The authors give an answer to some open questions about the definition of conserved quantities in Chern-Simons theory, with particular to \(SL(2, \mathbb R)\times SL(2,\mathbb R)\simeq SO(2, 2)\) Chern-Simons theory in dimension three (with this gauge choice, Chern-Simons theory is well suited to describe AdS gravity). The attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. The geometric framework for the Hamiltonian and symplectic formulation of the theory is illustrated.
A theory satisfying the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern-Simons Lagrangian and using a recipe developed earlier by the authors in G. Allemandi, M. Francaviglia and M. Raiteri [Classical Quantum Gravity 19, 2633-2655 (2002; Zbl 1004.83022), see also gr-qc/0110104] to calculate the variation of conserved quantities.
The problem of giving a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the Bañados-Teitelboim-Zanelli (BTZ) black hole. The first law of black-hole mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern-Simons AdS\(_3\) gravity theory and the variation of conserved quantities in general relativity is analyzed in detail.