Six-dimensional Methods for Four-dimensional Conformal Field Theories

TL;DR

This paper introduces six-dimensional methods to simplify the calculation of Green's functions in four-dimensional conformal field theories, revealing more general structures and aligning with AdS/CFT duality principles.

Contribution

It develops a six-dimensional projection approach for constructing four-dimensional fields, providing new insights into Green's functions and their conformal structures in CFTs.

Findings

Green's functions have more general structures than previously known.

Fields on AdS$_5$ approach boundary limits with specific conformal dimensions.

The methods align with and support the AdS/CFT duality framework.

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 AdS$_5$ approach finite limits on the boundary of 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.