# 2D ${\cal N}=(2,2)$ Super Yang-Mills with Superspace

I'm reading this famous paper by Witten. There is the expression of field strength for the abelian vector multiplet (eq. (2.16)):

$$\Sigma = \frac{1}{\sqrt{2}}\bar{D}_+D_- V\;.\tag{2.16}$$

I'm wondering what is the expression for a non-abelian vector multiplet, written explicitly.

Eq. (2.15) in principle give what I want:

$$\Sigma=\frac{1}{2\sqrt{2}}\{\bar{\mathcal{D}}_+,\mathcal{D}_-\}\;,\tag{2.15}$$

however I cannot see definition of $\mathcal{D}$ and $\bar{\mathcal{D}}$. Moreover eq. (2.8) is

$$\{\mathcal{D}_\alpha,\bar{\mathcal{D}}_{\dot{\alpha}} \} = -2i\sigma^m_{\alpha\dot{\alpha}}\mathcal{D}_m\;, \tag{2.8}$$

which, if plugged in $(2.15)$ seem to give not the right result.

Also in Mirror Symmetry Book only deals with the abelian case.

Do you know where can I find the general case? Or how can I extract by myself the field strength?

I tried some obvious generalization such as

$$\Sigma = \frac{1}{\sqrt{2}}\bar{D}_+e^{-V}D_-e^V\;,$$

which transforms correctly as

$$\Sigma \mapsto e^{-\Lambda}\Sigma e^{\Lambda}\;,$$

however, in this case

$$\bar{\Sigma}\Sigma\;,$$

does not transform correctly.

• I guess you need to be more careful with the notation. The calligraphic symbols $\mathcal{D}_\alpha$ and $\overline{\mathcal{D}}_{\dot\beta}$ are the covariant versions of $D_\alpha$ and $\overline{D}_{\dot\beta}$. – user178876 Jun 15 '18 at 17:27
• Maybe I made confusion with the sentence just above eq. (2.6). However I still do not understand how to interpret eq. (2.15). My final goal is just the expression of $\Sigma$ for the non-abelian case. – MaPo Jun 15 '18 at 17:34