I've had some exposure to vector bundles, but as good as none to principal bundles. I'm trying to get some orientation in the subjects concerning vector bundles and gauge theory.
On Wiki, there's a definition of a gauge transformation from a 'mathematical standpoint', namely that a gauge transformation of a vector bundle $\pi:E\rightarrow M$ is a diffeomorphism $\phi:E\rightarrow E$ commuting with $\pi$ and being a linear vector space isomorphism on each fiber.
[https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)#Gauge_transformations ]
What is meant by $\varphi$ and $\pi$ to commute?
Using this definition and given some vector field, in order to show that it is gauge invariant, would I have to apply $\phi$ and show that the object remains the same?
Is there a good source for someone interested in the above, to learn more about gauge transformations?
Thanks in advance!
The fact that the maps commute means that, given a point in a the fibre, $p\in E_x$ for $x\in M$, we have $\pi\circ\phi(p)=\phi\circ\pi(x)$. In other words, the gauge transformation $\phi$ preserves the fibre, it takes points in a fibre to points in the same fibre.
If you wish to read about mathematical gauge theory, there are two books that I've recently gone through that I would highly recommend: Principal Bundles by Sontz, and Mathematical Gauge Theory by Hamilton. Both can be accessed through Springer Link, so perhaps your university has given you free access.