In section 3 "M-theory, $G_2$-manifolds and four-dimensional physics"1, Acharya discusses how non-Abelian effective gauge theories arise locally from M-theory on spaces with ADE singularities $\mathbb{C}^2/\Gamma$, where $\Gamma\subset\mathrm{SU}(2)$ is an ADE subgroup. The description is purely geometric and does not actually tell us what kind of M-theory/11d SUGRA solution this phenomenon corresponds to.

In string theory, it is common to view such gauge enhancement - the appearance of non-Abelian gauge groups - as a symptom of coincident D-branes. In "A Note on Enhanced Gauge Symmetries in M- and String Theory" (this is also mentioned in e.g. the textbook by Becker, Becker and Schwarz), Sen shows how the correspondence between $D_6$-branes in type IIa and the Kaluza-Klein monopole and be used to understand A-type gauge groups, i.e. $\mathrm{SU}(n)$, as coincident Kaluza-Klein monopoles in Taub-NUT space, and D-type gauge groups, i.e. $\mathrm{SO}(2n)$, as coincident Kaluza-Klein monopoles together with an orientifold projection to a space he calls Atiyah-Hitchin space.

Notably, the E-type singularities are absent in this description, but occur unquestionably in the purely geometric description. So it is natural to wonder what the analogous dynamical M-theoretic and/or type IIa theoretic description of the emergence of an E-type gauge group is. What configuration of branes/monopoles in what geometry leads to E-type gauge groups?

1The reference is semi-randomly chosen, the same argument can be found more or less explicitly in most discussions of gauge enhancement.

  • $\begingroup$ Note that already for D-type gauge groups, you have orientifold planes, which are non-dynamical objects (as opposed to D-branes). Those non-dynamical objects implement the quotient $\mathbb{C^2}/\Gamma$. My guess would be that for E-type singularities, you can introduce an analogous (non-dynamical) E-fold in the brane system. Maybe the reason why this is not discussed in the literature is that these E-folds are not easier to manipulate than the geometric description as the quotient of $\mathbb{C}^2$ used to define them? $\endgroup$
    – Antoine
    Commented Mar 30, 2017 at 8:35

1 Answer 1


That's a good question. I can offer one hint:

As you probably know, there is a popular folkore that the way that the 11d KK-monopole exhibits an M-theory lift of the gauge enhancement on coincident D6-branes is that there are M2-branes wrapping the vanishing 2-cycles that appear in the blowup of the ADE-singularity:

enter image description here

(Graphics grabbed from HSS18)

Now by du Val's theorem, these 2-cycles are spheres that touch along the shape of an ADE-type Dynkin diagram. The picture above shows the A-type case, where the spheres, hence the M2-branes that wrap them, are all aligned in a row. As the picture indicates, this is the M-theory lift of strings stretching "linearly" between D6-branes

But in the D-type cases, as well as in the three E-type exceptional cases, the Dynkin diagram in addition has one triple node

enter image description here

Hence, if the picture about gauge enhancement by M2-branes wrapping vanishing cycles of the blowup of an ADE-singularity is correct, then the M-theory geometry of these more general DE-type singularities must be the famous, if maybe still somewhat mysterious... membrane triple junctions!


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