nLab ALE space

Idea

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 $\mathbb{C}^2/\Gamma$ for finite subgroup $\Gamma \hookrightarrow SU(2)$.

geometry transverse to KK-monopoles	Riemannian metric	remarks
Taub-NUT space: geometry transverse to $N+1$ distinct KK-monopoles at $\vec r_i \in \mathbb{R}^3 \;\; i \in \{1, \cdots, N+1\}$	$\array{d s^2_{TaubNUT} \coloneqq U^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U \coloneqq 1 + \underoverset{i = 1}{N+1}{\sum} U_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$	(e.g. Sen 97b, Sect. 2)
ALE space Taub-NUT close to $N$ close-by KK-monopoles e.g. close to $\vec r = 0$: $\frac{{\vert \vec r_i\vert}}{R/2}, \frac{{\vert \vec r\vert}}{R/2} \ll 1$	$\array{d s^2_{ALE} \coloneqq U'^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U' (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U' \coloneqq \underoverset{i = 1}{N+1}{\sum} U'_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U'_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$	e.g. via Euler angles: $\vec \omega = (N+1)R/2(\cos(\theta)-1) d\psi$ (e.g. Asano 00, Sect. 2)
$A_N$-type ADE singularity: ALE space in the limit where all $N+1$ KK-monopoles coincide at $vec r_i = 0$	$\array{d s^2_{A_N Sing} \coloneqq \frac{\vert\vec r\vert }{(N+1)R/2}(d x^4 + \vec \omega \cdot d \vec r)^2 + \frac{ (N+1)R/2}{\vert \vec r\vert} (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) }$	(e.g. Asano 00, Sect. 3)

In string theory

• In M-theory: KK-monopole

  In F-theory: D7-brane

References

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)

In

• Ken Intriligator, Nathan Seiberg, Mirror Symmetry in Three Dimensional Gauge Theories (arXiv:hep-th/9607207)

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

• Clifford Johnson, Robert Myers, Aspects of Type IIB Theory on ALE Spaces, Phys.Rev. D55 (1997) 6382-6393 (arXiv:hep-th/9610140)