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In the book (Supergravity - Daniel Z.Freedman & Antoine Van Proeyen - Cambridge), there is (Chapters 16-17) a presentation of pure supergravity or supergravity with matter, from a superconformal approach.

The "simplest" link, is to begin with a superconformal gauge multiplet coupled to a chiral multiplet, then gauge fix which will break the scale and special conformal transformation symmetries, and finally get a pure supergravity (in the same dimensional space-time). Here one speaks about $\mathcal N=1$ supersymmetry in a $D=3+1$ space-time.

I have some questions about this approach.

  1. Is it only a mathematical approach, or it is also a physical approach, that is, is it possible to associate some physical quantities of the $2$ theories in some way?
  2. Thinking about $\mathrm{AdS}_4$/$\mathrm{CFT}_3$, there is some regime, where supergravity is trustable. In this regime, with the above approach, we have $2$ sides of a triangle, so it may be tempting to look at the 3rd side of the triangle, that is a link between a superconformal theory in $3+1$ dimensions, with a superconformal theory in $2+1$ dimensions, or maybe a step further, that is looking at the superconformal theory in $3+1$ as a "mother" theory, as a united point of view of $\mathrm{AdS}_4$/$\mathrm{CFT}_3$, at least in the supergravity regime. Does all this makes sense ?
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  1. The "simplest" link, is NOT to begin with a Superconformal gauge multiplet coupled to a chiral multiplet, but to couple the Weyl multiplet to a superconformal chiral multiplet.

  2. It is just a mathematical tool to make your life easier. As a matter of fact, you can take the superconfromal action and make a field redefinition to get the Poincare action, thus the superconformal symmetry is nothing but a redundancy to be removed. You can see an xample of that for a five dimensional N=2 theory in http://arxiv.org/pdf/1107.2825.pdf see equation (3.1).

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