nLab Kaluza-Klein monopole




Riemannian geometry




In the context of Kaluza-Klein theory where an electromagnetic field in Einstein-Maxwell theory in dimension dd is modeled by a configuration of pure Einstein gravity in dimension d+1d+1, a Kaluza-Klein monopole is a configuration of gravity in dimension d+1d+1 which in dimension dd looks like a magnetic monopole (Sorkin 83, Gross-Perry 83).

In supergravity

This situation is of particular interest in the reduction of 11-dimensional supergravity (or M-theory, where one also speaks of the MK6-brane) where the Kaluza-Klein magnetic monopole charge is interpreted as D6-brane charge under duality between M-theory and type IIA string theory.

The Kaluza-Klein monopole (Han-Koh 85) is one type of solution of the equations of motion of 11-dimensional supergravity. It is given by the product N 4× 115,1N_4\times \mathbb{R}^{11-5,1} of Euclidean Taub-NUT spacetime with Minkowski spacetime. Upon Kaluza-Klein compactification this looks like a monopole, whence the name. (For discussion as an ADE-singularity see IMSY 98, section 9, Asano 00, section 3.)

Upon KK-compactification on a 6-dimensional fiber, with the 11d KK-monopole / D6-brane completely wrapping the fiber, the KK-monopole in 11d supergravity becomes the KK-monopole in 5d supergravity. Further compactifying on a circle leads to a black hole in 4d, incarnated as a D0/D6 bound state (e.g. Nelson 93).

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)


(For the moment the following is all about the KK-monopoles in 11d supergravity/M-theory.)

Far-horizon geometry

We discuss the far horizon geometry of coincident MK6-branes.

The metric tensor of NN coincident KK-monopoles in 11-dimensional supergravity in the limit that thN P0\ell_{th} \coloneqq N \ell_P \to 0 is

(1)g MK6=g 6,1+(dy) 2+y 2((dθ) 2+(sinθ) 2(dφ) 2+(cosθ) 2(dϕ) 2) g_{MK6} \;=\; g_{\mathbb{R}^{6,1}} + (d y)^2 + y^2 \big( (d \theta)^2 + (\sin \theta)^2 (d \varphi)^2 + (\cos \theta)^2 (d \phi)^2 \big)

subject to the identification

(2)(φ,ϕ)(φ,ϕ)+(2π/N,2π/N). (\varphi, \phi) \;\sim\; (\varphi, \phi) + (2\pi/N ,2\pi/N) \,.

This is equation (47) in IMSY 98, which applies subject to the condition

U/(Ng YM 2/3)=U/(N(2π) 4/3 P)1 U/\left(\frac{N}{g^{2/3}_{YM}}\right) \;=\; U/\left(\frac{N}{(2\pi)^{4/3} \ell_P}\right) \;\gg\; 1

from a few lines above. Inserting this condition into the definition y 22N P 3Uy^2 \coloneqq 2 N \ell^3_P U right above (47) shows that

y 2 =2N P 3U =2(2π) 4/3N 2 P 2(U/(N(2π) 4/3 P))1 \begin{aligned} y^2 & = 2 N \ell^3_P U \\ & = 2(2\pi)^{-4/3} N^2 \ell_P^2 \; \underset{ \gg 1 }{ \underbrace{ \left(U/\left(\frac{N}{ (2 \pi)^{4/3} \ell_P}\right)\right) }} \end{aligned}

hence that the distance yy from the locus of the MK6-brane is large in units of

th=2(2π) 2/3N P. \ell_{th} \;=\; \sqrt{2} (2\pi)^{-2/3} N \ell_P \,.

The identification (2) means that this is the orbifold metric cone 6,1×( 4/( N))\mathbb{R}^{6,1} \times \left( \mathbb{R}^4/(\mathbb{Z}_N)\right), hence an A-type ADE-singularity. To make this more explicit, introduce the complex coordinates

vye iφsinθwye iϕcosθ v \;\coloneqq\; y \, e^{i \varphi} \sin \theta \;\;\; w \;\coloneqq\; y \, e^{i \phi} \cos \theta

on 4 2\mathbb{R}^4 \simeq \mathbb{C}^2, in terms of which (1) becomes

g MK6dvdv¯+dwdw¯ g_{MK6} \;\coloneqq\; d v d \overline v + d w d \overline w

and which exhibit the identification (2) as indeed that of the A-type N\mathbb{Z}_N-action (Asano 00, around (18)).

Relation to the D6-brane in type IIA string theory

Under the relation between M-theory and type IIA superstring theory an ADE orbifold of the 11d KK-monopole corresponds to D6-branes combined with O6-planes (Townsend 95, p. 6, Atiyah-Witten 01, p. 17-18 see also e.g. Berglund-Brandhuber 02, around p. 15).

By (Townsend 95, (1), Sen 97c (1)-(4)) the 11d spacetime describing the KK-monopole lift of a plain single D6 brane is 6,1× 3×S 1\mathbb{R}^{6,1}\times \mathbb{R}^3\times S^1 with metric tensor away from the origin of the 3\mathbb{R}^3-factor (which is the locus of the lifted D6/monopole) being

ds 11 2=dt 2+ds 6 2+(1+μ/r)ds 3 2+(1+μ/r) 1(dx 11A idx i) 2, d s_{11}^2 = - d t^2 + d s_{\mathbb{R}^6}^2 + (1+\mu/r) d s_{\mathbb{R}^3}^2 + (1+ \mu/r)^{-1} (d x^{11} - A_i d x^i)^2 \,,


  • A=A idx iA = A_i d x^i is any 3-form on 3\mathbb{R}^3 satisfying d 3A=d(1μ/r)d_{\mathbb{R}^3} A = \star d (1-\mu/r);

  • rr denotes the distance in 3\mathbb{R}^3 from the origin.

  • μ\mu is the charge of the monopole.

Away from {0} 3\{0\} \in \mathbb{R}^3 this gives a circle bundle with first Chern class measured by the integral of R 0dAR_0 \coloneqq d A (the RR-field of the D0-brane) over any sphere surrounding the singular locus in 3{0}\mathbb{R}^3 - \{0\}. By the electric-magnetic duality between D0-branes and D6-branes (due to the self-duality of the RR-field) this means, from the 10-dimensional perspective, that at 0 30 \in \mathbb{R}^3 a D6-brane is located.

graphics grabbed from Acharya-Gukov 04

Notice that the circle bundle away from its degeneration locus as a bundle over S 2 3{0}S^2 \hookrightarrow \mathbb{R}^3 -\{0\} is necessarily of the form S 3S 2S^3 \to S^2, a multiple of the complex Hopf fibration (see also Atiyah-Maldacena-Vafa 00, p. 10).

Discussion as an A-type ADE singularity is in (Sen 97c, section 2). Generalization to D-type singularities and hence D6-branes in orientifolds is in (Sen 97c ,section 3). Discussion as the fixed point of the circle group-action on the M-theory circle fibers is in (Townsend 95, p. 6, Atiyah-Witten 01, pages 17-18). Witten emphasizes that it is important that the location of the D6 is not just a cyclic group orbifold singularity but really a circle group-action fixed point conical singularity:

Chiral fermions arise when the locus of A−D−E singularities passes through isolated points at which X has an isolated conical singularity that is not just an orbifold singularity (Witten 01, p. 2).

Discussion dealing carefully with the perspective where the locus of the D6-brane is not remived: Gaillard & Schmude 09.

For more on this see at D6-brane – Relation to other branes and at M-theory lift of gauge enhancement on D6-branes.

Relation to the D7-brane in type IIB string theory/F-theory

Under further T-duality to type IIB superstring theory/F-theory these D6-branes become the D7-branes.

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on S A 1×S B 1S^1_A \times S^1_B-elliptic fibrationKK-compactification on S A 1S^1_Atype IIA string theoryT-dual KK-compactification on S B 1S^1_Btype IIB string theorygeometrize the axio-dilatonF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping S A 1S_A^1double dimensional reduction \mapstotype IIA superstring\mapstotype IIB superstring\mapsto\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane\mapsto
M2-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp strings and qq D2-branes\mapsto(p,q)-string\mapsto
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane\mapsto
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\mapsto\mapstoNS5-brane
M5-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp D4-brane and qq NS5-branes\mapsto(p,q)5-brane\mapsto
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane\mapsto
KK-monopole/A-type ADE singularity (degeneration locus of S A 1S^1_A-circle fibration, Sen limit of S A 1×S B 1S^1_A \times S^1_B elliptic fibration)\mapstoD6-brane\mapstoD7-branes\mapstoA-type nodal curve cycle degeneration locus of elliptic fibration ADE 2Cycle (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity\mapstoD6-brane with O6-planes\mapstoD7-branes with O7-planes\mapstoD-type nodal curve cycle degeneration locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

Other Brane charges (?)

In (Hull 97) it was argued that the KK-monopole in 11-dimensional supergravity is the object which carries the 6-form charge Poincaré dual to the time-component of the 5-form charge of the M5-brane as appearing in the M-theory super Lie algebra via

5( 10,1) * 5( 10) * 6 10. \wedge^5 (\mathbb{R}^{10,1})^\ast \simeq \wedge^5 (\mathbb{R}^{10})^\ast \oplus \wedge^6 \mathbb{R}^{10} \,.

(The same kind of relation identifies the time-component of the M2-brane charge with the charge of the M9-brane, see there.)


In 5d gravity

Original articles:

Review includes

  • Emre Sakarya, Kaluza-Klein monopole, 2007 (pdf)

Discussion of topological T-duality for KK-monopoles is in

  • Ashwin S. Pande, Topological T-duality and Kaluza-Klein Monopoles, Adv. Theor. Math. Phys., 12, (2007), pp 185-215 (arXiv:math-ph/0612034)

In 11d supergravity

Original articles

Discussion for the massive case, reducing to the D6-brane in massive type IIA string theory:

Discussion in terms of G-structures:

Discussion of a BFSS-like matrix model for MK6-branes:

Relation to massive type IIA string theory:


Relation to black holes

Relation to black holes in string theory

  • William Nelson, Kaluza-Klein Black Holes in String Theory, Phys. Rev. D49:5302-5306, 1994 (arXiv:hep-th/9312058)

Last revised on June 30, 2023 at 18:22:11. See the history of this page for a list of all contributions to it.