A so(2,2) extension of JT gravity via the Virasoro-Kac-Moody semidirect product

We consider a bulk plus boundary extension of Jackiw-Teitelboim Gravity (JT) coupled with non-abelian gauge fields. The generalization is performed in the Poisson Sigma Model formulation and it is derived as a dimensional reduction of the AdS3 Chern-Simons theory with WZW boundary terms. We discuss the role of boundary conditions in relation to the symmetries of the boundary dynamics and we show that the boundary action can be written in terms of coadjoint orbits of an appropriate Virasoro Kac-Moody group. We obtain a Schwarzian action and interaction terms with additional edge modes that match the effective low energy action of recent SYK-like tensor models.