# Projective superspace: why extra bosonic coordinates

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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?

(My background knowledge is still very low since I'm just a master student physics so I'm looking for a simple argumentation...)

This post imported from StackExchange Physics at 2015-07-12 18:27 (UTC), posted by SE-user cherzieandkressy

Mainly arXiv:1207.1241 and old ones like sciencedirect.com/science/article/pii/0370269388908593 and sciencedirect.com/science/article/pii/0550321384905923. After some more thinking about it, I guess the main thing is that you want to describe the $\sigma$-model with your Lagrangian (just like in the N = (2,2) case where the Lagrangian is the generalised Kahler potential from which you can derive the metric and b-field). And constructing a consistent N = (4,4) Lagrangian with $[\mathcal{L}]$=-2 will be very hard I guess...
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