Where does $E_8$ come from in M-Theory?

Question

Where does the $E_8$ symmetry comes from in M-Theory?

For example when you compactify one of the dimensions on a line you get E8xE8 heterotic string theory.

Or if you compactify 11D Supergravity leaving just 3 dimensions the theory has $E_8$ symmetry.

Is it a mystery? Like maybe M-Theory has some $E_{11}$ symmetry that nobody knows about?

Or is there a simple explanation?

(For example heterotic string theory has a "simple" explanation that 16 of the 26 dimensions of the left-handed bosonic modes are compactified on an $E_8\times E_8$ torus lattice.)

Answer 1

Compactifying M-Theory on an interval (a line segment) is usually thought of as a $\mathbb{Z}_{2}$ quotient acting on 'ordinary' M-Theory on $\mathbb{R}^{1,9} \times S^1$. The $\mathbb{Z}_2$ action on $S^{1}$ is what produces an interval with two fixed points. On the two fixed points you have two 10-dimensional boundaries (which so far is just because the fixed points have something like an $\mathbb{R}^{1,9}$ fibered over them). The distance between the two boundaries equals half the circumference of the original $S^1$. But you can do more.

The low-energy effective theory of M-Theory is 11-dimensional supergravity. Compactifying this as above leads to a bulk supergravity theory which far away from the boundaries "looks" pretty much like 11-dimensional supergravity but with a modification of the supersymmetry transformation laws and the Bianchi identity, due to the presence of the boundaries. On the (even-dimensional) boundaries, now one can have chiral fermions. So gauge and gravitational anomalies are possible (gauge anomalies occur in even spacetime dimensions, and gravitational anomalies arise in $d = 4k+2$ spacetime dimensions.)

In fact, requiring that the theory be (1) supersymmetric and (2) anomaly-free, leads us to two possible gauge groups for the gauge bosons in the theory: $SO(32)$ or $E_8 \times E_8$. The argument is roughly similar to the one made to decide what the permissible gauge groups are for $\mathcal{N}=1$ supergravity coupled to $\mathcal{N}=1$ super Yang-Mills theory. [I say roughly because the details are different: you have to study something called anomaly inflow which describes how the gauge and gravitational anomalies on the boundaries are canceled by contributions from the bulk, through a Green-Schwarz like coupling of bulk-boundary fields, and contributions from the Chern-Simons term which is already present in the action of 11-dimensional supergravity.]

The reason to pick $E_8 \times E_8$ is that the anomalies must be canceled on both boundaries, and there's no way to distribute $SO(32)$ between two boundaries (it is a simple group with no factors).

The above setup is called Horava-Witten theory [1,2]. It is the strongly coupled version of the $E_8 \times E_8$ heterotic string. Specifically, the distance between the two boundaries (Horava-Witten walls, "end of the world branes", "9-branes") is related to the heterotic coupling. It is also sometimes called heterotic M theory.

Note: Usually, one is not so interested in the compactification to $R^{1,9} \times (S^{1}/\mathbb{Z}_2)$, but instead in $R^{1,3} \times CY_3 \times (S^{1}/\mathbb{Z}_2)$, which produces $\mathcal{N} = 1$ supersymmetry in the $(1+3)$ spacetime dimensions. Here $CY_3$ is a Calabi-Yau 3-fold.

References:

1. P. Horava and E. Witten, "Heterotic and Type I String Dynamics from Eleven Dimensions," Nucl. Phys. B460:506-524, 1996. [arXiv:hep-th/9510209]

2. P. Horava and E. Witten, "Eleven Dimensional Supergravity on a Manifold with a Boundary," Nucl. Phys. B475:94-114, 1996. [arXiv:hep-th/9603142]

3. B. Ovrut, "Lectures on Heterotic M-Theory," [arXiv:hep-th/020103]

Sidenote: For $E_{11}$, you might want to see https://ncatlab.org/nlab/show/E11, and references therein. I don't know enough about the subject to comment in any meaningful way.