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I am trying to understand how the Hodge decomposition is affected by gauge transformations in non-abelian in gauge theory (eg $\mathrm{SU}(N)$). In particular, I am searching for a way to generalise the below splitting of the connection in the abelian case. Please correct me if I'm wrong about any of the below.

In abelian gauge theory, e.g. $\mathrm{U}(1)$, we know that our connection $A$ transforms by an exact shift $A\to A+d\lambda$, or (non-infinitesimally) $A\to A+gdg^{-1}$. Because we have $d^2=0$, on a closed surface we can perform a Hodge decomposition into exact and co-exact parts. For the Laplacian $\Delta=\delta d+d\delta$, one can in theory extract the exact part by $$A_{\mathrm{exact}}=\int_\Sigma \Delta^{-1} \delta A.$$ Under gauge transformation, only $A_{exact}$ is modified, and $A_{coexact}=A-A_{\mathrm{exact}}$ is unchanged.

My question is then how this is modified in the non-abelian case. Now, gauge transformations act as $A\to A+d_A\lambda$, or $A\to gAg^{-1}+gdg^{-1}$. We can formally define $\delta_A$ I suppose, however in general $d_A^2\neq 0$ so we don't have a complex such that we can form a Hodge decompositon. Is there an analogous decomposition to split $A$ into the part that is unchanged by gauge transformations, and a part that captures all of the changes? Is there anything interesting at all we can say about the non-Abelian case?

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  • $\begingroup$ A quick comment: as Budjum's answer implies, to do anything like this in the non-trivial case, you'd have to work relative to a fixed background connection—in the trivial case, there's a canonical choice of background connection, namely, the trivial flat connection. $\endgroup$
    – Branimir Ćaćić
    Commented Nov 1 at 11:12

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Absence of an explicit gauge-fixing condition that selects precisely one representative in each gauge orbit is a serious problem of non-abelian Yang-Mills gauge theories and is known as `Gribov ambiguity'.

As you ask this as a mathematical question, I want also to underline that even in the $U(1)$ case a decomposition like you suggest is possible for trivial vector bundles only (or, if you have a non-trivial vector bundle, you can restrict it to a subset of the base space on which it trivialises). Otherwise, there is no globally-defined differential form $A$ at all, and only the operator $d_A$ makes sense. In this case looking for a global gauge fixing is a lost cause, because no globally-defined frame (`choice of gauge') exists.

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  • $\begingroup$ Gribov ambiguity is an issue even for abelian Yang–Mills gauge theory over compact manifolds at least: in the case of $U(1)$-gauge theory, you get Gribov ambiguity iff the first Betti number is non-zero. $\endgroup$
    – Branimir Ćaćić
    Commented Nov 1 at 11:03
  • $\begingroup$ OK thankyou, I had been reading about the Gribov obstruction just before this. Suppose then that I'm considering a boundary of some finite region in a slice of $\mathbb{R}^4$. I only care about the value of $A$ on this closed curve, rather than all of $\mathbb{R}^4$. I understand that I can't have a global section, but what can I say about $A$ in this case? Is there a well-defined splitting on this submanifold? $\endgroup$
    – b0bgary
    Commented Nov 5 at 1:28

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