# Is gauge connection unique?

In QFT, given a gauge group and matter field, is the form of the gauge field unique? In other words, given a principal G-bundle and its associated vector bundle, is the construction of the principle G-connection unique?

This is related to the other question (here: Gauge Field Tensor from Wilson Loop) where the author implies that the gauge field is natural/unique given the matter field. May be it is, but I wanted to confirm (edit: From answer below, the gauge connection is not unique)

Because a gauge connection (or a gauge field) can exist independent of the vector matter field (as in pure gauge theories), non-uniqueness of the connection would imply a symmetry on the connection itself.

• Could you clarify what precisely the given data are? In general, there are many choices for the gauge connection, leading to the idea of moduli spaces of connections. Also, you will of course always retain the gauge freedom - gauge equivalent connections will not be physically different. – ACuriousMind Aug 17 '14 at 19:30

The gauge connection is not unique, and this has nothing to do with the presence of matter fields. Let $\Sigma$ be our space-time, $P$ a principal $G$-bundle, and $\mathcal{A}$ the space of connections on $P$. Then, gauge transformations $t : P \to G$, forming the group of gauge transformations $\mathcal{G}$ have an action on $\mathcal{A}$ given by

$$A \overset{t}{\mapsto} tAt^{-1} + t \mathrm{d}t$$

and the space of physically different connections is $\mathcal{A}/\mathcal{G}$.

Side note: Unfortunately, the naive way of taking this quotient does not quite succeed in producing a manifold we could integral the path integral over, since there are so-called reducible connections corresponding to "corners" in the resulting almost-manifold (I think it is technically an orbifold), and since the action of $\mathcal{G}$ on $\mathcal{A}$ is not free if the center of $G$ is non-trivial.

• Thanks! So, 'Moduli space of connections' is what I should be looking for. I was instead searching for 'connection of connection' etc :) – GuSuku Aug 17 '14 at 20:03
• @crackjack: You could indeed look at a "connection of connections", since $\mathcal{A}$ (modulo the technicalities of reducibility and such I alluded to) is, in fact, a $\mathcal{G}$-principal bundle over $\Sigma$. I cannot recall if there is any reason to look at such a connection in general contexts, though. – ACuriousMind Aug 17 '14 at 21:42
• Yes, I knew it had to be that (~ 'connection of connection) before I posted my question, but it didn't yield any leads, at least on google. Now with your pointer to 'moduli of connection', I could dig out many leads! :) I am not sure either if there is any reason to. – GuSuku Aug 18 '14 at 3:25
• Also, as the n-lab link you posted above said, I dont see a lot of resources on moduli spaces of non-flat connections. If 'flat' there refers to what I know from physics (vanishing field strength), then most of those results on moduli of connections will not be applicable to most cases in phenomenological physics? – GuSuku Aug 18 '14 at 3:38
• @crackjack: Flat connections are, as you say, those with $F = 0$. They mostly show up in examining certain limits of Yang-Mills theories, and are not very phenomenological, I'm afraid. As an overview, though it focuses on 2D theories, this set of lecture notes should also contain many leads. In the papers on 2D gauge from Witten, there is some discussion of the full space of connections and of certain elliptic operators on it, but even this is not very "real", as it is in the context of 2D gauge. – ACuriousMind Aug 18 '14 at 12:40