Definition of Gauge group

Question

I have a problem with an example of Gauge group. I'm reading ""Yang-Mills equations over Riemann surfaces"" (Atiyah, Bott). Let $P$ be a principal $G$-bundle over $X$. We define the adjoint bundle $AdP:= P \times_G G$. For example we can consider $$ S^1 \to S^3 \to \mathbb{C}P^1 ,$$ so $AdS^3=S^3 \times_{S^1} S^1$. Then $$ pt \to S^3 \times_{S^1} S^1 \to S^3 .$$ But in this way the fibre of $AdP$ is always a point. Is it true? How can I describe the sections of $AdS^3$?

Answer 1

You're not applying the definition of $\mathrm{Ad}(P)$ correctly.

Recall the associated bundle construction: If $\pi: P \rightarrow X$ is a principal $G$-bundle over $X$, $F$ is a topological space, and we have a homomorphism $\rho: G \longrightarrow \mathrm{Homeo}(F)$, we can form a new fiber bundle $F \hookrightarrow P \times_\rho F \xrightarrow{~\pi_\rho~} X$ where $$E \times_\rho F = P \times F/\langle(p.g, f) \sim (p,\rho(g)(f))\rangle,$$ and the projection $\pi_\rho: P \times_\rho F \longrightarrow X$ is defined by $$\pi_\rho([p, f]) = \pi(p).$$ Note that $P \times_\rho F$ is a fiber bundle with structure group $G$ and fiber $F$.

For the construction of $\mathrm{Ad}(P)$, one takes $F = G$ and $\rho$ is defined by $$\rho(g)(h) = ghg^{-1}.$$ Then $\mathrm{Ad}(P) = P \times_\mathrm{Ad} G$ is a fiber bundle with fiber $G$ and structure group $G$.

In particular, for your example $P$ is the Hopf fibration $S^1 \hookrightarrow S^3 \to S^2$, and $\mathrm{Ad}(P)$ is a circle bundle over $S^2 = \Bbb C P^1$. Note that the fiber is a circle, not a point!

In general, sections of $\mathrm{Ad}(P)$ can be identified with $\mathrm{Ad}$-equivariant maps $f: P \longrightarrow G$, i.e. maps $f$ satisfying $$f(p.g) = gf(p)g^{-1}.$$ Write $\Gamma(\mathrm{Ad}(P))$ for the space of sections of $\mathrm{Ad}(P)$. Then $\Gamma(\mathrm{Ad}(P))$ has the structure of a group if we define the product of $f, g: P \longrightarrow G$ pointwise: $$(fg)(p) = f(p)g(p),$$ where the product on the right-hand side is the multiplication in $G$. Clearly $fg$ still satisfies the $\mathrm{Ad}$-equivariance property. We call this group $\Gamma(\mathrm{Ad}(P))$ the gauge group of $P$ and denote it by $\mathscr{G}(P)$.