Bundle Automorphisms, Structure Groups and Gauge Groups

Question

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.

1. 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 $x \in M$.

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

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

4. We call $G$ the gauge group.

5. 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!

Answer 1

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.