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

This paper delves into the exploration of suitable boundary conditions for the asymptotically flat scenario of general relativity presented in terms of Ashtekar-Barbero variables. While the standard parity conditions have been extensively studied by Thiemann [Classical Quantum Gravity 12, 181 (1995)] and Campiglia [Classical Quantum Gravity 32, 145011 (2015)], it turns out that they fail to produce nontrivial supertranslations at spatial infinity. We propose new parity conditions for the Ashtekar-Barbero variables that do yield nontrivial supertranslation charges at spatial infinity. We compare our findings with those of Henneaux and Troessaert [J. High Energy Phys. 03 (2018) 147] and demonstrate that the new boundary conditions ensure the finiteness of the symplectic structure. Moreover, when embarking on the quest for appropriate parity conditions, it is essential to ensure that the selected parities remain invariant under hypersurface deformations. Given that working with Ashtekar-Barbero variables provides more asymptotic structure as compared to the Arnowitt-Deser-Misner variables, it is shown that by fixing the Lagrange multiplier corresponding to the Gauss constraint, the invariance of certain parity conditions can be guaranteed.