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,
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This definition is specific to 2 dimensions. Is there an analogue of this definition for higher dimensions?
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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"!
Best Answer
The conformal group is defined for any spacetime you want. The conformal group of d-dimensional Euclidean space, which has isometry group SO(d), is SO(d+1,1). The conformal group of d+1 dimensional Minkowski space, whose isometry group is SO(d,1), is SO(d+1,2). The defining property of the conformal group is that its the set of transformations that leave the metrix $g_{\mu\nu}$ invariant up to a scaling factor $e^{\omega(x)}$.
A more general definition for CFTs in d>2 dimensions is that a CFT is a QFT whose Hilbert space breaks up into representations of the conformal group and whose correlation functions are invariant under any conformal transformation (I don't believe we have to restrict to primary operators).
The special case of two dimensions is that the conformal algebra is infinite dimensional. The group of globally defined conformal transformations is still finite, but there are an infinite number of local conformal transformations. So 2d CFTs are a lot more restricted then higher dimensional CFTs.
For a good introduction look at https://sites.google.com/site/slavarychkov/