OPE in a generally covariant form

TL;DR

This paper explores the operator product expansion in conformal field theories on curved manifolds, revealing universal curvature terms that influence calculations in conformal perturbation theory.

Contribution

It introduces a covariant organization of the OPE using geodesic distances and identifies universal curvature terms in the expansion.

Findings

Curvature terms appear in the OPE on curved spaces.

Leading curvature term involves the Schouten tensor for D>2.

Curvature contributions are universal and relevant for perturbation theory.

Abstract

We discuss the general covariance of operator product expansion in D-dimensional Euclidean conformal field theories. We propose to organise the expansion in powers of geodesic distance between two insertion points and to use the tangent vector to the geodesic for contractions with tensor operators. For conformally flat manifolds we show by explicit calculation that certain curvature terms arise in the OPE. For example for D>2 the leading term of this type in the identity channel of OPE of two scalar primaries is proportional to the Schouten tensor. We further argue that the terms we found are present for a general metric and are thus universal but there may be curvature terms at higher order in the expansion whose coefficients are not determined by the flat space OPE. The curvature terms we discuss are of practical interest in conformal perturbation theory calculations on curved spaces.