# What is the M-theory description of an E-type gauge enhancement?

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.

• 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? Mar 30, 2017 at 8:35