On the Boundary Dynamics of Chern-Simons Gravity

TL;DR

This work provides a covariant, off-shell derivation of boundary dynamics for three-dimensional gravity formulated as Chern-Simons theory with either complex $G_{{\fam\msbfam C}}$ or real $G\times G$ gauge groups. By fixing half of the boundary gauge fields and enforcing strict $G$-invariance, the authors derive a unique boundary action that is a coset WZW model, either $G_{{\fam\msbfam C}}/G$ or $(G\times G)/G$, coupled to the boundary gauge data. The boundary theory has target space equal to the bulk gravity geometry ($dS_3$ or $AdS_3$), and on pure gauge configurations it reduces to a WZW model for $h=g\bar{g}^{-1}$, with PW identities ensuring off-shell recombination of chiral sectors. The results illuminate the role of boundary dynamics in 3D gravity, relate to worldsheet-like interpretations, and clarify how bulk and boundary theories interrelate without invoking a strict holographic duality. Overall, the paper unifies previous non-covariant approaches and presents a robust framework for boundary coset WZW actions in CS gravity.

Abstract

We study Chern-Simons theory with a complex G_C or a real G x G gauge group on a manifold with boundary - this includes Lorentzian and Euclidean (anti-) de Sitter (E/A)dS gravity for G=SU(2) or G=SL(2,R). We show that there is a canonical choice of boundary conditions that leads to an unambiguous, fully covariant and gauge invariant, off-shell derivation of the boundary action - a G_C/G or G WZW model, coupled in a gauge invariant way to the boundary value of the gauge field. In particular, for (E/A)dS gravity, the boundary action is a WZW model with target space (E/A)dS_3, reminiscent of a worldsheet for worldsheet mechanism. We discuss in some detail the properties of the boundary theories that arise and we confront our results with various related constructions in the literature.