# Construction of the supersymmetric Faraday tensor

When I first learned gauge theories in my introductory quantum field theory course, I was taught that the Faraday (field-strength) tensor can be constructed by computing the commutator of the gauge-covariant derivative:

$$[D_\mu,D_\nu]=-ieF_{\mu\nu}$$

Now, I am studying supersymmetry following Martin's SUSY primer, and in chapter 4.8, the author immediately writes down the super-symmetric field strength chiral superfield out of the vector superfield $V$:

$$\mathcal{W}_\alpha=-\frac{1}{4}D^\dagger D^\dagger D_\alpha V.$$

I would have liked a more gentle introduction to this in terms of something I am already familiar with: is there a way for me to have constructed this using the commutator of some 'gauge super-covariant derivative'?

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As far as I know, it is defined in that form in order to satisfy chirality $$D^{\dagger}_{\dot{\alpha}}W_{\alpha}=0$$ and gauge invariance $$\delta W_\alpha=0.$$
Ordinary covariant derivatives are bad in SUSY because the $\partial_\mu$ derivative is no longer the fundamental "minimal" derivative. Instead, one may find a square root of it - and the superderivatives $D_\alpha$ etc. are square roots of the ordinary derivatives, and they're therefore more fundamental. So you haven't really started to think in the SUSY way if you still want to place ordinary derivatives everywhere. – Luboš Motl Jan 20 '13 at 10:25