# Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

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 gauge fields can only be written as pure gauge in simply-connected regions of the underlying manifold. (This is because to gauge transform a gauge field, $$A$$, to zero, the group element, $$U$$, required is a Wilson line 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?

• without knowing too much about the subject, from a quick read of the passage in the paper, I think that they just use this to count the degrees of freedom. So they basically are sloppy not mentioning that it should be true patch by patch, but it doesn't affect the outcome. – ɪdɪət strəʊlə Mar 4 at 8:28