Supersymmetric boundary algebras in AdS$_3$ gravity: constraints, Dirac brackets and non-locality
We study supersymmetric boundary dynamics in AdS$_3$ gravity by treating boundary conditions as constrained reductions of a generic Chern-Simons boundary phase space. We focus on how the constraint structure controls the locality and physical field content of the reduced asymptotic symmetry algebra. For supersymmetric Kac--Moody boundary conditions, the surviving bosonic currents remain local, whereas the fermion--fermion Dirac bracket contains the inverse of a field-dependent first-order differential operator and is therefore intrinsically non-local. Imposing an additional fermionic constraint reveals a further structure: after the bosonic second-class reduction, it becomes first class and generates a fermionic gauge redundancy. Generically the remaining fermionic field can be gauged away, while special backgrounds with periodic zero modes support non-trivial physical fermionic sectors. We further employ the Batalin--Fradkin--Tyutin formalism to eliminate the non-locality of the reduced symmetry algebra by converting the second-class constraints into an equivalent local first-class system on an enlarged phase space.