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

TL;DR

The paper shows that the $SL(2,\mathrm{Z})$ action on 3D CFTs, previously understood for a $U(1)$ current, extends to two-point functions of the energy-momentum tensor and higher-spin conserved currents; the nontrivial $S$-operation is realized by an irrelevant current-current (or stress-tensor) double-trace deformation, and the $T$-operation corresponds to shifting a conformal contact term. Explicit results are derived for spin-1 and spin-2, and generalized to arbitrary spin, with the dual currents exhibiting the same modular transformation of their two-point function coefficients $(C_s,W_s) \to (C_s/(C_s^2+W_s^2), -W_s/(C_s^2+W_s^2))$. A bulk-AdS$_4$ interpretation is proposed where the $S$- and $T$-operations arise from a gravity action of MacDowell-Mansouri form with a Pontryagin term, and possibly extended to higher spins via Vasiliev-type theories, suggesting that many 3D CFTs have AdS$_4$ duals with $SL(2,\mathrm{Z})$ duality at the linearized level. The work links the holography of free field theories to discrete dualities and points to rich further structure in 3D CFTs and their higher-spin holographic avatars.

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.