I am reading the paper arXiv:9906126. https://arxiv.org/abs/gr-qc/9906126 on the symmetry algebra at horizon (see also well known work done by Brown and Henneaux about the asymptotic algebra of AdS$_3$ in 1986).

In this paper the symmetry of horizon is defined by the Poisson brackets of diffeomorphism generators $H[\xi]$ who compose of volume term (sum of constraints) and boundary terms $J$, as eq(3.1) in that paper.

When on shell, the constraints vanish, so only $J$ contribute to $H[\xi]$, so the algebra take the form as eq(3.4). However, I do not know why the Possion bracket in (3.1) turns to Dirac bracket in (3.4)?

P.S. I know that in the Hamiltonian analysis, the Dirac bracket is define by subtracting the Possion bracket the terms involving the second class constraints. While the first class constraints correspond gauge transformations.


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