End of the World Brane Dynamics in Holographic 4d $\mathcal{N}=4$ SU(N) with 3d $\mathcal{N}=2$ Boundary Conditions

We consider 4d $\mathcal{N}=4$ $SU(N)$ super Yang-Mills on half-space with boundary conditions defined by configurations of Gaiotto-Witten NS5- and D5-branes modded out by a rotated orientifold 5-plane breaking an extra half of the supersymmetries. We focus on configurations in which this O5'-brane is split by an NS5-brane into two oppositely charged halves, leading to a breaking of supersymmetry down to 3d $\mathcal{N}=2$ at the local level. The 5-brane configurations turn into non-trivial $(p,q)$ webs and lead to non-trivial brane and gauge theory phenomena, including a 5d $\mathbf{Z}_2$ global gauge anomaly cancelled by a novel 6d global anomaly inflow, and the appearance of localized 3d $\mathcal{N}=2$ Chern-Simons couplings and the 3d parity anomaly at the boundary of the 4d theory. The systems have explicit gravity duals, given by orientifolds of the AdS$_4\times\mathbf{S}^2\times\mathbf{S}^2$ fibrations over a Riemann surface dual to the Gaiotto-Witten setups. They describe solutions with an asymptotic AdS$_5\times\mathbf{S}^5$ terminating at AdS$_4$ End of the World boundary configurations, whose rich dynamics reproduces the above physical properties. They also provide the SymTFT of the 4d $\mathcal{N}=4$ on half-space with a new class of boundary conditions. We explore the interplay of bulk and boundary topological couplings and draw general lessons for the classification of cobordism defects of bulk quantum gravity theories in the swampland program.