Energy Momentum Tensor in Conformal Field Theories Near a Boundary

TL;DR

This paper investigates the form of the energy momentum tensor's two-point function near a boundary in conformal field theories, determining its functional dependence and boundary operators across various dimensions and theories.

Contribution

It derives the general form of the energy momentum tensor two-point function near a boundary, including dependence on a conformally invariant variable and boundary operators, extending to curved manifolds.

Findings

Functional dependence on variable v determined for free fields and in epsilon expansion.

Unique expression for correlation functions involving energy momentum tensor and scalar fields.

Boundary operators related to energy momentum tensor components via diffeomorphisms.

Abstract

The requirements of conformal invariance for the two point function of the energy momentum tensor in the neighbourhood of a plane boundary are investigated, restricting the conformal group to those transformations leaving the boundary invariant. It is shown that the general solution may contain an arbitrary function of a single conformally invariant variable $v$, except in dimension 2. The functional dependence on $v$ is determined for free scalar and fermion fields in arbitrary dimension $d$ and also to leading order in the $\vep$ expansion about $d=4$ for the non Gaussian fixed point in $\phi^4$ theory. The two point correlation function of the energy momentum tensor and a scalar field is also shown to have a unique expression in terms of $v$ and the overall coefficient is determined by the operator product expansion. The energy momentum tensor on a general curved manifold is further discussed by considering variations of the metric. In the presence of a boundary this procedure naturally defines extra boundary operators. By considering diffeomorphisms these are related to components of the energy momentum tensor on the boundary. The implications of Weyl invariance in this framework are also derived.