We may think of tensors as sections of an associated vector bundle to a principal $\mathrm{GL}(n,\mathbb R)$ bundle, with a fibre chosen to be $\mathbb R^m \times (\mathbb R^*)^n$ - these play a role in general relativity.

Similarly, given a manifold with a $\mathrm{Spin}(n)$ bundle, we can construct an associated vector bundle whose sections correspond to spinors.

My question is now: suppose we take a principal bundle for an exceptional Lie group - say for example an $E_8$ bundle. Is there any associated vector bundle we can construct, which has sections of some physical significance? In the same way spinors and tensors have been fruitful.


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