Hello once again. I'm trying to understand the relation between the superspace representation of the SUSY generators [itex]Q_\alpha,\overline Q_{\dot\beta}[/itex] and the covariant derivatives on superspaces [itex]D_\alpha, \overline D_{\dot\beta}[/itex]:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]Q_\alpha = \frac{\partial}{\partial\theta^\alpha} - i\sigma^{\mu}_{\phantom\mu\alpha\dot\alpha}\overline\theta^{\dot\alpha}\partial_\mu[/itex]

[itex]\overline Q_{\dot\alpha} = \frac{\partial}{\partial\overline\theta^{\dot\alpha}} - i\theta^\alpha\sigma^{\mu}_{\phantom\mu\alpha\dot\alpha}\partial_\mu[/itex]

[itex]D_\alpha = \frac{\partial}{\partial\theta^\alpha} + i\sigma^{\mu}_{\phantom\mu\alpha\dot\alpha}\overline\theta^{\dot\alpha}\partial_\mu[/itex]

[itex]\overline D_{\dot\alpha} = -\frac{\partial}{\partial\theta^\alpha} - i\sigma^{\mu}_{\phantom\mu\alpha\dot\alpha}\overline\theta^{\dot\alpha}\partial_\mu[/itex]

These look very similar. The difference between the [itex]Q[/itex]s and the [itex]D[/itex] is the sign of their anticommutators:

[itex]\{Q_\alpha,\overline Q_{\dot\alpha}\} = +2i\sigma^\mu_{\phantom\mu\alpha\dot\alpha}\partial_\mu = -2\sigma^{\mu}_{\phantom\mu\alpha\dot\alpha}P_\mu[/itex]

[itex]\{D_\alpha,\overline D_{\dot\alpha}\} = -2i\sigma^\mu_{\phantom\mu\alpha\dot\alpha}\partial_\mu = +2\sigma^{\mu}_{\phantom\mu\alpha\dot\alpha}P_\mu[/itex]

My question is this: we have two pairs of differential operators which both induce a motion in superspace. Both pairs of operators transform appropriately under Lorentz transformations. Why is it, then, that one pair of operators get called the generators of the symmetry while the other set get called the covariant derivatives?

Could we have swapped them, so that we call the [itex]Q[/itex]'s the covariant derivatives and the [itex]D[/itex]'s the SUSY generators?

Wess & Bagger Chapter IV suggests that this has something to do with left- and right-multiplication... but it's not clear to me what the significance of this is.

I'm struggling to find an analogy for this. For gauge symmetries we never talk about the gauge covariant derivative and a differential representation of the gauge symmetry generators. For translations in flat space we never talk about the momentum generator versus the translation covariant derivative.

How are the [itex]Q[/itex]s and [itex]D[/itex]s related and why is one considered a generator while the other a covariant derivative? (And what does this all have to do with right- and left- multiplication?)

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# What's the relation between SUSY Generators and Covariant Derivatives?

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