Suppose $E$ is a vector bundle with structure group $G$ and let $P = F(E)$ be the frame bundle. Let $\mathfrak{g}\_P$ denote the associated bundle to the adjoint representation of $G$ on its Lie algebra (i.e. $\mathfrak{g}\_P = P \times\_G \mathfrak{g}$ where $\mathfrak{g}$ is the Lie algebra of $G$). Given a section of $\mathfrak{g}\_P$, I should be able to "exponentiate" it pointwise to get a gauge transformation. How is this defined? I couldn't come up with anything well-defined.


For some context, I was reading Chapter 2 of Donaldson and Kronheimer's "The Geometry of 4-Manifolds," and they mention this pointwise exponential in passing on p. 33. I'm guessing they assume the reader is familiar with it from a more elementary text, but I looked in a few other books and couldn't find it.



How do you exponentiate a section of the adjoint bundle to get a gauge transformation?
I'm not sure what it is that you tried, but the exponential map should work. First of all, let $U\_i$ be a trivialising cover for $P$ and its associated bundles. A section through $\mathfrak{g}\_P$ is given by functions $\omega\_i: U\_i \to \mathfrak{g}$ which, on overlaps, transform according to


$$\omega\_i(p) = \operatorname{Ad}\_{g\_{ij}(p)} \omega\_j(p) \qquad \forall p \in U\_i \cap U\_j,$$


where I use $\operatorname{Ad}$ to mean the adjoint representation of $G$ on its Lie algebra $\mathfrak{g}$.


Now simply compose $\omega\_i$ with the exponential map $\exp: \mathfrak{g} \to G$, resulting in functions $\exp\omega\_i : U\_i \to G$ which, on overlaps, transform according to


$$\exp\omega\_i(p) = g\_{ij}(p) \exp\omega\_j(p) g\_{ij}(p)^{-1} \qquad \forall p \in U\_i \cap U\_j.$$


But this is just a section through the associated fibre bundle usually denoted $\operatorname{Ad} P$, and that is the same thing as a gauge transformation.


