Conformal manifolds: ODEs from OPEs

TL;DR

<p>The paper tackles how exactly marginal deformations in a conformal field theory constrain local operator data and OPE coefficients. It derives a two-loop beta-function sum rule from conformal perturbation theory and recasts the resulting constraints as a dynamical system that would flow operator dimensions $\Delta_i(g)$ and OPE coefficients $\lambda_{ijk}(g)$ along a conformal manifold. In one dimension, the authors obtain an explicit, closed set of evolution equations and prove a robust avoidance of level crossing for operators with the same quantum numbers, illustrating the framework with the compact free boson as an exact realization. These results offer a principled route to systematically track CFT data along exactly marginal directions and point toward numerical implementations for higher-dimensional theories and more complex manifolds.

Abstract

The existence of an exactly marginal deformation in a conformal field theory is very special, but it is not well understood how this is reflected in the allowed dimensions and OPE coefficients of local operators. To shed light on this question, we compute perturbative corrections to several observables in an abstract CFT, starting with the beta function. This yields a sum rule that the theory must obey in order to be part of a conformal manifold. The set of constraints relating CFT data at different values of the coupling can in principle be written as a dynamical system that allows one to flow arbitrarily far. We begin the analysis of it by finding a simple form for the differential equations when the spacetime and theory space are both one-dimensional. A useful feature we can immediately observe is that our system makes it very difficult for level crossing to occur.

Figures (5)

• Figure 1: Once we send our four points to $(0, z, 1, \infty)$, $\mathcal{R}_{12}$, $\mathcal{R}_{23}$ and $\mathcal{R}_{13}$ map to the blue, red and yellow $z$-plane regions respectively. We have used a change of variables to give all integrals the blue domain, which we denote by $\mathcal{R}$.
• Figure 2: Plots showing how a primary operator in $\hat{\mathcal{O}} \times \hat{\mathcal{O}}$ contributes to the beta function. These follow from a numerical integral but they may be obtained analytically in one dimension.
• Figure 3: We represent $\mathbb{R}^d$ as a blob with the function to be integrated inside it. The left and right choices both compute the two-loop beta function. Because $\mathbb{R}^d$ has no boundary, the integrated power-laws being subtracted are equal to their divergent parts.
• Figure 4: The cartoons obtained by splitting Figure \ref{['circle1']} into $s$, $t$ and $u$ channel regions. In both cases, the identity block is not annihilated. A divergence subtraction is now no longer the same as a full subtraction. In particular, removing $\mathrm{div}$ everywhere in the left blob would not compute a physical quantity.
• Figure 5: Continuing to complex $z$, our blocks have one branch cut from $-\infty$ to $0$ and another from $1$ to $\infty$. This differs from higher-dimensional blocks which are analytic on $\mathbb{C} \setminus (1, \infty)$. As an example, we may multiply two $SL(2 ; \mathbb{R})$ blocks to get an $SL(2 ; \mathbb{C})$ block. This causes the left cut to cancel and the right cut to double.