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).

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 97 (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 97, section 2). Generalization to D-type singularities and hence D6-branes in orientifolds is in (Sen 97 ,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 taken out is in (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 theoryF-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\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane
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
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\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
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane
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-branesA-type nodal curve cycle degenertion 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-planesD-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

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 include

  • Rafael Sorkin, Kaluza-Klein monopole Phys. Rev. Lett. 51 (1983) 87 (publisher)

  • David Gross and M. Perry, Nucl. Phys. B226 (1983) 29.

  • A. L. Cavalcanti de Oliveira, E. R. Bezerra de Mello, Kaluza-Klein Magnetic Monopole in Five-Dimensional Global Monopole Sapcetime, Class.Quant.Grav. 21 (2004) 1685-1694 (arXiv:hep-th/0309189)

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 in terms of G-structures:

  • Ulf Danielsson, Giuseppe Dibitetto, Adolfo Guarino, KK-monopoles and G-structures in M-theory/type IIA reductions, JHEP 1502 (2015) 096 (arXiv:1411.0575)


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 August 25, 2018 at 12:00:01. See the history of this page for a list of all contributions to it.