I have a question about the Wess-Zumino gauge in non-Abelian supersymmetric gauge theory. I am following BUSSTEPP Lectures on Supersymmetry.

The spinortial field strength is defined as


where $V(x,\theta,\bar{\theta})$ is a vector superfield. The vector superfield $V$ is $\mathfrak{g}$-valued, and can be written as


where $\left\{T_{i}\right\}$ is a fixed basis for the Lie algebra $\mathfrak{g}$.

The gauge transformation is given by

$$e^{V}\rightarrow e^{-\Lambda^{\dagger}}e^{V}e^{-\Lambda},$$

where $\Lambda(x,\theta,\bar{\theta})$ is a $\mathfrak{g}$-valued chiral superfield.

Using Baker-Campbell-Hausdorff-formula, one finds that the above gauge tranformation is equivalent to the following infinitesimal gauge transformation

$$V\rightarrow V-(\Lambda^{\dagger}+\Lambda)-\frac{1}{2}[V,\Lambda-\Lambda^{\dagger}]+\cdots.$$

On page 52, section 5.6, exercise V.11 says that from the above equation, one concludes that $V$ can be put into the Wess-Zumino gauge

$$V(x,\theta,\bar{\theta})\equiv\bar{\theta}\bar{\sigma}^{\mu}\theta v_{\mu}(x)+\bar{\theta}^{2}\theta\lambda(x)+\theta^{2}\bar{\theta}\bar{\lambda}(x)+\bar{\theta}^{2}\theta^{2}D(x),$$

where $v_{\mu}(x)$, $\lambda(x)$, and $D(x)$ are $\mathfrak{g}$-valued.

1. My first question is how to prove the above statement that Wess-Zumino gauge in non-Abelian gauge theory is always possible by a suitable choice of $\Lambda$?

Next, since in Wess-Zumino gauge $V^{3}=0$, the infinitesimal gauge transformation in Wess-Zumino gauge reduces to the following finite sum

$$V\rightarrow V-(\Lambda^{\dagger}+\Lambda)-\frac{1}{2}[V,\Lambda-\Lambda^{\dagger}]-\frac{1}{12}[V,[V,\Lambda+\Lambda^{\dagger}]]$$

The author concludes that the gauge transformation that preserves the Wess-Zumino gauge must take the form


where $\omega(x)$ is a $\mathfrak{g}$-valued scalar obeying $\omega^{\dagger}=-\omega$.

2. My last question is how to prove the above statement that the $\Lambda$ preserving the Wess-Zumino gauge must take the above form?

I just realized that the first question is answered here. My real question is the last one.


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