SL(2, ℤ) action on three-dimensional CFTs and holography

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.