Holography from Conformal Field Theory (0907.0151v3)

Abstract: The locality of bulk physics at distances below the AdS length is one of the remarkable aspects of AdS/CFT duality, and one of the least tested. It requires that the AdS radius be large compared to the Planck length and the string length. In the CFT this implies a large-N expansion and a gap in the spectum of anomalous dimensions. We conjecture that the implication also runs in the other direction, so that any CFT with a planar expansion and a large gap has a local bulk dual. For an abstract CFT we formulate the consistency conditions, most notably crossing symmetry, and show that the conjecture is true in a broad range of CFT's, to first nontrivial order in 1/N^2: any CFT with a gap and a planar expansion is generated via the AdS/CFT dictionary from a local bulk interaction. We establish this result by a counting argument on each side, and also investigate various properties of some explicit solutions.

• The paper demonstrates that a large spectral gap in low-dimension operators of large-N CFTs ensures emergence of a local AdS bulk dual.
• It employs scalar field models and rigorous crossing symmetry constraints to map CFT correlators to bulk interaction couplings.
• The analysis in the Regge limit provides clear, gauge-invariant criteria for verifying bulk locality from CFT singularities.

Insights into "Holography from Conformal Field Theory"

This paper investigates the conditions under which a holographic, locally interacting bulk theory emerges from a higher-dimensional Conformal Field Theory (CFT) with a large-$N$ expansion. The paper is conducted in the context of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) duality, a pivotal concept connecting quantum gravity in an AdS spacetime to a CFT on its boundary. The authors propose a conjecture that any CFT with a large-$N$ expansion and a significant gap in dimensions has a dual description in terms of a local bulk theory.

Key Findings and Arguments

1. Duality and Locality: In AdS/CFT duality, fields of a gravitational theory in bulk AdS are mapped to operators in a boundary CFT. Locality in the bulk is mysterious since operators that interact at large separations on the boundary may encode interactions at very small scales in the bulk.
2. Gap in Dimensions: For a sharp AdS-bulk locality, the corresponding CFT must have a large gap in the spectrum of dimensions between a small number of low-dimension operators and the rest of the spectrum. The authors argue that if CFT operators of spin greater than two have parametrically large dimensions, a local bulk theory results.
3. Scalar Field Model: As a test case, the researchers consider a scalar CFT where the only low-dimension operator is a scalar. This can be viewed as a toy model. They show through rigorous constraint analysis (e.g., crossing symmetry) that locality in the bulk emerges from two-dimensional and four-dimensional CFTs with specific scalar correlator properties.
4. Constraints and Solutions: By solving conformal partial wave expansions subject to crossing symmetry and matching these solutions to possible bulk interactions, they map possible CFT solutions directly to bulk field couplings, establishing a one-to-one correspondence.
5. Bulk Perspective: The paper explores calculating bulk interaction amplitudes and matching them to boundary correlators. These are dominated by so-called Witten diagrams and exhibit consistency with crossing-symmetric expansions.
6. Regge Limit and Locality: Through analyzing the Regge limit in CFT correlators, the authors provide criteria demonstrating how bulk locality manifests as specific singularities in CFT amplitudes. This offers a gauge-invariant test of locality for the existence of a well-defined bulk dual.

Implications and Future Directions

The confirmed conjecture that a large gap in operator dimensions ensures a local bulk dual solidifies a central aspect of the holographic correspondence. This provides theoretical underpinning for the observed correspondence between low-energy quantum gravity and large-$N$ gauge theories.

Practically, these findings could extend the range of CFTs known to have local bulk duals and inform the construction of new dualities relevant to high-energy physics and condensed matter systems. Future work could address incorporating T_{\mu\nu}, gravitating systems, and examining non-conformal field theories, thereby broadening the landscape of holographic correspondences beyond those explicitly connected by known string theories.

Oversight could include understanding correlations in weakly coupled regimes, strong interactions, or precisely how the CFT mirror of higher-derivative corrections imposes locality constraints on possible non-local formats. Overall, this paper lays important groundwork for exploring quantum field theories through the lens of holography, constructing novel field-theoretic pathways for analyzing quantum gravitational phenomena.