M-theory lift of gauge enhancement on D6-branes




It is well understood that in type IIA string theory there appears

The physical picture of this effect is that the gauge bosons of these gauge fields are the modes of the open strings stretching between these D-branes which become massless as the branes coincide.

Now, as the situation is lifted to M-theory, the D0-branes, D2-branes and D4-branes lift to M2-branes and M5-branes, and the gauge enhancement is thought to be similarly reflected on these M-branes (as exhibited for the M2-branes by the BLG-model and ABJM-model).

But for the D6-brane the situation is different: the D6-brane lifts not to an M-brane, but to a configuration of the field of 11-dimensional supergravity: the 11d Kaluza-Klein monopole.

Here we discuss the picture of how gauge enhancement on D6-branes in type IIA string theory is reflected on Kaluza-Klein monopoles at ADE-singularities. The explicit realization reviewed below is due to (Sen 97).

ADE 2Cycle

graphics grabbed from HSS18


A N1A_{N-1}-singularity and SU(N)SU(N)-gauge enhancement

The 11d multi-centered Kaluza-Klein monopole spacetime has metric tensor of the form

ds 2dt 2+n=510dy ndy n+ds TN 2, ds^2 \coloneqq - d t^2 + \underoverset{n = 5}{10}{\sum} d y^n d y^n + ds^2_{TN} \,,


ds TN 2V 1(dx 4+ωdr) 2+Vdr 2 ds^2_{TN} \coloneqq V^{-1}( d x^4 + \vec \omega \cdot d \vec r )^2 + V d \vec r^2

is the metric tensor of the multi-centered Taub-NUT space, with x 4x^4 the canonical coordinate on a circle and with

V1+I=1N4m|rr I| V \coloneqq 1 + \underoverset{I = 1}{N}{\sum} \frac{4m}{{\vert \vec r - \vec r_I\vert}}


×ω=V \vec \nabla \times \vec \omega = - \vec \nabla V

for {r I} I=1 N\{\vec r_I\}_{I =1}^N the set of positions of the KK-monopoles of mass mm.

Notice that the radius 16πmV 1/216 \pi m V^{-1/2} of the x 4x^4-circle vanishes precisely at the positions r I\vec r_I. Nevertheless, as long as all the r I\vec r_I are distinct, then the above is a smooth spacetime also at the positions of the r I\vec r_I – if the periodicity of x 4x^4 is taken to be 16πm16 \pi m. But with this periodicity fixed, then as the NN monopole positions r I\vec r_I coincide, then the resulting metric has a conical singularity at that point, of ADE-singularity type A N1A_{N-1}.

ADE 2Cycle

To see where the gauge enhancement arises from in this singular case, observe that in the non-singular configuration there are N1N-1 linearly indepent 2-cycles S ijS_{i j} in the multi-center KK-monopole spacetime, represented by the the 2-spheres that are swept out by the circle fiber as it moves from r i\vec r_i to r j\vec r_j (remembering that the circle fiber radius vanishes precisely at the positions r I\vec r_I).

For the canonical choice of straight path between r i\vec r_i and r j\vec r_j (and arbitrary fixed position in the remaining 7 dimensions) then the surface area of these 2-spheres is

vol(S ij)=16πm|r ir j|. vol(S_{i j}) = 16 \pi m {\vert \vec r_i - \vec r_j\vert} \,.

For any other choice of path the surface area will be larger. Hence an M2-brane with tension T M2T_{M2} swrapping the 2-cycle S ijS_{i j} has minimal tension energy when in the configuration of these spheres, namely

m ij=16πmT M2|r ir j|. m_{i j} = 16 \pi m T_{M2} {\vert \vec r_i - \vec r_j\vert} \,.

The type IIA limit is given by m0m \to 0. In this limit the M2-branes wrapping the above cycles become the type IIA superstring by double dimensional reduction, the KK-monopoles become the D6-branes, and it is evident from the geometry that the membrane warpping S ijS_{i j} becomes an open string strentching between the iith and the jjth D6-brane.

In the limit m0m \to 0 the D6-branes coincide,the strings stretching between them become massless (in accord with the above formula for the wrapped M2-brane mass), and become the gauge bosons of an SU(N)SU(N) Chan-Paton gauge field.

(Sen 97, section 2)

D ND_{N}-singularity and SO(2N)SO(2N)-gauge enhancement

Now consider the above setupmodified by replacing the Taub-NUT space with coordinates (r,x 4)(\vec r, x^4) by its /2\mathbb{Z}/2-orbifold given by the /2\mathbb{Z}/2-action with nontrivial operation given by

(r,x 4)(r,x 4) (\vec r, x^4) \mapsto (- \vec r, - x^4)

and in the Taub-NUT metric replace VV by

V116mr+i=1N(4m|rr i|+4m|r+r i|). V \coloneqq 1 - \frac{16m}{r} + \underoverset{i = 1}{N}{\sum} \left( \frac{4m}{\vert \vec r - \vec r_i\vert} + \frac{4m}{\vert \vec r + \vec r_i\vert} \right) \,.

The type IIA image of the origin of this configuration is an orientifold plane.

Now as the r i\vec r_i all approach 0 we get an 11d spacetime with a D ND_N-type ADE-singularity, whose type IIA image is NN D6-branes coincident on an orientifold plane.

(Sen 97, section 3)

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)


Last revised on May 17, 2018 at 05:14:40. See the history of this page for a list of all contributions to it.