Effective conformal theory and the flatspace limit of AdS
Abstract
We develop the idea of an effective conformal theory describing the lowlying 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 lowestdimension singletrace operator is a scalar, mathcal{O} , we consider the anomalous dimensions, γ( n, l), of the doubletrace 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. Nonrenormalizable 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 resonancelike 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 flatspace Smatrix 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 ddimensional CFTs and flatspace Smatrix elements in d + 1 dimensions. We comment on the emergence of flatspace locality from the CFT perspective.
 Publication:

Journal of High Energy Physics
 Pub Date:
 July 2011
 DOI:
 10.1007/JHEP07(2011)023
 arXiv:
 arXiv:1007.2412
 Bibcode:
 2011JHEP...07..023F
 Keywords:

 AdSCFT Correspondence;
 Field Theories in Higher Dimensions;
 1/N Expansion;
 High Energy Physics  Theory
 EPrint:
 46 pages, 2 figures. v2: JHEP published version