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

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.