On the same page of the paper, $\mathcal{N}$ is defined as a connection on a bundle $\sigma^* E$. The claim that its BRST variation is
$$\delta_{\text{BRST}} \,\mathcal{N} = \mathrm{d} \alpha + [\mathcal{N},\alpha],~~~ (\alpha = \omega \eta + T),$$
means that $\delta_\text{BRST} \mathcal{N}$ is just a gauge transformation of $\mathcal{N}$, $\delta_\alpha \mathcal{N} = \mathrm{d}_\mathcal{N}\, \alpha$, with $\mathrm{d}_\mathcal{N} = \mathrm{d} + [\mathcal{N}, \cdot]$ the gauge-covariant derivative. The trace of the holonomy around a curve $C$ is just a Wilson loop,
$$W_C(\mathcal{N}) = \mathrm{str} \,\mathrm{P}\, e^{ \oint_C \mathcal{N}},$$
which is of course a gauge invariant operator. It is therefore also BRST invariant.

This post imported from StackExchange Physics at 2016-08-14 09:23 (UTC), posted by SE-user user81003