Conformal manifolds: ODEs from OPEs

TL;DR

This paper explores the structure of conformal manifolds by deriving differential equations from OPE data, revealing constraints on operator dimensions and OPE coefficients in the presence of exactly marginal deformations.

Contribution

It introduces a dynamical system framework for understanding how CFT data varies along conformal manifolds, with explicit equations in one-dimensional cases.

Findings

Derived sum rule for CFT data on conformal manifolds

Formulated differential equations governing CFT data flow

Observed that level crossing is unlikely in the system

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.