Abstract
We discuss the constraints that a conformal field theory should enjoy to admit exactly marginal deformations, i.e. to be part of a conformal manifold. In particular, using tools from conformal perturbation theory, we derive a sum rule from which one can extract restrictions on the spectrum of low spin operators and on the behavior of OPE coefficients involving nearly marginal operators. We then focus on conformal field theories admitting a gravity dual description, and as such a large-N expansion. We discuss the relation between conformal perturbation theory and loop expansion in the bulk, and show how such connection could help in the search for conformal manifolds beyond the planar limit. Our results do not rely on supersymmetry, and therefore apply also outside the realm of superconformal field theories.
Highlights
JHEP11(2017)167 take any of the known SCFTs belonging to a conformal manifold and truncate the spectrum of operators, excluding all operators with half-integer spins, while leaving CFT data of integer-spin operators unmodified
We discuss the constraints that a conformal field theory should enjoy to admit exactly marginal deformations, i.e. to be part of a conformal manifold
We focus on conformal field theories admitting a gravity dual description, and as such a large-N expansion
Summary
Given a CFT and a deformation as that in eq (1.1), one expects that a β function for the coupling g is generated and that conformal invariance is lost. Expanding (2.2) in g one gets a perturbative expansion in terms of integrals of n-point functions of O These are generically plagued by logarithmic divergences, which can be absorbed by demanding that the coupling g runs with scale μ in a way that the final result is μ-independent. This, in turns, lets one extract the β function Proceeding this way one gets for the β function at two loops (which to this order is universal, independent of the renormalization scheme) the following expressions. At one loop in CPT, the persistence of a conformal manifold under the deformation (1.1) implies the following constraints on the OPE coefficients of the CFT. (2.6) and (2.7) are the two constraints the existence of a conformal manifold under the deformation (1.1) imposes on the CFT at two-loop order in CPT. Eqs. (2.6) and (2.7) are the two constraints the existence of a conformal manifold under the deformation (1.1) imposes on the CFT at two-loop order in CPT.
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