The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 \end{pmatrix}. $$ There is one famous application in physics. Which is that a symmetric bilinear $K_{E_8}$-Chern-Simons theory describes the low energy physics of a so-called $E_8$-quantum Hall state (with a $U(1)^8$ gauge group). The field theory partition function $Z$ is given by $$ Z= \int[DA]\exp(\frac{(K_{E_8})_{IJ}}{2 \pi}\int A_I dA_J). $$ This $E_8$-quantum Hall state occurs in a 2-dimensional spatial condensed matter system. This describes an invertible TQFT. Since the $|Z|$ on any closed 3-manifold of $M^2 \times S^1$ has $|Z|=|\det K_{E_8}|=1$. Which means the Hilbert space on any spatial closed $M^2$ is always 1-dimensional.
The above is what I am familiar already. Now I am asking a different question about the application of $E_8$ manifold.
My question here is that: Is there some real-world application of $E_8$ manifold, such as in physics or in any branch of science, or in the engineer?
The $E_8$ manifold is the unique compact, simply connected topological 4-manifold with intersection form the $E_8$ lattice.
The $E_8$ manifold has no smooth structure.
The $E_8$ manifold is not triangulable as a simplicial complex.
The $E_8$ manifold is constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for $E_8$. This results in P$E_8$, a 4-manifold with boundary equal to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the $E_8$ manifold.
Some refs and introductory level of explanations are welcome! Thanks.
