Abstract
Abstract We study the contact terms that appear in the correlation functions of exactly marginal operators using the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. It is known that CFT with an exactly marginal deformation requires the existence of the contact terms with their coefficients having a geometrical interpretation in the context of conformal manifolds. We show that the AdS/CFT correspondence captures properly the mathematical structure of the correlation functions that is expected from the CFT analysis. For this purpose, we employ a holographic renormalization group to formulate a most general setup in the bulk for describing an exactly marginal deformation. The resultant bulk equations of motion are nonlinear and solved perturbatively to obtain the on-shell action. We compute three- and four-point functions of the exactly marginal operators using the GKP–Witten prescription, and show that they match the expected results precisely. The cut-off surface prescription in the bulk serves as a regularization scheme for conformal perturbation theory in the boundary CFT. As an application, we examine a double OPE limit of the four-point functions. The anomalous dimensions of double trace operators are written in terms of the geometrical data of a conformal manifold.
Highlights
Marginal operators of CFT are defined as a scalar operator that has the conformal dimension∆ = d at a conformal fixed point with d being the spacetime dimension
It is argued that the gravity dual of the exactly marginal operator is given by a massless scalar in AdSd+1 with a trivial potential term
We fix the bulk metric to be an AdSd+1 background metric with d even, and treat only the bulk scalars as a dynamical field. This implies that the exchange diagrams due to the stress tensor are missed from the results in the four-point function
Summary
Marginal operators of CFT are defined as a scalar operator that has the conformal dimension. It is argued that the gravity dual of the exactly marginal operator is given by a massless scalar in AdSd+1 with a trivial potential term This guarantees that the RG β-function for the marginal couplings vanishes.. For an analysis of exactly margical deformation within the context of the AdS/CFT correspondence, see [15,16,17,18] and [9] Starting with this model, we work out the on-shell action by solving the equations of motion of the bulk scalars. We fix the bulk metric to be an AdSd+1 background metric with d even, and treat only the bulk scalars as a dynamical field This implies that the exchange diagrams due to the stress tensor are missed from the results in the four-point function.
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