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.