On non-supersymmetric conformal manifolds: field theory and holography

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 consider 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.

Figures (9)

• Figure 1: Integration in the $(z, \bar{z})$ plane. The fundamental domain $D_1$ is the violet region. The regions $D_2, D_3$ and $D_4$ are defined in \ref{['domains']} and are easily recognizable in the figure.
• Figure 2: Integrated conformal blocks $G$ as a function of operator dimensions for $l=0, ~2, ~4, ~6, ~8$ spin in $d=4$ dimensions.
• Figure 3: The estimate $\Sigma(\Delta_*)$ as a function of $\Delta_*$.
• Figure 4: Witten diagrams contributing to $C_{\cal OOO}$. Violet lines correspond to propagation of $\phi$ fields and may have spacetime derivatives acting on them, depending on the specific structure of the operators \ref{['scalintgen1']}. At tree-level, only cubic couplings can contribute to the three-point function. At loop level, also couplings with $n>3$ can contribute, e.g., the quintic coupling shown in the figure.
• Figure 5: Structure of Witten diagrams contributing to the two-loop coefficient of $\beta(g)$, after integration in $d^dx$. Conventions are as in figure \ref{['Wdt3all']}.