Lightcone Commutator and Stress-Tensor Exchange in $d>2$ CFTs

Motivated by developing a field-theoretic algebraic approach to the universal part of the stress-tensor sector of a scalar four-point function in a class of higher-dimensional CFTs, we construct a mode operator, ${\mathcal{L}}_m$, near the lightcone in $d=4$ CFTs and show that it leads to a Virasoro-like commutator, including a regularized central-term. As an example, we describe how to reproduce the $d=4$ single-stress tensor exchange contribution in the lightcone limit by a mode summation. A general-$d$ extension is included. We comment on possible generalizations.