In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the following equations of motion are derived from it:
$$ F_{\bar{m}\bar{n}}=0,~~~~~\epsilon^{mn\bar{m}\bar{n}}D_{\bar{n}}B_{mn}=0. $$
The authors claim that these equations imply that classically, $A_m$ is undetermined, $A_{\bar{m}}$ is pure gauge, and $B_{mn}$ is holomorphic.
My question is, why is $A_{\bar{m}}$ definitely pure gauge?
Now, the pure gauge configuration $A_{\bar{m}}=-\partial_{\bar{m}}gg^{-1}$ certainly appears to be a solution of $F_{\bar{m}\bar{n}}=0$.
However, in general, flat connections can only be written as pure gauge in simply-connected regions of the underlying manifold. (This is because to gauge transform a connection, $A$, to zero, the group element, $U$, required is the parallel transport of $A$ along a curve, and to show that $U$ does not depend on the curve, the manifold must be simply-connected. Further details can be found in the answer to this question.)
The authors do not seem to be assuming that the Kahler manifold at hand is simply-connected.
Therefore, why is $A_{\bar{m}}$ pure gauge? Does this follow somehow because they use complex coordinates?
