Chern-Simons corner phase spaces in BF-BB theories

We investigate an approach to determine corner Poisson brackets of fields restricted to codimension 2 and 3 surfaces in 4D theories, without making use of boundary conditions. Instead, within the example of BF-BB type theories, we show that the constraint/generator algebra on a partial Cauchy slice is enough to determine the Poisson brackets and a corner symplectic form, useful in extended phase space constructions or holography. The codimension 2 phase space turns out to be of Chern-Simons type, but lacking a flatness constraint. In a setting with codimension 3 surfaces, there is also a further Wess-Zumino-type current algebra. We apply this approach also to a specific formulation of full 4D gravity, based on the Maxwell algebra $\text{gfrak}=\mathfrak{s}\mathfrak{o}(1,3)⋉({\text{rbb}}^{1,3}\tilde{\oplus }\mathfrak{s}\mathfrak{o}(1,3{)}^{\ast })$. This realises the corner Poisson bracket of the spin connection for the first time and shows it is off-shell commutative, while the corner metric is noncommutative.