Axiomatic conformal theory in dimensions >2 and AdS/CT correspondence

TL;DR

This paper develops axioms for higher-dimensional conformal theories, establishing a correspondence with weakly local quantum field theories on hyperbolic space, and relates correlation functions between these frameworks.

Contribution

It formulates axioms for conformal theories in dimensions greater than two and establishes a bijective correspondence with weakly local quantum field theories on hyperbolic space.

Findings

Derived AdS/CT correspondence for local theories on AdS.

Constructed a bijection between conformal theories on spheres and weakly local QFTs on hyperbolic space.

Expressed AdS correlation functions in terms of conformal theory correlation functions.

Abstract

We formulate axioms of conformal theory (CT) in dimensions $>2$ modifying Segal's axioms for two-dimensional CFT. (In the definition of higher-dimensional CFT one includes also a condition of existence of energy-momentum tensor.) We use these axioms to derive the AdS/CT correspondence for local theories on AdS . We introduce a notion of weakly local quantum field theory and construct a bijective correspondence between conformal theories on the sphere $S^d$ and weakly local quantum field theories on $H^{d+1}$ that are invariant with respect to isometries. (Here $H^{d+1}$ denotes hyperbolic space= Euclidean AdS space.) We give an expression of AdS correlation functions in terms of CT correlation functions. The conformal theory has conserved energy-momentum tensor iff the AdS theory has graviton in its spectrum.