SymTFT Fans: The Symmetry Theory of 4d N=4 Super Yang-Mills on spaces with boundaries

The SymTFT construction is an efficient way of studying the symmetries of Quantum Field Theories. In this paper we initiate the study of the SymTFT construction for interacting theories on spaces with boundaries, in the concrete example of $d=4$ $\mathcal{N}=4$ Super Yang-Mills theory with Gaiotto-Witten boundary conditions. The resulting SymTFT has a number of peculiarities, most prominently a fan-like structure with a number of different SymTFT sectors joined by gapless interfaces that merge at the Gaiotto-Witten boundary SCFT.