r\vert} (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) } (e.g. Asano 00, Sect. 3)

In string theory


An ADE classification of 4d ALE-spaces is due to

  • Peter Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. (Euclid)


this result is interpreted physically as describing the moduli space of vacua of gauge theories with spontaneously broken symmetry (“Higgs branches”). See at 3d mirror symmetry for more on this.

For application in string theory see at KK-monopole and see

Last revised on December 30, 2019 at 18:29:17. See the history of this page for a list of all contributions to it.