Approximate Symmetries in $d=4$ CFTs with an Einstein Gravity Dual

TL;DR

This paper uncovers approximate Virasoro and Kac-Moody algebraic structures in four-dimensional conformal field theories with Einstein gravity duals, revealing new symmetries in the stress tensor sector at large scalar dimensions.

Contribution

It demonstrates that certain rescaled stress tensor modes in 4D CFTs satisfy Virasoro-like and Kac-Moody-like algebras, extending the understanding of symmetries in holographic CFTs.

Findings

Rescaled stress tensor modes satisfy Virasoro-like algebra.

Inclusion of $T^{--}$ enhances to Kac-Moody algebra.

Finite central terms are obtained in the algebraic structures.

Abstract

By applying the stress-tensor-scalar operator product expansion (OPE) twice, we search for algebraic structures in $d=4$ conformal field theories (CFTs) with a pure Einstein gravity dual. We find that a rescaled mode operator defined by an integral of the stress tensor $T^{++}$ on a $d=2$ plane satisfies a Virasoro-like algebra when the dimension of the scalar is large. The structure is enhanced to include a Kac-Moody-type algebra if we incorporate the $T^{--}$ component. In our scheme, the central terms are finite. It remains challenging to directly compute the stress-tensor sector of $d=4$ scalar four-point functions at large central charge, which, based on holography and bootstrap methods, were recently shown to have a Virasoro/${\cal W}$-algebra vacuum block-like structure.