# Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

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.