8
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • $\begingroup$ 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

Reset to default
5
$\begingroup$

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!

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.