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Axiomatic conformal theory in dimensions \(>2\) and AdS/CT correspondence. (English) Zbl 1347.81058

Summary: 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.

MSC:

81T05 Axiomatic quantum field theory; operator algebras
81T20 Quantum field theory on curved space or space-time backgrounds
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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