# 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.

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!

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.

• Henry, many thanks for your answer. Just to check, am I right in identifying the local gauge transformations with the automorphisms of the bundle over a neighbourhood $U$ then? I imagine this is correct since one can identify a change of local trivialisation with an automorphism, in the same way that one identifies a change of coordinates with a diffeomorphism in GR. Or is there something more subtle going on? – Edward Hughes Mar 25 '13 at 9:26
• @EdwardHughes: That is correct. My original discussion of local gauge transformations was very poor. Check out my edited answer to see some details about why your comment is correct. – Henry T. Horton Mar 28 '13 at 23:27

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.