# Fields transforming under an exceptional Lie group

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.