Dimensional reduction of AdS3 Chern-Simons gravity: Schwarzian and affine boundary theories

Abstract

We study a symmetry-reduced sector of $AdS_3/\mathbb Z_2$ gravity formulated as an $SO(2,2)$ Chern--Simons theory on a 3D-manifold with toroidal boundary. The reduction is implemented by requiring a globally defined symmetry and restricting to the sector in which the gauge connection is invariant along the symmetry flow. The resulting theory reduces to a two-dimensional BF-like model together with an induced one-dimensional boundary action. We show that the reduced theory admits two inequivalent boundary sectors, originated by two different boundary conditions for the parent 3d theory at the level of the variational principle. On the boundary subspace $A_τ=Φ$, the universal one-dimensional action reproduces the standard Drinfel'd--Sokolov reduction in JT gravity which captures the Schwarzian boundary dynamics. On the generalized boundary $A_τ=λ'Φ+u^{-1}\partial_τu$, the same action instead yields a deformed Schwarzian functional with affine residual symmetry, naturally associated with a non-extremal or Rindler-type regime. We further show how the $\mathfrak{so}(2,2)$ algebra of the 3D Chern--Simons model naturally leads to current-dressed Kac--Moody extensions of both sectors.