Effective conformal theory and the flat-space limit of AdS

By A. Fitzpatrick, E. Katz, David Poland, David Simmons-Duffin. 2010

208 citations recorded by the source corpus.

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, $ \mathcal{O} $, we consider the anomalous dimensions, γ(n, l), of the double-trace operators of the form $ \mathcal{O}{\left( {{\partial^2}} \right)^n}{\left( \partial \right)^l}\mathcal{O} $. Purely from the CFT we find that perturbative unitarity places a bound on these dimensions of |γ(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 γ(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 ofanomalous 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.

Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c007e9c154544ce7580bd58e2bb46642449a08ea

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Citations in the corpus

Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents (2012, unknown): “, in [50, 51].”
Universality of long-distance AdS physics from the CFT bootstrap (2014, result): “[11, 29])”
Operator product expansion convergence in conformal field theory (2012, methodology): “In the AdS/CFT context, a CFT truncated to just the subsector of low-dimension operators has been dubbed “Effective CFT” [29].”
A natural language for AdS/CFT correlators (2011, background): “One can obtain a holographic description of the flat space S-Matrix [3, 27–33] by taking a limit of the AdS/CFT correspondence in which the curvature of AdS goes to zero.”
Causality constraints in conformal field theory (2015, background): “(See [25] for the boundary point of view.”
Quartic AdS interactions in higher-spin gravity from Conformal Field Theory (2015, background): “Criteria for a CFT to have a local bulk dual at the these scales have been proposed in [62, 65–67].”
Looking for a bulk point (2015, background): “For large n, these are given by [11,40]”
The analytic bootstrap and AdS superhorizon locality (2012, background): “The calculation of anomalous dimensions of double-trace operators starting with a perturbative AdS description is usually simplest for the lowest-spin states [16, 55], and becomes increasingly more involved for higher spins.”
Unitarity and the holographic S-Matrix (2011, methodology): “Furthermore, special conformal generators in the flat-space limit become the momentum operators [16, 15].”
OPE Convergence in Conformal Field Theory (2012, methodology): “In the AdS/CFT context, a CFT truncated to just the subsector of low-dimension operators has been dubbed “Effective CFT” [29].”
Carving out the space of 4D CFTs (2011, background): “Moreover, studying conformal field theories (CFTs) can also give us important insights into quantum gravity and string theory via the AdS/CFT correspondence [38–40], which in turn provides a simple framework for describing many new physics scenarios via effective field theories in AdS [41–43] (dual ”
Analyticity and the holographic S-matrix (2011, methodology): “2 of [17] using an inner product on the space of propagators”
Emergent spacetime and holographic CFTs (2011, background): “From this point of view the hologram seems to be an effective representation of a small part of the Hilbert space of the CFT (a similar point of view is espoused in [12]).”
Wormholes, emergent gauge fields, and the weak gravity conjecture (2015, background): “This is in some tension with the philosophy of “effective conformal field theory” advocated in [36–38], but it is not a contradiction with the validity of ECFT near the vacuum since we are”
Projectors, shadows, and conformal blocks (2012, background): “Finally, let us note that the technology developed here should work equally well in Mellin space [45, 46], which is proving to be a convenient setting for understanding effective CFTs [47, 48] dual to weakly coupled theories in AdS [37, 49–51].”
Spinning conformal blocks (2011, background): “Additional progress has been made at understanding the constraints of unitarity in this context [9], as well as in exploring the recently found analogy between CFT correlation functions in the Mellin representation and scattering amplitudes [10–13].”