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I'm studying the projective superspace formalism for N = 4 supersymmetric $\sigma$-models in two dimensions. My question is: why do we need the extra bosonic coordinates for the manifest action?

I can see that the action (with $\sigma$ the worldsheet coordinates and $\theta$ the fermionic coordinates),

$$S = \int\mathrm{d}^{2}\sigma\mathrm{d}^{8}\theta \mathcal{L},$$

would have a Lagrangian with negative mass dimension since $[\mathrm{d}\sigma]=-1$ and $[\mathrm{d}\theta]=+1/2$ so $[\mathcal{L}]=-2$ and this would be the reason to introduce (two) extra bosonic coordinates in the superspace such that the Lagrangian would become dimensionless.

But what's wrong with a Lagrangian with negative mass dimension in this very abstract superspace? If we would reduce to normal space then there wouldn't be a problem I guess?

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