The global symmetries of a D-dimensional QFT can, in many cases, be
captured in terms of a (D+1)-dimensional symmetry topological field theory
(SymTFT). In this work we construct a (D+1)-dimensional theory which governs
the symmetries of QFTs with multiple sectors which have connected correlators
that admit a decoupling limit. The associated symmetry field theory decomposes
into a SymTree, namely a treelike structure of SymTFTs fused along possibly
non-topological junctions. In string-realized multi-sector QFTs, these
junctions are smoothed out in the extra-dimensional geometry, as we demonstrate
in examples. We further use this perspective to study the fate of higher-form
symmetries in the context of holographic large M averaging where the
topological sectors of different large M replicas become dressed by
additional extended operators associated with the SymTree.Comment: 69 pages + appendices + ref