Abstract
Abstract We discuss universal properties of conformal field theories with holographic duals. A central feature of these theories is the existence of a low-lying sector of operators whose correlators factorize. We demonstrate that factorization can only hold in the large central charge limit. Using conformal invariance and factorization we argue that these operators are naturally represented as fields in AdS as this makes the underlying linearity of the system manifest. In this class of CFTs the solution of the conformal bootstrap conditions can be naturally organized in structures which coincide with Witten diagrams in the bulk. The large value of the central charge suggests that the theory must include a large number of new operators not captured by the factorized sector. Consequently we may think of the AdS hologram as an effective representation of a small sector of the CFT, which is embedded inside a much larger Hilbert space corresponding to the black hole microstates.
Highlights
One of the most important ideas considered in the last decades is the possibility that space and time may be emergent concepts
We defined a “large c generalized free CFT” by assuming it has a sector of “single-particle operators” {Oi} defined by the fact that their correlators factorize in an expansion in 1/c
From the dual gravity point of view, the factorized nature of the large c CFT is due to the fact that the correlators do satisfy a linear differential equation in d + 1 dimensions and, as such, constitute a genuinely free theory implying a Fock space structure of the Hilbert space
Summary
One of the most important ideas considered in the last decades is the possibility that space and time may be emergent concepts. To summarize the main points, in this paper we argue that a CFT is naturally described holographically in AdS space if it has the following basic properties: i) It has large central charge i.e. many degrees of freedom. We will try to argue that in theories with these properties the effective interactions of the low-lying operators are naturally described in terms of a dual gravitational theory in antide Sitter space These conditions, as well as the meaning of the holographic dual theory, will be made more precise in the rest of the paper. The reason is that in order to have factorization of correlators it is necessary for the central charge of the CFT to be large This implies that at large conformal dimension the theory must have a large number of operators that do not appear in the naive spectrum of generalized free fields. Appendix C reviews the different notions of central charges in d > 2
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