{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"8b612a04dce549bc49163f86f5341ade5e44e0a95ecad8131f168249962aa68b","created":"2026-10-03T07:18:09Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"f8a032110f9a27d63e0f32391732ccadd8d07a32bc98cfa51dc22093437913a2","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"classification","assisted_by":[],"external_id":"qft.generalized-symmetries.symtft-classification","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"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.","posed_since":"","precise":"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 $d = 3\\ \\text{and}\\ 4$.","problem_ref":null,"references":"","settled_by":"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_note":"The symmetry TFT framework was developed in 2021-2025; classification of fusion higher categories is incomplete.","title":"Mathematical structure of finite generalized symmetries above two dimensions","topic_ref":"5eb0476a75858c5dd523d1082965cb71e335c77a7a5cf121927138b7c11c6ef2"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"f1ab6124c8b44f0070fc1b09a710af934533acd4a8fc97dd0be7b2737226fbfa","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"a21fbd7576d91dc7522f7c6e211027163f53d5406df68cb34b81362c0a5a554e","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"yMJodpkJuTW3Hd211i0zmRnFLD2eB5Ofi_QRq0xjHF1fqWxnMQuwhDLqHaGJ-ogh3fWP-RD2_ZguqspwaIs0BA"},"schema":"pubphys.envelope/1"},"record_hash":"f1ab6124c8b44f0070fc1b09a710af934533acd4a8fc97dd0be7b2737226fbfa","leaf_index":1913}