SL(2,Z) Action on Three-Dimensional CFTs and Holography

TL;DR

This paper explores how SL(2,Z) transformations act on two-point functions in 3D conformal field theories and discusses the holographic implications for dual AdS4 theories with SL(2,Z) duality.

Contribution

It introduces a natural SL(2,Z) action on two-point functions in 3D CFTs and links the S-operation to an irrelevant current-current deformation, suggesting duality properties in holographic models.

Findings

SL(2,Z) acts on energy-momentum tensor two-point functions.

The S-operation corresponds to an irrelevant current-current deformation.

Many 3D CFTs may have AdS4 duals with SL(2,Z) duality at the linearized level.

Abstract

We show that there is a natural action of SL(2,Z) on the two-point functions of the energy momentum tensor and of higher-spin conserved currents in three-dimensional CFTs. The dynamics behind the S-operation of SL(2,Z) is that of an irrelevant current-current deformation and we point out its similarity to the dynamics of a wide class of three-dimensional CFTs. The holographic interpretation of our results raises the possibility that many three-dimensional CFTs have duals on AdS4 with SL(2,Z) duality properties at the linearized level.