Dual Gravitons in AdS4/CFT3 and the Holographic Cotton Tensor

TL;DR

The paper shows that gravity in AdS$_4$ can be holographically dual to two distinct CFT$_3$ theories: a Dirichlet CFT$_1$ with a fixed boundary graviton sourcing the stress-energy tensor, and a dual CFT$_2$ with a fixed dual graviton sourcing a dual stress-energy tensor identified with the Cotton tensor of CFT$_1$. The dual theories are related by a gravitational Legendre transformation generated by a gravitational Chern–Simons coupling, mirroring electric–magnetic duality, and the duality operates at all radial slices via a bulk $S$-duality that exchanges $ angle T_{ij} angle$ and $C_{ij}[ar{h}]$. Robin and mixed boundary conditions reveal dynamical boundary gravity, yielding Cotton-gravity or topologically massive gravity on the boundary and enabling a nonzero vev for the dual stress-energy tensor even when the original source vanishes. The framework extends beyond linear order, showing a non-linear dual graviton with a CS-induced coupling between the two gravitons, and provides a comprehensive holographic dictionary for duality of two-point functions and the nontrivial boundary dynamics, with potential implications for three-dimensional CFTs and condensed-matter systems.

Abstract

We argue that gravity theories in AdS4 are holographically dual to either of two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton acts as a source for the stress-energy tensor, and a dual CFT2 with a fixed dual graviton which acts as a source for a dual stress-energy tensor. The dual stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The two CFT's are related by a Legendre transformation generated by a gravitational Chern-Simons coupling. This duality is a gravitational version of electric-magnetic duality valid at any radius r, where the renormalized stress-energy tensor is the electric field and the Cotton tensor is the magnetic field. Generic Robin boundary conditions lead to CFT's coupled to Cotton gravity or topologically massive gravity. Interaction terms with CFT1 lead to a non-zero vev of the stress-energy tensor in CFT2 coupled to gravity even after the source is removed. We point out that the dual graviton also exists beyond the linearized approximation, and spell out some of the details of the non-linear construction.