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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?

Addendum

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.

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  • $\begingroup$ 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 $\overline{D}_{\dot\beta}$. $\endgroup$ – user178876 Jun 15 '18 at 17:27
  • $\begingroup$ 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. $\endgroup$ – MaPo Jun 15 '18 at 17:34
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Witten defines it in equation 4.5 of https://arxiv.org/pdf/hep-th/9312104.pdf

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