# A Primaries, descendents and transformation properties in CFT

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1. Feb 27, 2017

### Kurret

I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of operators:

Primaries: Defined by the fact that they are annihilated (commute with) the lowering operators of the theory.

Descendants: Obtained by acting on primaries with raising operators.

Set 3: Operators that scale as $\mathcal{O'}(x')=\lambda^{-\Delta/d}\mathcal{O}(x)$ under a scaling (dilatation).

Set 4: Operators that transform as $\mathcal{O'}(x')=|\partial x'/\partial x|^{-\Delta/d}\mathcal{O}(x)$ under any conformal transformation.

What are the relation between the above sets? I think all descendants are also part of set 3 (but not the converse, since there could probably also exist linear combinations of descendants that are part of set 3). All primaries should also be part of set 4, but I am not sure about the converse.

2. Mar 4, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.