Effective Conformal Theory and the Flat-Space Limit of AdS (1007.2412v2)

Abstract: We develop the idea of an effective conformal theory describing the low-lying spectrum of the dilatation operator in a CFT. Such an effective theory is useful when the spectrum contains a hierarchy in the dimension of operators, and a small parameter whose role is similar to that of 1/N in a large N gauge theory. These criteria insure that there is a regime where the dilatation operator is modified perturbatively. Global AdS is the natural framework for perturbations of the dilatation operator respecting conformal invariance, much as Minkowski space naturally describes Lorentz invariant perturbations of the Hamiltonian. Assuming that the lowest-dimension single-trace operator is a scalar, O, we consider the anomalous dimensions, gamma(n,l), of the double-trace operators of the form O (del^2)^n (del)^l O. Purely from the CFT we find that perturbative unitarity places a bound on these dimensions of |gamma(n,l)|<4. Non-renormalizable AdS interactions lead to violations of the bound at large values of n. We also consider the case that these interactions are generated by integrating out a heavy scalar field in AdS. We show that the presence of the heavy field "unitarizes" the growth in the anomalous dimensions, and leads to a resonance-like behavior in gamma(n,l) when n is close to the dimension of the CFT operator dual to the heavy field. Finally, we demonstrate that bulk flat-space S-matrix elements can be extracted from the large n behavior of the anomalous dimensions. This leads to a direct connection between the spectrum of anomalous dimensions in d-dimensional CFTs and flat-space S-matrix elements in d+1 dimensions. We comment on the emergence of flat-space locality from the CFT perspective.

• The paper introduces the Effective Conformal Theory (ECT) framework for analyzing the low-lying spectrum of operator dimensions in Conformal Field Theories.
• The paper shows that perturbative unitarity in Conformal Field Theory mirrors that of an effective theory in AdS space.
• The research has theoretical implications for understanding the emergence of flat space locality in quantum theories of gravity.

Effective Conformal Theory and the Flat-Space Limit of AdS

The paper "Effective Conformal Theory and the Flat-Space Limit of AdS" by Fitzpatrick, Katz, Poland, and Simmons-Duffin introduces the conceptual framework of an Effective Conformal Theory (ECT) that describes a low-lying spectrum of the dilatation operator in a Conformal Field Theory (CFT). The motivation for this framework stems from the AdS/CFT correspondence, a pivotal hypothesis in theoretical physics connecting a CFT on the boundary of a space and a gravitational theory in an Anti-de Sitter (AdS) space.

Key Concepts and Framework

The authors propose the ECT as a tool to manage CFTs that exhibit operator dimensions with a hierarchical structure and a small parameter analogous to $1/N$ in large $N$ gauge theories. In such cases, perturbative modifications to the dilatation operator can be feasible, facilitating an effective description. The paper extends the notion of perturbative unitarity bounds, common in quantum field theories, to CFTs by mounting constraints on the anomalous dimensions $\gamma(n, l)$ of specific double-trace operators. The bounds are set at $|\gamma(n, l)|<4$, with violations potentially indicating deviations such as non-renormalizable AdS interactions.

The core analysis revolves around investigating effective theories in AdS space designed to respect conformal invariance, alongside exploring the implications of integrating a heavy scalar field in AdS. This approach unveils "resonance-like" behavior for specific values of operator dimensions, providing insights into the formation of anomalous dimensions.

Implications and Insights

The results demonstrate that ECTs can characterize the correlations of low-dimension operators in the CFT effectively. Moreover, the authors highlight a mechanism for bulk flat-space S-matrix elements to manifest from the behavior of $\gamma(n,l)$ at large dimensions. This link establishes an intriguing tie between higher-dimensional CFT spectra and flat-space scattering processes, as perturbative unitarity in CFT mirrors that of an AdS effective field theory.

Theoretical and Practical Impact

Theoretical implications of this paper extend to understanding the emergence of flat space locality in quantum theories of gravity. Practically, the framework may have applications in scenarios where CFTs describe physical systems ranging from high-energy physics to condensed matter, contingent on identifying a paradigm where such hierarchies in operator dimensions are evident.

Future Directions

Looking ahead, further exploration into the flat-space limit of AdS may unravel additional insights into nonperturbative phenomena in quantum gravity and the refinement of CFT techniques to analyze dynamics beyond tree level. Bridging these findings with empirical data, especially in systems displaying conformal behavior without supersymmetry, remains a fertile ground for future research endeavors.

In summary, this paper lays substantial groundwork for utilizing ECTs to bridge theories in curved and flat-space contexts, contributing to a deeper comprehension of the AdS/CFT duality's capacity for describing physical phenomena across scales.