Equations 2-7 on page 21 of these notes, http://www.math.ias.edu/QFT/fall/NewGaw.ps seems to give a fairly compact definition of what a CFT is.
But I have two questions,
This definition is specific to 2 dimensions. Is there an analogue of this definition for higher dimensions?
I want to know if there is an equivalent definition of a CFT in terms of the conformal group in the specific dimension.
Like I would like to know if I can make precise a statement like this (for any dimension), "CFT is a QFT such that its Hilbert space splits into Verma modules and the correlation function of its primary fields is invariant under the conformal group in that dimension."
We do have odd-dimensional CFTs and there I don't know what is a "conformal group"!