This is a great question, but I don't think a reasonable answer can be given in this short space. So I wrote an expository note jointly with a colleague who was trained as a physicist. You can read it by following the link above -- comments are welcome.
But here are a couple of highlights:
- It's not true that they measured this one number, and so claimed to have detected E8.* There is a lot more history! Back around 1990, there was a series of theoretical "deductions" (in the weak sense of physics) investigating what the appropriate theoretical model should be for the situation in the magnet experiment. This led to a unique candidate for a model, one built out of E8. I would say that the experiment corroborated the series of deductions, with the sensational bonus that the deductions led to E8.
- Which E8 appears in the theoretical model? The obvious answer is that it is the compact real E8 and not just the root system or root lattice. For example, even though the masses of the 8 particles are given as entries in an eigenvector for the Cartan matrix (which makes it sound like it's just the root system), the proof of this statement is a calculation within the compact Lie algebra.
One can argue about both of these points, of course. But these seem to be what the physicists claim and what they use in their papers.
To address Wadim's concerns about fringe science: Whether or not you find the E8 angle interesting or plausible, it seems that the experiment is interesting for entirely different reasons. The experimenters themselves claim that their main achievement is realizing this 1-dimensional quantum Ising model in the laboratory in a situation where the external field can be tuned to be above, below, or at the critical point. The Physics Today article on the subject paraphrases Subir Sachdev:
only recently could researchers reach the high fields and low temperatures needed to access the critical point and have high enough instrumental resolution to resolve the masses of at least some of the quasiparticles they excited. (Temperatures have to be low enough to suppress any impact of thermal fluctuations.)
- Footnote: To be precise: Coldea, the author of this particular article in Science, uses the more-cautious "detected evidence of E8 symmetry" as opposed to the stronger "detected E8".