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KK-monopole geometries -- table

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)

Last revised on October 27, 2020 at 13:29:24. See the history of this page for a list of all contributions to it.