Six-dimensional methods for four-dimensional conformal field theories

Abstract

The calculation of both spinor and tensor Green's functions in four-dimensional conformally-invariant field theories can be greatly simplified by six-dimensional methods. For this purpose, four-dimensional fields are constructed as projections of fields on the hypercone in six-dimensional projective space, satisfying certain transversality conditions. In this way some Green's functions in conformal field theories are shown to have structures more general than those commonly found by use of the inversion operator. These methods fit in well with the assumption of AdS/CFT duality. In particular, it is transparent that if fields on ${\mathrm{AdS}}_{5}$ approach finite limits on the boundary of ${\mathrm{AdS}}_{5}$, then in the conformal field theory on this boundary these limits transform with conformal dimensionality zero if they are tensors (of any rank), but with conformal dimension $1/2$ if they are spinors or spinor-tensors.