Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action.

There is an associated fibre bundle $E\rightarrow M$ with $E=P\times_\alpha F=(P\times F)/G$.

As it is well known, one may either treat sections of the associated fibre bundle "directly", or consider maps $\psi:P\rightarrow F$ which satisfy the equivariance property $\psi(pg)=g^{-1}\cdot\psi(p)$, where $\cdot$ denotes the left action. Let us refer to this latter method as the "equivariant bundle approach".

I am interested in describing the gauge field theories of physics using global language with appropriate rigour. However, most references I know treat this topic using the "direct approach", and not with the equivariant approach, the chief exception being *Gauge Theory and Variational Principles* by David Bleecker.

Bleecker's book however doesn't go far enough for my present needs.

- Bleecker only uses linear matter fields, eg. the case where $F$ is a vector space and $\alpha$ is a linear representation. Some things are easy to generalize, others appear to be highly nontrivial to me.
Bleecker treats only first-order Lagrangians. The connection between a higher-order variational calculus based on the equivariant bundle approach and between the more "standard" one built on the jet manifolds $J^k(E)$ of the associated bundle is highly unclear to me. Example: If $\bar\psi:M\rightarrow E$ is a section of an associated bundle, its $k$-th order behaviour is represented by the jet prolongation $j^k\bar\psi:M\rightarrow J^k(E)$, but if instead I use the equivariant map $\psi:P\rightarrow F$, what represents its $k$-th order behaviour? I assume it is related to something like $J^k(P\times F)/G$, but the specifics are unclear to me.

In Bleecker's approach, connections are $\mathfrak g$-valued, $\text{Ad}$-equivariant 1-forms on $P$, however I am interested in treating them on the same footing as matter fields. Connections however are higher order associated objects in the sense that they are associated to $J^1P$. Bleecker absolutely doesn't treat higher order principal bundles.

In short, I am interested in references that consider gauge theories, gauge natural bundles, including nonlinear and higher-order associated bundles and calculus of variations/Lagrangian field theory from the point of view where fields are fixed space-valued objects defined on the principal bundle (equivariant bundle approach), rather than using associated bundles directly.