Bundle Automorphisms, Structure Groups and Gauge Groups I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true.


*

*If $E$ is a $G$-principle bundle over $M$ then we call $G$ the structure group of $E$ and it is the group of transition functions at any point $x \in M$.

*The group of bundle automorphisms is precisely $G$ at any point $x \in M$.

*We call the group of bundle automorphisms the group of (global) gauge transformations. 

*We call $G$ the gauge group.

*We call the group of transition functions on some neighbourhood $U$, the group of (local) gauge transformations. It is the group of bundle automorphisms over $U$.
I feel that I may be mixing up global and local notions here a bit, particularly with regard to automorphisms and transition functions. Is there a better way to phrase the interaction between the group of transition functions and the automorphism group?
I'd appreciate mathematical precision, since the hand-wavy arguments in physics books are precisely what confuses me on these points! Many thanks in advance! 
 A: This is not really an answer. It seems that I can't add a comment (low reputation). Just wanted to correct a small (but crucial) inaccuracy in Henry T. Horton's fine answer: in general $\mathscr{G}$ can be identified with $C^\infty(M,G)$ only when $G$ is Abelian or the principal bundle $E$ is trivializable. See e.g. Husemoller, Fibre bundles, Proposition 1.7 p. 81.
A: I'm not sure what you mean by "group of transition functions." If we have a local trivialization $\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}$ of the principal $G$-bundle $\pi: E \longrightarrow M$, then the $\varphi_\alpha$ are diffeomorphisms
$$\varphi_\alpha: \pi^{-1}(U_\alpha) \longrightarrow U_\alpha \times G$$
such that
$$\pi = \mathrm{proj}_{U_\alpha} \circ \varphi_\alpha.$$
Then the collection of transition functions $\{\theta_{\beta\alpha}\}_{\alpha, \beta \in A}$ (with respect to the bundle trivialization $\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}$) are smooth maps
$$\theta_{\beta\alpha} : U_\alpha \cap U_\beta \longrightarrow G$$
determined by
$$(\varphi_\beta \circ \varphi_\alpha^{-1})(x, g) = (x, \theta_{\beta\alpha}(x) g).$$
The structure group $G$ just tells you what group the transition functions map overlaps to.
I'm not sure what precisely you mean in 2. If you're saying the group of gauge transformations on $E|_{\{x\}}$ (the bundle restricted to a point) is $G$, then this is true. In general, the group of gauge transformations $\mathscr{G}$ can be identified with smooth functions from $M$ to $G$, i.e. $\mathscr{G} = C^\infty(M, G)$.
A local gauge is the physicist's term for a choice of trivialization of $\pi:E \longrightarrow M$ over a neighborhood $U \subset M$. A local gauge transformation is a physicist's term for changing this choice of trivialization. A choice of trivialization of $\pi^{-1}(U)$ is equivalent to the choice of a local section $s: U \longrightarrow \pi^{-1}(U)$. Then an equivalent definition of a local gauge transformation is a change in choice of local section.
We can construct local gauge transformations of a given local gauge $s: U \longrightarrow \pi^{-1}(U)$ as follows. Take a smooth map $g: U \longrightarrow G$, and define a new local gauge $s^g$ by
$$s^g(x) = s(x) \cdot g(x) \text{ for all } x \in U.$$
All possible local gauges arise in this way from some choice of $g$. Now given a local gauge $s$ and a local gauge transform $s^g$ of $s$, we can define a bundle automorphism $f$ of $\pi^{-1}(U)$ by
$$f(s(x) \cdot h) = s^g(x) \cdot h \text{ for all } h \in G.$$
Conversely, given a local gauge $s$ and a bundle automorphism $f$ of $\pi^{-1}(U)$, we can define a gauge transform of $s$ by
$$\tilde{s}: U \longrightarrow \pi^{-1}(U),$$
$$x \mapsto f^{-1}(s(x)).$$
Now, since $\tilde{s}$ is another local section, there exists a map $g: U \longrightarrow G$ such that $\tilde{s}(x) = s(x) \cdot g(x)$ for all $x \in U$. One can show that $g$ is smooth, and hence $\tilde{s} = s^g$ as before. This establishes the correspondence
$$\text{local gauge transformations} \,\leftrightarrow\, \text{bundle automorphisms of $\pi^{-1}(U)$}.$$
Finally, mathematicians often call the group of gauge transformations $\mathscr{G}$ just "the gauge group," in contrast to the physicist's terminology.
