It is well understood that in type IIA string theory there appears
$SU(N)$ enhanced gauge symmetry on $N$-coincident D-branes;
$SO(2N)$ enhanced gauge symmetry on $N$-coincident D-branes at an orientifold plane.
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).
graphics grabbed from HSS18
The 11d multi-centered Kaluza-Klein monopole spacetime has metric tensor of the form
where
is the metric tensor of the multi-centered Taub-NUT space, with $x^4$ the canonical coordinate on a circle and with
and
for $\{\vec r_I\}_{I =1}^N$ the set of positions of the KK-monopoles of mass $m$.
Notice that the radius $16 \pi m V^{-1/2}$ of the $x^4$-circle vanishes precisely at the positions $\vec r_I$. Nevertheless, as long as all the $\vec r_I$ are distinct, then the above is a smooth spacetime also at the positions of the $\vec r_I$ – if the periodicity of $x^4$ is taken to be $16 \pi m$. But with this periodicity fixed, then as the $N$ monopole positions $\vec r_I$ coincide, then the resulting metric has a conical singularity at that point, of ADE-singularity type $A_{N-1}$.
To see where the gauge enhancement arises from in this singular case, observe that in the non-singular configuration there are $N-1$ linearly indepent 2-cycles $S_{i j}$ 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 $\vec r_i$ to $\vec r_j$ (remembering that the circle fiber radius vanishes precisely at the positions $\vec r_I$).
For the canonical choice of straight path between $\vec r_i$ and $\vec r_j$ (and arbitrary fixed position in the remaining 7 dimensions) then the surface area of these 2-spheres is
For any other choice of path the surface area will be larger. Hence an M2-brane with tension $T_{M2}$ swrapping the 2-cycle $S_{i j}$ has minimal tension energy when in the configuration of these spheres, namely
The type IIA limit is given by $m \to 0$. 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 $S_{i j}$ becomes an open string strentching between the $i$th and the $j$th D6-brane.
In the limit $m \to 0$ 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 $SU(N)$ Chan-Paton gauge field.
Now consider the above setupmodified by replacing the Taub-NUT space with coordinates $(\vec r, x^4)$ by its $\mathbb{Z}/2$-orbifold given by the $\mathbb{Z}/2$-action with nontrivial operation given by
and in the Taub-NUT metric replace $V$ by
The type IIA image of the origin of this configuration is an orientifold plane.
Now as the $\vec r_i$ all approach 0 we get an 11d spacetime with a $D_N$-type ADE-singularity, whose type IIA image is $N$ D6-branes coincident on an orientifold plane.
from M-branes to F-branes: superstrings, D-branes and NS5-branes
(e.g. Johnson 97, Blumenhagen 10)
Last revised on September 9, 2019 at 15:35:03. See the history of this page for a list of all contributions to it.