I'm studying Nakahara's Geometry, topology and physics chapter 10.
In section 10.4 on page 391, the book writes:
In physics, we often need to differentiate sections of a vector bundle which is associated with a certain principle bundle. For example, a charged scalar field in QED is regarded as a section of a complex line bundle associated with a $U(1)$ bundle $P(M,U(1))$. Differentiating sections covariantly is very important in constructing gauge-invariant actions.
My questions: what about fermions like electrons? Are they sections of a certain bundle? And how would we define covariant derivative $D_\mu=\partial_\mu+ieA_\mu(x)$ mathematically for fermions then?
