Operator product expansion of the energy momentum tensor in 2D conformal field theories on manifolds with boundary

TL;DR

This paper derives the operator product expansion of the energy-momentum tensor in 2D conformal field theories on manifolds with boundary, showing it remains unchanged from the boundary-free case due to the trace anomaly's properties.

Contribution

It demonstrates that the T·T OPE in 2D conformal theories with boundary is identical to the boundary-free case, relying on the trace anomaly proportional to the Gauss-Bonnet density.

Findings

OPE is unaffected by boundary conditions

Trace anomaly proportional to Gauss-Bonnet density is crucial

Results relate to open string sigma-model approach

Abstract

Starting from the well-known expression for the trace anomaly we derive the $T\cdot T$ operator product expansion of the energy-momentum tensor in 2D conformal theories defined in the upper halfplane $without$ making use of the additional condition of no energy-momentum flux across the boundary. The OPE turns out to be the same as in the absence of the boundary. For this result it is crucial that the trace anomaly is proportional to the Gau\ss-Bonnet density. Some relations to the $\sigma$ - model approach for open strings are discussed.