black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
In the context of Kaluza-Klein theory where an electromagnetic field in Einstein-Maxwell theory in dimension $d$ is modeled by a configuration of pure Einstein gravity in dimension $d+1$, a Kaluza-Klein monopole is a configuration of gravity in dimension $d+1$ which in dimension $d$ looks like a magnetic monopole (Sorkin 83, Gross-Perry 83).
This situation is of particular interest in the reduction of 11-dimensional supergravity/M-theory to type IIA string theory, where the Kaluza-Klein magnetic monopole charge is interpreted as D6-brane charge.
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 $N_4\times \mathbb{R}^{11-5,1}$ of Euclidean Taub-NUT spacetime with Minkowski spacetime. Upon Kaluza-Klein compactification this looks like a monopole, whence the name.
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 a black hole in 4d, incarnated as a D0/D6 bound state (e.g. Nelson 93).
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 $\mathbb{R}^{6,1}\times \mathbb{R}^3\times S^1$ with metric tensor away from the origin of the $\mathbb{R}^3$-factor (which is the locus of the lifted D6/monopole) being
where
$A = A_i d x^i$ is any 3-form on $\mathbb{R}^3$ satisfying $d_{\mathbb{R}^3} A = \star d (1-\mu/r)$;
$r$ denotes the distance in $\mathbb{R}^3$ from the origin.
$\mu$ is the charge of the monopole.
Away from $\{0\} \in \mathbb{R}^3$ this gives a circle bundle with first Chern class measured by the integral of $R_0 \coloneqq d A$ (the RR-field of the D0-brane) over any sphere surrounding the singular locus in $\mathbb{R}^3 - \{0\}$. 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 $0 \in \mathbb{R}^3$ 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 $S^2 \hookrightarrow \mathbb{R}^3 -\{0\}$ is necessarily of the form $S^3 \to S^2$, 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.
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
(e.g. Johnson 97, Blumenhagen 10)
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
(The same kind of relation identifies the time-component of the M2-brane charge with the charge of the M9-brane, see there.)
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
Seung Kee Han, I.G. Koh, $N = 4$ Remaining Supersymmetry in Kaluza-Klein Monopole Background in D=11 Supergravity Theory, Phys.Rev. D31 (1985) 2503, in Michael Duff (ed.), The World in Eleven Dimensions 57-60 (spire)
Paul Townsend, The eleven-dimensional supermembrane revisited, Phys.Lett.B350:184-187,1995 (arXiv:hep-th/9501068)
Chris Hull, Gravitational Duality, Branes and Charges, Nucl.Phys. B509 (1998) 216-251 (arXiv:hep-th/9705162)
Ashoke Sen, Kaluza-Klein Dyons in String Theory, Phys.Rev.Lett.79:1619-1621,1997 (arXiv:hep-th/9705212)
Ashoke Sen, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (arXiv:hep-th/9707123)
Michael Atiyah, Juan Maldacena, Cumrun Vafa, An M-theory Flop as a Large N Duality, J.Math.Phys.42:3209-3220,2001 (arXiv:hep-th/0011256)
Michael Atiyah, Edward Witten $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)
Edward Witten, Anomaly Cancellation On Manifolds Of $G_2$ Holonomy (arXiv:hep-th/0108165)
Per Berglund, Andreas Brandhuber, Matter From $G_2$ Manifolds, Nucl.Phys. B641 (2002) 351-375 (arXivLhep-th/0205184)
Jerome Gaillard, Johannes Schmude, The lift of type IIA supergravity with D6 sources: M-theory with torsion, JHEP 1002:032,2010 (arXiv:0908.0305)
Clifford Johnson, section 10.5 of D-brane primer (arXiv:hep-th/0007170)
Katrin Becker, Melanie Becker, John Schwarz, p. 333 of String Theory and M-Theory: A Modern Introduction, 2007
Luis Ibáñez, Angel Uranga, section 6.3.3 of String Theory and Particle Physics: An Introduction to String Phenomenology, Cambridge University Press 2012
Bobby Acharya, Sergei Gukov, p. 45 of M theory and Singularities of Exceptional Holonomy Manifolds, Phys.Rept.392:121-189,2004 (arXiv:hep-th/0409191)
Relation to black holes in string theory