$d>2$ Stress-Tensor OPE near a Line

TL;DR

This paper investigates the operator product expansion of the stress tensor in higher-dimensional conformal field theories with Einstein gravity duals, revealing an algebraic structure akin to a Virasoro-like algebra and connecting it to known 2D structures.

Contribution

It introduces a novel algebraic structure derived from the stress-tensor OPE in $d>2$ CFTs and explores its relation to 2D $ ext{W}$-algebras, providing insights into higher-dimensional conformal dynamics.

Findings

Derived an algebraic structure consistent with the Jacobi identity from the $TT$ OPE.

Identified a connection between higher-dimensional stress-tensor blocks and 2D $ ext{W}$-algebras.

Highlighted the role of transverse integrals and a curious central term in the algebra.

Abstract

We study the $TT$ OPE in $d>2$ CFTs whose bulk dual is Einstein gravity. Directly from the $TT$ OPE, we obtain, in a certain null-like limit, an algebraic structure consistent with the Jacobi identity: $[{\cal L}_m, {\cal L}_n]= (m-n) {\cal L}_{m+n}+ C m (m^2-1) \delta_{m+n,0}$. The dimensionless constant $C$ is proportional to the central charge $C_T$. Transverse integrals in the definition of ${\cal L}_m$ play a crucial role. We comment on the corresponding limiting procedure and point out a curiosity related to the central term. A connection between the $d>2$ near-lightcone stress-tensor conformal block and the $d=2$ $\cal W$-algebra is observed. This note is motivated by the search for a field-theoretic derivation of $d>2$ correlators in strong coupling critical phenomena.