Mathematical structure of finite generalized symmetries above two dimensions
In plain words
In two dimensions all finite symmetries, including those without an undo, are described by a known algebraic structure. In higher dimensions the right structure is conjectured but not established.
Precise statement
In $2\mathrm{D}$, finite internal symmetries are fusion categories. In d dimensions finite generalized symmetries are conjectured to be fusion $\left(d-1\right)$-categories, equivalently a $\left(d+1\right)$-dimensional topological field theory (the symmetry TFT) with a gapped boundary. Determine whether every finite generalized symmetry of a d-dimensional QFT arises from a symmetry TFT in this way, and classify the possible structures for $d = 3\ \text{and}\ 4$.
What would settle it
A proof that any finite collection of topological defects in a $d$-dimensional QFT defines a symmetry TFT with gapped boundary, together with a classification for $d = 3, 4$.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
The symmetry TFT framework was developed in 2021-2025; classification of fusion higher categories is incomplete.
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