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This paper delves into the exploration of suitable boundary conditions forthe asymptotically flat scenario of general relativity presented in terms ofAshtekar-Barbero variables. While the standard parity conditions have beenextensively studied in \cite{Thiemann, Campiglia}, it turns out that they failto produce non-trivial supertranslations at spatial infinity. We propose newparity conditions for the Ashtekar-Barbero variables that do yield non-trivialsupertranslation charges at spatial infinity. We compare our findings withthose presented in \cite{Henneaux} and demonstrate that the new boundaryconditions ensure the finiteness of the symplectic structure. Moreover, whenembarking on the quest for appropriate parity conditions, it is essential toensure that the selected parities remain invariant under hypersurfacedeformations. Given that working with Ashtekar-Barbero variables provides moreasymptotic structure as compared to the ADM variables, it is shown that byfixing the Lagrange multiplier corresponding to the Gauss constraint, theinvariance of certain parity conditions can be guaranteed.

Boundary conditions for Ashtekar-Barbero variables in the context of  asymptotically flat spacetimes which lead to supertranslations at spatial  infinity