{"as_of":"2026-08-07T16:40:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:1ca11bcc0db1cc7ec052d0403e0deb28b5fc2db6af871a955e68c81c109300c0","coverage":[{"denominator":122,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":100,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-06T21:09:17.365266Z","state":"measured"},{"denominator":109,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":109,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-07T06:34:17.273281+00:00","state":"measured"},{"denominator":9,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":9,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-04T05:19:23.122338Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"arxiv_reference","source_observed_at":"2026-07-02T13:57:00.363141Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2508.08639","last_updated":"2026-03-14T14:11:12Z","snapshot_observed_at":"2026-07-06T02:11:23.670680Z","submitted_at":"2025-08-12T05:05:35Z","title":"Extending fusion rules with finite subgroups: A general construction of $Z_{N}$ extended conformal field theories and their orbifoldings","version":5},"reference_index":204,"source":"pdf_text","source_observed_at":"2026-05-21T23:53:04.901248Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2508.08639"},"observation_digest":"sha256:e684e45a3a2fac731c393e58d2e5c125fb660f60c8a9367a684964eace8c1b37","observation_id":"51588b6d-c500-4d7f-b1bf-6aa61b0837e2","resolution":{"observed_at":"2026-05-21T23:54:28.448724Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2605.07734","last_updated":"2026-08-03T13:30:05Z","snapshot_observed_at":"2026-08-06T23:31:04.621272Z","submitted_at":"2026-05-08T13:41:03Z","title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","version":1},"reference_index":120,"source":"pdf_text","source_observed_at":"2026-05-11T03:23:34.161611Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2605.07734"},"observation_digest":"sha256:1238319629ac0775992ab6b8f25c92cef5fc25b43c2ee30be62607f113036440","observation_id":"c88eda1c-dd7f-470a-9fb7-04350acf30b6","resolution":{"observed_at":"2026-05-11T03:25:56.779845Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2605.07734","last_updated":"2026-08-03T13:30:05Z","snapshot_observed_at":"2026-08-06T23:31:04.621272Z","submitted_at":"2026-05-08T13:41:03Z","title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","version":2},"reference_index":122,"source":"pdf_text","source_observed_at":"2026-05-20T22:59:26.266718Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2605.07734"},"observation_digest":"sha256:f8bc077514b26596d959b04c6ddd6a09a0c5554e127c9f6d7b2b30f69f03b329","observation_id":"603b1bc9-d124-47e7-9672-0654b8e23b5a","resolution":{"observed_at":"2026-05-20T23:03:51.697515Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2605.07734","last_updated":"2026-08-03T13:30:05Z","snapshot_observed_at":"2026-08-06T23:31:04.621272Z","submitted_at":"2026-05-08T13:41:03Z","title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","version":3},"reference_index":115,"source":"pdf_text","source_observed_at":"2026-06-30T23:13:18.266390Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2605.07734"},"observation_digest":"sha256:4a741182a4f4ba2d67f20b897b2f6bb8db7253f8acee1aad8da37f6f976b3b8b","observation_id":"242d8e78-5337-4444-bfcc-c27452461bbc","resolution":{"observed_at":"2026-06-30T23:15:08.452776Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-08-04T05:19:23.122338Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2605.07734","last_updated":"2026-08-03T13:30:05Z","snapshot_observed_at":"2026-08-06T23:31:04.621272Z","submitted_at":"2026-05-08T13:41:03Z","title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","version":4},"reference_index":114,"source":"pdf_text","source_observed_at":"2026-08-04T05:19:23.122338Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2605.07734"},"observation_digest":"sha256:b8f05589f1460b098cdf5c6572ba37af32d8cecaeabd6411b0861b633ee6dd2a","observation_id":"0696cb3d-2ff8-447e-b831-16d2ffbb0ac7","resolution":{"observed_at":"2026-08-04T05:19:23.122338Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2605.28688","last_updated":"2026-06-22T14:35:22Z","snapshot_observed_at":"2026-08-03T10:57:10.209932Z","submitted_at":"2026-05-27T16:17:28Z","title":"Topological lattice gauge theory enriched by non-invertible symmetry","version":2},"reference_index":59,"source":"arxiv_source","source_observed_at":"2026-06-29T09:55:51.803708Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2605.28688"},"observation_digest":"sha256:3d9bdc35ffc07b972f2f086a01929a3a87f5f72a2808d45067fca79d9816315c","observation_id":"99388a45-07d7-46d2-a3c4-4e8e47388df5","resolution":{"observed_at":"2026-06-29T10:03:17.231103Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2606.05279","last_updated":"2026-06-03T18:00:00Z","snapshot_observed_at":"2026-08-02T13:11:18.915766Z","submitted_at":"2026-06-03T18:00:00Z","title":"Hypergroup Symmetry in Relative Quantum Field Theories and Chiral Algebras","version":1},"reference_index":95,"source":"pdf_text","source_observed_at":"2026-06-28T05:12:12.178791Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2606.05279"},"observation_digest":"sha256:82ef2dd819cbc25769fb8239d7a9047d8c6fac9e390818bc4725306be62b3398","observation_id":"77ec221f-c46b-4ac4-8a1a-d748d03093e4","resolution":{"observed_at":"2026-07-02T10:16:52.047309Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":"2507.01142","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-07-02T13:57:00.363141Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders,","venue":null,"work_id":"c8d1befe-3772-4ef1-afc0-e68642de7cb8","year":null},"citing_paper":{"arxiv_id":"2606.05543","last_updated":"2026-07-15T02:48:01Z","snapshot_observed_at":"2026-08-02T12:26:57.733415Z","submitted_at":"2026-06-04T00:58:08Z","title":"Notes on (-2)-form symmetries","version":1},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-06-28T00:46:19.482598Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2606.05543"},"observation_digest":"sha256:8411749b4d9e9a27716f7958bb499c59f93cc4ca2093299af6a144ac78488ec8","observation_id":"228891f9-e712-4c57-91fe-92f908e17f8a","resolution":{"observed_at":"2026-07-02T13:57:00.364918Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.01142","snapshot_observed_at":"2026-08-02T12:26:59.980699Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.05543","last_updated":"2026-07-15T02:48:01Z","snapshot_observed_at":"2026-08-02T12:26:57.733415Z","submitted_at":"2026-06-04T00:58:08Z","title":"Notes on (-2)-form symmetries","version":2},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-08-02T12:26:59.980699Z"},"links":{"cited_paper":"/paper/2507.01142","citing_paper":"/paper/2606.05543"},"observation_digest":"sha256:4f927da5e804f471ebbc4cafd58baa3a533a25ee7bc9d276680707aa5c50f6a1","observation_id":"3be72644-7cc6-4f93-a4bb-020c03f73d82","resolution":{"observed_at":"2026-08-02T12:26:59.980699Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"links":{"evidence":"/evidence","html":"/paper/2507.01142/citation-record","integrity":"/paper/2507.01142/integrity","json":"/paper/2507.01142/citation-record.json","paper":"/paper/2507.01142"},"outbound":[{"citation":{"cited_paper":{"arxiv_id":"1809.01483","last_updated":"2020-04-27T15:22:34Z","snapshot_observed_at":"2026-07-06T06:59:18.703646Z","submitted_at":"2018-09-05T13:22:58Z","title":"Orbifolds of Reshetikhin-Turaev TQFTs","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1809.01483","snapshot_observed_at":"2026-08-06T21:09:05.799548Z","title":"Orbifolds of Reshetikhin-Turaev TQFTs","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:05.799548Z"},"links":{"cited_paper":"/paper/1809.01483","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:f1cfe29bca19911edc4ab711a9940467bbaa8bedf33bf094ba6e4d3aa9038cc2","observation_id":"ba27f8e3-ea57-44f4-aadc-f379b2891ae2","resolution":{"observed_at":"2026-08-06T21:09:05.799548Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:05.846186Z","title":"On generalized symmetries and structure of modular categories","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:05.846186Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:1e6d6c684a59e5416c6fd18bcb39f860c7fd43e9cf98fac931aacfb6f33ff1a6","observation_id":"e3a12dff-e4cf-452c-88fc-6b549c158f77","resolution":{"observed_at":"2026-08-06T21:09:05.846186Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2010.00932","last_updated":"2020-10-02T11:51:46Z","snapshot_observed_at":"2026-07-06T10:00:53.952758Z","submitted_at":"2020-10-02T11:51:46Z","title":"Fibonacci-type orbifold data in Ising modular categories","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2010.00932","snapshot_observed_at":"2026-08-06T21:09:05.942937Z","title":"Fibonacci-type orbifold data in Ising modular categories","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:05.942937Z"},"links":{"cited_paper":"/paper/2010.00932","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8391afac57f2e4d420a9c0d2b3f520725919840d410485d9aae5eeee3f5cd7ef","observation_id":"1078e71b-b880-4050-9a3c-4ef0cadd3a58","resolution":{"observed_at":"2026-08-06T21:09:05.942937Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2206.02611","last_updated":"2022-06-06T13:31:38Z","snapshot_observed_at":"2026-07-06T13:17:51.474784Z","submitted_at":"2022-06-06T13:31:38Z","title":"Condensation inversion and Witt equivalence via generalised orbifolds","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2206.02611","snapshot_observed_at":"2026-08-06T21:09:06.049724Z","title":"Condensation inversion and Witt equivalence via generalised orbifolds","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.049724Z"},"links":{"cited_paper":"/paper/2206.02611","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:1869bc9238c4d49f0e5315bd1908ee00f5f91f309c9c45c5f3065fd7476f9f76","observation_id":"1a866b16-c042-42aa-b27c-fe5758225c53","resolution":{"observed_at":"2026-08-06T21:09:06.049724Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2301.01259","last_updated":"2024-02-15T10:40:47Z","snapshot_observed_at":"2026-07-06T14:37:09.104197Z","submitted_at":"2023-01-03T17:56:40Z","title":"Higher categorical symmetries and gauging in two-dimensional spin systems","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2301.01259","snapshot_observed_at":"2026-08-06T21:09:06.130710Z","title":"Higher categorical symmetries and gauging in two- dimensional spin systems","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.130710Z"},"links":{"cited_paper":"/paper/2301.01259","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:5ed2062c2607fd2495badfbafdf2d9b050a9d9c1e02da4b7ba1308542dcf9aa6","observation_id":"2aeca688-68f9-4510-b420-2f7b9b517c32","resolution":{"observed_at":"2026-08-06T21:09:06.130710Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2405.20401","last_updated":"2025-05-07T18:17:26Z","snapshot_observed_at":"2026-07-06T18:22:57.400982Z","submitted_at":"2024-05-30T18:12:15Z","title":"Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.20401","snapshot_observed_at":"2026-08-06T21:09:06.303969Z","title":"Fractionalization of Coset Non-Invertible Sym- metry and Exotic Hall Conductance","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.303969Z"},"links":{"cited_paper":"/paper/2405.20401","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:e323fc6e02635dc9bad630b46e6d13a8d1c7c68c6da162ec9c468a056754a4f8","observation_id":"94469623-5899-4417-88b3-052d537277a8","resolution":{"observed_at":"2026-08-06T21:09:06.303969Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2503.13685","last_updated":"2025-03-17T19:48:18Z","snapshot_observed_at":"2026-07-06T20:54:18.695103Z","submitted_at":"2025-03-17T19:48:18Z","title":"Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2503.13685","snapshot_observed_at":"2026-08-06T21:09:06.442080Z","title":"Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.442080Z"},"links":{"cited_paper":"/paper/2503.13685","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:2431099bd6f5614165886383a532b62cabb16e8672a6f549a3b01a88232c3cb4","observation_id":"d8bc520a-d700-4be3-906e-b8902d8bd81a","resolution":{"observed_at":"2026-08-06T21:09:06.442080Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:06.568114Z","title":"Anomalies of Coset Non-Invertible Symmetries","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.568114Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:72d3d9c293653a442ca2f3c3d38990e11f8fbfe4b7b1afb1a1d901f7158a4054","observation_id":"06808fea-83dc-40be-b73a-857c98564072","resolution":{"observed_at":"2026-08-06T21:09:06.568114Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:06.719592Z","title":"(2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.719592Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:cd0765c0dac44cd17cbac87862121a274364fed515b2eb6095622f8c477eef57","observation_id":"9737b9a2-be26-4552-bfd3-c5ae985393f3","resolution":{"observed_at":"2026-08-06T21:09:06.719592Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2503.02925","last_updated":"2025-07-19T19:50:25Z","snapshot_observed_at":"2026-07-06T20:46:46.095651Z","submitted_at":"2025-03-04T19:00:00Z","title":"Gauging non-invertible symmetries on the lattice","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2503.02925","snapshot_observed_at":"2026-08-06T21:09:06.810815Z","title":"Gauging non-invertible symmetries on the lattice","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.810815Z"},"links":{"cited_paper":"/paper/2503.02925","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:428e60863d6a1c19ccb73d929fb78c86226a2580b7fe81af40a7298e42aa1f4c","observation_id":"c415d25c-c223-4450-ad83-1ed1b2b235c7","resolution":{"observed_at":"2026-08-06T21:09:06.810815Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2302.00689","last_updated":"2023-10-31T22:32:17Z","snapshot_observed_at":"2026-07-06T14:47:23.410556Z","submitted_at":"2023-02-01T19:00:03Z","title":"Universal lower bound on topological entanglement entropy","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2302.00689","snapshot_observed_at":"2026-08-06T21:09:06.915373Z","title":"Universal Lower Bound on Topological Entanglement Entropy","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:06.915373Z"},"links":{"cited_paper":"/paper/2302.00689","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:3c3f44944b90fbea5af562a6b61b7fe1d8fa530624ec53b55bc00d63ce4e4630","observation_id":"356ffc0c-5985-4307-b4b2-02bfa9303000","resolution":{"observed_at":"2026-08-06T21:09:06.915373Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2112.03061","last_updated":"2022-01-06T17:31:14Z","snapshot_observed_at":"2026-07-06T12:15:47.711366Z","submitted_at":"2021-12-02T18:58:34Z","title":"Efficiently preparing Schr\\\"odinger's cat, fractons and non-Abelian topological order in quantum devices","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2112.03061","snapshot_observed_at":"2026-08-06T21:09:07.037333Z","title":"Efficiently preparing Schr¨ odinger’s cat, fractons and non-Abelian topological order in quantum devices","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.037333Z"},"links":{"cited_paper":"/paper/2112.03061","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8f25e1e692f2019ecae513f95f424b7ac02290f4cee6d3d1c45bf44a4ecccd9b","observation_id":"972f86e6-63f9-4e2b-87d2-c2e41674a00b","resolution":{"observed_at":"2026-08-06T21:09:07.037333Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:07.157858Z","title":"Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.157858Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:39871d15f977e54785461eda992e03ca155262c513c19622465ab96dcd0ce391","observation_id":"a7c77b5e-4351-4e6b-adde-8b45e18b1e7a","resolution":{"observed_at":"2026-08-06T21:09:07.157858Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:07.334656Z","title":"Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.334656Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:ca93c9b976132dd2dafd9a7fbb4734f0e7b077a39be0949f0cbc266a7bcea299","observation_id":"88e8699a-a505-4e25-b8bf-37bfbdc979dc","resolution":{"observed_at":"2026-08-06T21:09:07.334656Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2411.04985","last_updated":"2025-08-29T17:29:57Z","snapshot_observed_at":"2026-07-06T19:46:54.707852Z","submitted_at":"2024-11-07T18:55:09Z","title":"Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2411.04985","snapshot_observed_at":"2026-08-06T21:09:07.453828Z","title":"Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.453828Z"},"links":{"cited_paper":"/paper/2411.04985","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:c1d576a9aa019236e64d7aaa774d59e142a9f14771065c5361c4d87259410c2e","observation_id":"ec74c508-34b4-4079-88ef-3cc0cd196199","resolution":{"observed_at":"2026-08-06T21:09:07.453828Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:07.555114Z","title":"Higher Gauging and Non-invertible Condensation Defects","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.555114Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:4f010f2d6a03f9d49304706cf0c421a48308466b838dc98bc9dc73b4e87c344f","observation_id":"6f24aac7-1de8-41e0-8736-3d1a047a84db","resolution":{"observed_at":"2026-08-06T21:09:07.555114Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:07.638174Z","title":"Higher condensation theory","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.638174Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7ef2500e60933c369f1ec3e4407ddad2c13efa5f7c95055e7db998bdbea3fe0d","observation_id":"690a004f-b330-46e9-b5d4-bf750d8d2fc7","resolution":{"observed_at":"2026-08-06T21:09:07.638174Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2408.00866","last_updated":"2024-08-01T18:37:24Z","snapshot_observed_at":"2026-07-06T18:55:49.577776Z","submitted_at":"2024-08-01T18:37:24Z","title":"On the Classification of Bosonic and Fermionic One-Form Symmetries in $2+1$d and 't Hooft Anomaly Matching","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2408.00866","snapshot_observed_at":"2026-08-06T21:09:07.756679Z","title":"On the Classification of Bosonic and Fermionic One-Form Symmetries in 2+1d and ’t Hooft Anomaly Matching","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.756679Z"},"links":{"cited_paper":"/paper/2408.00866","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:eefc878bab401d7a22bb3d27054739e01837687b7ade0fd56f70446ae12f8412","observation_id":"acc9569d-37fa-4848-afbe-8126f50b26c3","resolution":{"observed_at":"2026-08-06T21:09:07.756679Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1410.4540","last_updated":"2019-10-15T16:12:53Z","snapshot_observed_at":"2026-08-05T15:38:30.137639Z","submitted_at":"2014-10-16T19:21:13Z","title":"Symmetry Fractionalization, Defects, and Gauging of Topological Phases","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1410.4540","snapshot_observed_at":"2026-08-06T21:09:07.884955Z","title":"Symmetry Fractionalization, Defects, and Gauging of Topological Phases","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:07.884955Z"},"links":{"cited_paper":"/paper/1410.4540","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:15eac904372761fec15b99f01ed513e4c258977d09fb624f97f398f9cc230e78","observation_id":"7302ce66-17ca-4e01-9245-75d4fd050179","resolution":{"observed_at":"2026-08-06T21:09:07.884955Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2208.14018","last_updated":"2022-08-30T06:47:04Z","snapshot_observed_at":"2026-08-05T00:08:25.560653Z","submitted_at":"2022-08-30T06:47:04Z","title":"Composing topological domain walls and anyon mobility","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2208.14018","snapshot_observed_at":"2026-08-06T21:09:08.057441Z","title":"Composing topological domain walls and anyon mobility","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:08.057441Z"},"links":{"cited_paper":"/paper/2208.14018","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:6f2623acdbc15dfe0e3ad770b597440ef9172b9040c9bbdc9b101ac7214b9eea","observation_id":"5e15cdc5-ac63-47d3-aef8-1b0cba1cf81a","resolution":{"observed_at":"2026-08-06T21:09:08.057441Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2309.15181","last_updated":"2023-09-26T18:37:54Z","snapshot_observed_at":"2026-07-06T16:24:01.868957Z","submitted_at":"2023-09-26T18:37:54Z","title":"Invertibility of Condensation Defects and Symmetries of 2 + 1d QFTs","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2309.15181","snapshot_observed_at":"2026-08-06T21:09:08.182768Z","title":"Invertibility of Condensation Defects and Symme- tries of 2 + 1d QFTs","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:08.182768Z"},"links":{"cited_paper":"/paper/2309.15181","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:3fffd99dd0976692d34c5fe07bade1f38a2562dcfd577bfb0d77138dc76e3189","observation_id":"babe66b3-9f3b-422b-993a-fc16020f2b20","resolution":{"observed_at":"2026-08-06T21:09:08.182768Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:08.285817Z","title":"On Gauging Symmetry of Mod- ular Categories","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:08.285817Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:d45c6bcfff29e963dc48ea50367057ed13da667026ae989126a1085492030bf9","observation_id":"859e9b13-e3fe-4403-b828-56570e30d771","resolution":{"observed_at":"2026-08-06T21:09:08.285817Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.22544","last_updated":"2025-06-27T18:00:03Z","snapshot_observed_at":"2026-08-06T22:01:30.151891Z","submitted_at":"2025-06-27T18:00:03Z","title":"An Algebraic Theory of Gapped Domain Wall Partons","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.22544","snapshot_observed_at":"2026-08-06T21:09:08.608053Z","title":"An Algebraic Theory of Gapped Domain Wall Partons","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:08.608053Z"},"links":{"cited_paper":"/paper/2506.22544","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:e703e2b23a77f68061e841b3ff186cfa09831813bd78a1c68669a4ba136e3022","observation_id":"70267e21-ab87-47ae-8755-f6f4f92907ed","resolution":{"observed_at":"2026-08-06T21:09:08.608053Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1408.6514","last_updated":"2014-11-26T01:59:16Z","snapshot_observed_at":"2026-07-06T03:52:48.428145Z","submitted_at":"2014-08-27T19:59:25Z","title":"Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1408.6514","snapshot_observed_at":"2026-08-06T21:09:08.709030Z","title":"Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:08.709030Z"},"links":{"cited_paper":"/paper/1408.6514","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:31f714a2807dff7dc179c968897472090e9abeff1cddd12cb6b4eba78af041fa","observation_id":"f0fa2209-c9a5-4c40-be7b-241f9d4e360c","resolution":{"observed_at":"2026-08-06T21:09:08.709030Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:08.876406Z","title":"On the structure of the Witt group of braided fusion categories","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:08.876406Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:32babc030bd4707fa2715a2614af73c59643eff08c48e20c4419c49656d8022a","observation_id":"ca7908a7-2f9b-4906-8832-1b7e1701c331","resolution":{"observed_at":"2026-08-06T21:09:08.876406Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"hep-th/0609042","last_updated":"2006-09-05T18:33:34Z","snapshot_observed_at":"2026-07-07T04:58:27.165568Z","submitted_at":"2006-09-05T18:33:34Z","title":"D-branes and K-theory in 2D topological field theory","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"hep-th/0609042","snapshot_observed_at":"2026-08-06T21:09:09.008775Z","title":"D-branes and K-theory in 2D topological field theory","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.008775Z"},"links":{"cited_paper":"/paper/hep-th/0609042","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:b73aa2cf709e7d9cee02fbe6d106bc7e264e0613a129c81423a6b55015e1d5ac","observation_id":"4ae2d71e-bbb3-4b07-b55d-af3fd0df2fb4","resolution":{"observed_at":"2026-08-06T21:09:09.008775Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2110.02958","last_updated":"2021-10-28T21:57:07Z","snapshot_observed_at":"2026-08-01T15:01:22.128854Z","submitted_at":"2021-10-06T18:00:00Z","title":"Construction of two-dimensional topological field theories with non-invertible symmetries","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2110.02958","snapshot_observed_at":"2026-08-06T21:09:09.114063Z","title":"Construction of two-dimensional topo- logical field theories with non-invertible symmetries","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.114063Z"},"links":{"cited_paper":"/paper/2110.02958","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:9a263b5a5ef5e6a8dad57e0cdfe414d935e5a24362f0df94f1041e1d33c5a531","observation_id":"46ce1c14-514f-4aaf-9fcb-c7bfea7b8301","resolution":{"observed_at":"2026-08-06T21:09:09.114063Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"math/0203255","last_updated":"2002-03-24T20:39:02Z","snapshot_observed_at":"2026-08-02T17:37:13.805061Z","submitted_at":"2002-03-24T20:39:02Z","title":"Fusion categories of rank 2","version":1},"cited_work":{"arxiv_id":"math/0203255","doi":null,"metadata_source":"pith","pith_arxiv_id":"math/0203255","snapshot_observed_at":"2026-08-06T21:09:22.504662Z","title":"Fusion categories of rank 2","venue":"math.QA","work_id":"14ed84f9-15c0-4d32-b478-ae11853d120e","year":2002},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.208355Z"},"links":{"cited_paper":"/paper/math/0203255","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:cc6e9e9ac71759a74c387c16ff9abc5251cddf35880dc226c6827ea94d2950b8","observation_id":"6ef11836-dc61-4718-98f7-43b5c2f8001e","resolution":{"observed_at":"2026-08-06T21:09:22.510448Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:09.313068Z","title":"Tensor categories","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.313068Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:ed7cb574cdceda3e899bf2269ad7e69e16a243a0d51fb29da16c5b6cee5b5094","observation_id":"2dcb329a-abe6-46b2-a5d8-8af1fa7ef5d3","resolution":{"observed_at":"2026-08-06T21:09:09.313068Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:09.418717Z","title":"Rigid and separable algebras in fusion 2-categories","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.418717Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8eb30e2395cb7a6abcdc99881b99422d4c629e16208c6f27c9623db4e65b0825","observation_id":"3427680a-48cf-4e4c-b5e3-f1e9770362ca","resolution":{"observed_at":"2026-08-06T21:09:09.418717Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2311.16827","last_updated":"2024-08-18T21:30:57Z","snapshot_observed_at":"2026-08-04T09:46:52.959918Z","submitted_at":"2023-11-28T14:45:56Z","title":"On the Dualizability of Fusion 2-Categories","version":3},"cited_work":{"arxiv_id":"2311.16827","doi":null,"metadata_source":"pith","pith_arxiv_id":"2311.16827","snapshot_observed_at":"2026-08-06T21:09:22.477203Z","title":"On the Dualizability of Fusion 2-Categories","venue":"math.CT","work_id":"3f815248-8bda-4752-884b-9e52f2a14653","year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.507894Z"},"links":{"cited_paper":"/paper/2311.16827","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:6338bb50e56d9ce757421617599c9e9425386c396b258292d8da762022725d62","observation_id":"f69f2b23-06a0-411e-8ac8-9d3bd2b3c917","resolution":{"observed_at":"2026-08-06T21:09:22.484122Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2305.09713","last_updated":"2023-12-07T14:56:47Z","snapshot_observed_at":"2026-08-07T10:02:53.725251Z","submitted_at":"2023-05-16T18:00:00Z","title":"Remarks on Boundaries, Anomalies, and Noninvertible Symmetries","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2305.09713","snapshot_observed_at":"2026-08-06T21:09:09.606844Z","title":"Remarks on boundaries, anomalies, and noninvertible symmetries","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.606844Z"},"links":{"cited_paper":"/paper/2305.09713","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8e8a7cd5d9c4701a7f0d70851e609f4f4484901659859d4e43b8c598e3bade4e","observation_id":"ca324992-9424-47ef-8b6c-87876a7f0543","resolution":{"observed_at":"2026-08-06T21:09:09.606844Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2503.13633","last_updated":"2025-03-17T18:29:00Z","snapshot_observed_at":"2026-08-04T16:05:57.571322Z","submitted_at":"2025-03-17T18:29:00Z","title":"Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2503.13633","snapshot_observed_at":"2026-08-06T21:09:09.710829Z","title":"Braidings on topological operators, anomaly of higher- form symmetries and the SymTFT","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.710829Z"},"links":{"cited_paper":"/paper/2503.13633","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8e48c449477cf6c48ba41063d8c3b021b7d9738d5bb13d52df2183a14a7aeb5a","observation_id":"72bd179d-5c6a-43af-951c-ff086bfda0cb","resolution":{"observed_at":"2026-08-06T21:09:09.710829Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1707.04564","last_updated":"2018-01-30T21:01:41Z","snapshot_observed_at":"2026-08-01T18:33:26.829012Z","submitted_at":"2017-07-14T17:06:15Z","title":"Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1707.04564","snapshot_observed_at":"2026-08-06T21:09:09.819594Z","title":"Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.819594Z"},"links":{"cited_paper":"/paper/1707.04564","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:c03fda9e7c3b2a77045aeeed7f6e01a5ff7a7351f03de6cd04025584d49ccea1","observation_id":"35f05881-a98a-4ada-8b1c-ad50d5a150ee","resolution":{"observed_at":"2026-08-06T21:09:09.819594Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"math/0309465","last_updated":"2005-06-08T15:09:32Z","snapshot_observed_at":"2026-07-06T02:11:23.670680Z","submitted_at":"2003-09-29T18:28:39Z","title":"Correspondences of ribbon categories","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"math/0309465","snapshot_observed_at":"2026-08-06T21:09:09.949751Z","title":"Correspondences of ribbon categories","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:09.949751Z"},"links":{"cited_paper":"/paper/math/0309465","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:fd8347e1a15090772c26c14d9efc2e0e10e5462009a57f18d267d63bd8a3a3ef","observation_id":"817d100b-d36f-4ecf-a478-4a2a37fc59d5","resolution":{"observed_at":"2026-08-06T21:09:09.949751Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1307.8244","last_updated":"2021-11-09T17:26:26Z","snapshot_observed_at":"2026-08-05T05:48:30.938493Z","submitted_at":"2013-07-31T08:04:28Z","title":"Anyon condensation and tensor categories","version":7},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1307.8244","snapshot_observed_at":"2026-08-06T21:09:10.051308Z","title":"Anyon condensation and tensor categories","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.051308Z"},"links":{"cited_paper":"/paper/1307.8244","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7439e9893a948f1f3b81a5bbfc38ffc001e75e231e8693077324be9818393ee5","observation_id":"af974245-0fae-4465-8ed4-a3dbb53d018e","resolution":{"observed_at":"2026-08-06T21:09:10.051308Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"cond-mat/0404617","last_updated":"2004-04-27T13:24:42Z","snapshot_observed_at":"2026-08-04T19:23:46.056516Z","submitted_at":"2004-04-26T15:50:47Z","title":"String-net condensation: A physical mechanism for topological phases","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"cond-mat/0404617","snapshot_observed_at":"2026-08-06T21:09:10.144662Z","title":"String net condensation: A Physical mechanism for topological phases","venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.144662Z"},"links":{"cited_paper":"/paper/cond-mat/0404617","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:2528923bc1262aa2f8f066870a87a5980513a52334a8f188595cbd9616dfa5c4","observation_id":"b7439112-0ccb-45e0-8939-3f7e2742d760","resolution":{"observed_at":"2026-08-06T21:09:10.144662Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1106.6033","last_updated":"2011-06-29T19:30:55Z","snapshot_observed_at":"2026-07-06T02:30:06.437232Z","submitted_at":"2011-06-29T19:30:55Z","title":"String-net model of Turaev-Viro invariants","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1106.6033","snapshot_observed_at":"2026-08-06T21:09:10.224741Z","title":"String-net model of Turaev-Viro invariants","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.224741Z"},"links":{"cited_paper":"/paper/1106.6033","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:15c9838715751cc4a7c4f071cc3647c839622cfd036e4c5bd7261c18ffd7c473","observation_id":"70ac8f69-e397-4baf-8668-7083fd075680","resolution":{"observed_at":"2026-08-06T21:09:10.224741Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1812.11933","last_updated":"2018-12-31T18:16:00Z","snapshot_observed_at":"2026-07-06T07:24:20.780719Z","submitted_at":"2018-12-31T18:16:00Z","title":"Fusion 2-categories and a state-sum invariant for 4-manifolds","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1812.11933","snapshot_observed_at":"2026-08-06T21:09:10.334210Z","title":"Fusion 2-categories and a state-sum invariant for 4-manifolds","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.334210Z"},"links":{"cited_paper":"/paper/1812.11933","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7f8c3ada7461615d9959adc605c53279759791854726e6b394a04bb29ba9977c","observation_id":"56f6e1fc-5bc7-4150-9850-5cf7fe889de0","resolution":{"observed_at":"2026-08-06T21:09:10.334210Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1905.09566","last_updated":"2025-04-04T14:20:29Z","snapshot_observed_at":"2026-07-06T07:54:54.802338Z","submitted_at":"2019-05-23T10:07:25Z","title":"Condensations in higher categories","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1905.09566","snapshot_observed_at":"2026-08-06T21:09:10.443395Z","title":"Condensations in higher categories","venue":null,"work_id":null,"year":1905},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.443395Z"},"links":{"cited_paper":"/paper/1905.09566","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8417258892b269df5095cdb498f5934bc0f2e08d8f5c5a543609ec972611592d","observation_id":"be0ee8ac-e704-4174-ab95-8e26c0220c96","resolution":{"observed_at":"2026-08-06T21:09:10.443395Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2204.02407","last_updated":"2023-11-01T01:55:13Z","snapshot_observed_at":"2026-07-06T02:11:23.670680Z","submitted_at":"2022-04-05T18:00:00Z","title":"Higher Gauging and Non-invertible Condensation Defects","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2204.02407","snapshot_observed_at":"2026-08-06T21:09:10.579384Z","title":"Higher Gauging and Non-invertible Condensation Defects","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.579384Z"},"links":{"cited_paper":"/paper/2204.02407","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:0ab40b1c72a6ee5e7449dd5c00a82ca68cdee6190363c5b09aa5276f350223ba","observation_id":"752c740d-8856-4c15-8780-8f6c3ba01719","resolution":{"observed_at":"2026-08-06T21:09:10.579384Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2208.05982","last_updated":"2022-08-22T21:56:34Z","snapshot_observed_at":"2026-07-06T13:41:07.345036Z","submitted_at":"2022-08-11T18:00:02Z","title":"Decomposition, condensation defects, and fusion","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2208.05982","snapshot_observed_at":"2026-08-06T21:09:10.670321Z","title":"Decomposition, Condensation Defects, and Fusion","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.670321Z"},"links":{"cited_paper":"/paper/2208.05982","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:513fa0e6ce69611d5c17a1d2142546ca798d6af158fa20acbe9220c4b710e4df","observation_id":"f9189357-7c2f-4d7b-96c0-c9220576fb16","resolution":{"observed_at":"2026-08-06T21:09:10.670321Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2204.09025","last_updated":"2022-06-20T02:34:09Z","snapshot_observed_at":"2026-08-06T12:25:56.923031Z","submitted_at":"2022-04-19T17:35:30Z","title":"Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2204.09025","snapshot_observed_at":"2026-08-06T21:09:10.808027Z","title":"Non-invertible Condensa- tion, Duality, and Triality Defects in 3+1 Dimensions","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.808027Z"},"links":{"cited_paper":"/paper/2204.09025","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:f3911c14b4435a56a0d9ec382b0061e25ddb1098d9dea1fcfd3b0c3553d48a3a","observation_id":"863d4707-85a9-4e99-a681-3d934bd5d4a3","resolution":{"observed_at":"2026-08-06T21:09:10.808027Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2405.13105","last_updated":"2024-05-21T18:00:00Z","snapshot_observed_at":"2026-07-06T18:17:34.028834Z","submitted_at":"2024-05-21T18:00:00Z","title":"Non-invertible and higher-form symmetries in 2+1d lattice gauge theories","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.13105","snapshot_observed_at":"2026-08-06T21:09:10.918917Z","title":"Non-invertible and higher-form symmetries in 2+1d lattice gauge theories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:10.918917Z"},"links":{"cited_paper":"/paper/2405.13105","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:e184bc79a9aea938e1c835207e852fa68a332bc6338ca4a81c9f2ac5de242f7a","observation_id":"d73f1d3b-0698-406a-8a4e-41e8ccc931e9","resolution":{"observed_at":"2026-08-06T21:09:10.918917Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2208.05993","last_updated":"2023-05-05T15:58:05Z","snapshot_observed_at":"2026-07-06T13:41:07.345036Z","submitted_at":"2022-08-11T18:01:48Z","title":"Non-invertible Symmetries and Higher Representation Theory I","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2208.05993","snapshot_observed_at":"2026-08-06T21:09:11.045084Z","title":"Non-invertible Symmetries and Higher Representation Theory I","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.045084Z"},"links":{"cited_paper":"/paper/2208.05993","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:02cc6f0888f347c2aa63f733a814e8869b70f4568dfd8ed094ff5aede760d347","observation_id":"c8dff1aa-5445-41f5-bb90-0768c428e0e3","resolution":{"observed_at":"2026-08-06T21:09:11.045084Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2212.07393","last_updated":"2023-07-14T09:57:41Z","snapshot_observed_at":"2026-08-04T14:56:08.844916Z","submitted_at":"2022-12-14T18:22:06Z","title":"Non-invertible Symmetries and Higher Representation Theory II","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2212.07393","snapshot_observed_at":"2026-08-06T21:09:11.156371Z","title":"Non-invertible Symmetries and Higher Representation Theory II","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.156371Z"},"links":{"cited_paper":"/paper/2212.07393","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:615a5b2e35ac6c87d4a05080ce8b4198c19a0e23c2640ad390dd47dfd010d12f","observation_id":"3011924f-d403-4751-84ab-289c8fd0d50c","resolution":{"observed_at":"2026-08-06T21:09:11.156371Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2204.06564","last_updated":"2022-09-07T15:17:02Z","snapshot_observed_at":"2026-08-02T18:33:45.809543Z","submitted_at":"2022-04-13T18:00:00Z","title":"Non-Invertible Higher-Categorical Symmetries","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2204.06564","snapshot_observed_at":"2026-08-06T21:09:11.273772Z","title":"Non-invertible higher- categorical symmetries","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.273772Z"},"links":{"cited_paper":"/paper/2204.06564","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:bc334f3a285e61cbaf9d6cd061f63e249eb858e30fc94a4c724fa8019c34e84e","observation_id":"191aaa96-500e-4583-b3d9-98fb93730c4f","resolution":{"observed_at":"2026-08-06T21:09:11.273772Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2112.08324","last_updated":"2022-08-15T14:04:48Z","snapshot_observed_at":"2026-08-06T03:38:42.479337Z","submitted_at":"2021-12-15T18:16:10Z","title":"Tensor network approach to electromagnetic duality in (3+1)d topological gauge models","version":2},"cited_work":{"arxiv_id":"2112.08324","doi":null,"metadata_source":"pith","pith_arxiv_id":"2112.08324","snapshot_observed_at":"2026-08-06T21:09:22.180645Z","title":"Tensor network approach to electromagnetic duality in (3+1)d topological gauge models","venue":"cond-mat.str-el","work_id":"1cea727c-753c-4748-9500-b2b4e4b27b3d","year":2021},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.421793Z"},"links":{"cited_paper":"/paper/2112.08324","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:ddac5152468ba309ac3a9b3298ab6cb33da62004a7f3bfbaa9daa2e176a9a7de","observation_id":"d93bb270-6caa-404b-8aac-4224cc6c5d16","resolution":{"observed_at":"2026-08-06T21:09:22.186941Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2501.16301","last_updated":"2025-08-27T12:43:28Z","snapshot_observed_at":"2026-08-06T10:20:44.918476Z","submitted_at":"2025-01-27T18:39:10Z","title":"Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories","version":2},"cited_work":{"arxiv_id":"2501.16301","doi":null,"metadata_source":"pith","pith_arxiv_id":"2501.16301","snapshot_observed_at":"2026-08-06T21:09:22.152349Z","title":"Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories","venue":"cond-mat.str-el","work_id":"27609647-97ec-4091-baf0-31b33fda7de7","year":2025},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.525730Z"},"links":{"cited_paper":"/paper/2501.16301","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:2cff68b9a954b6cd54d91f7213bc5727fdeb1dcc8b7c5608c35568b40b0cb2bc","observation_id":"aae3c3f9-8da1-4058-ad4f-0db012871846","resolution":{"observed_at":"2026-08-06T21:09:22.158753Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"0909.3140","last_updated":"2009-12-29T02:10:39Z","snapshot_observed_at":"2026-08-01T06:58:45.443831Z","submitted_at":"2009-09-17T02:17:33Z","title":"Fusion categories and homotopy theory","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"0909.3140","snapshot_observed_at":"2026-08-06T21:09:11.659458Z","title":"Fusion categories and homotopy theory","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.659458Z"},"links":{"cited_paper":"/paper/0909.3140","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8ac936e72345dfd15edc93a98912663c8f8f7c82032f83a5cb8a4c647ab21c5f","observation_id":"ade9d20b-f304-4ee3-900c-1b175b912411","resolution":{"observed_at":"2026-08-06T21:09:11.659458Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:11.799070Z","title":"Monoidal 2-structure of bimodule cat- egories","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.799070Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:6e4d0bc370e9f19cf2a360b1364ae524f4944bb84560060514e6aceb2ed60d01","observation_id":"22a1e1b3-24fe-480f-99b1-aaa7108a0f88","resolution":{"observed_at":"2026-08-06T21:09:11.799070Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.jalgebra.2022.12.029","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:19.622406Z","title":"The relative Deligne tensor product over pointed braided fusion categories","venue":null,"work_id":"8a0e0d7c-ec02-4274-8694-087ef851460d","year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:11.908918Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:26ec48b67876e02d11e18a91cac688ea7a5533bc64e076009132000243a06d75","observation_id":"2db16f2a-d155-46a3-8b01-879199c04614","resolution":{"observed_at":"2026-08-06T21:09:19.677870Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:12.038887Z","title":"On dualizability of braided tensor categories","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.038887Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:924c77cd30224787c944e361f778e713ead009235e6df49721296168984fc471","observation_id":"fbf65683-f8a1-49cd-bd8f-b1af71a51ad7","resolution":{"observed_at":"2026-08-06T21:09:12.038887Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2107.11037","last_updated":"2023-06-05T06:10:31Z","snapshot_observed_at":"2026-08-04T10:46:22.930322Z","submitted_at":"2021-07-23T05:53:33Z","title":"Finite Semisimple Module 2-Categories","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2107.11037","snapshot_observed_at":"2026-08-06T21:09:12.147968Z","title":"Finite Semisimple Module 2-Categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.147968Z"},"links":{"cited_paper":"/paper/2107.11037","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:dea8f91354be6e546f23dcacf6e1db52bc0be1c4a808c9e00544b704a3f51cf0","observation_id":"cd8af414-49fa-489c-a04b-63233a7336ea","resolution":{"observed_at":"2026-08-06T21:09:12.147968Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2408.13302","last_updated":"2025-08-23T17:41:43Z","snapshot_observed_at":"2026-07-06T19:05:18.260091Z","submitted_at":"2024-08-23T18:00:00Z","title":"Fusion 3-Categories for Duality Defects","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2408.13302","snapshot_observed_at":"2026-08-06T21:09:12.264840Z","title":"Fusion 3-Categories for Duality Defects","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.264840Z"},"links":{"cited_paper":"/paper/2408.13302","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:e1caa2ff1a44b3fdcbf7f9a0763570c92bb209315cdb770c3c7a392ce73b6901","observation_id":"32853d2d-8cec-4f56-a8e7-f9e67e557e1e","resolution":{"observed_at":"2026-08-06T21:09:12.264840Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:12.423218Z","title":"Dualities between 2+1d fusion surface models from braided fusion categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.423218Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:5e18cede3f9f686c097eba844e37e6dde343c04a7ddf4958d60fd031c7d3ebe0","observation_id":"70996c59-7e1f-434b-b872-85d1aabe6f0c","resolution":{"observed_at":"2026-08-06T21:09:12.423218Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2401.13838","last_updated":"2024-01-24T22:29:45Z","snapshot_observed_at":"2026-07-06T17:20:07.930031Z","submitted_at":"2024-01-24T22:29:45Z","title":"Levin-Wen is a gauge theory: entanglement from topology","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2401.13838","snapshot_observed_at":"2026-08-06T21:09:12.571102Z","title":"Levin-Wen is a Gauge Theory: Entanglement from Topology","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.571102Z"},"links":{"cited_paper":"/paper/2401.13838","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:bd51929ef5af893f52ba324f28a5955de39eb52dcb37469c485d8faa2ce2ab64","observation_id":"afab97dd-69ef-4d62-8842-1a9165039ee6","resolution":{"observed_at":"2026-08-06T21:09:12.571102Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:12.739136Z","title":"The Witt group of non- degenerate braided fusion categories","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":59,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.739136Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7a52b8e9b0615d6928062a364b32dfd69319fdbbb9af19576610ce17bdbf0768","observation_id":"8e4e04f6-1241-41e1-b427-be1490691ed7","resolution":{"observed_at":"2026-08-06T21:09:12.739136Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1805.02738","last_updated":"2018-05-14T05:26:41Z","snapshot_observed_at":"2026-08-06T00:10:35.502697Z","submitted_at":"2018-05-07T20:38:57Z","title":"A study of time reversal symmetry of abelian anyons","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1805.02738","snapshot_observed_at":"2026-08-06T21:09:12.858969Z","title":"A study of time reversal symmetry of abelian anyons","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":60,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.858969Z"},"links":{"cited_paper":"/paper/1805.02738","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7a5976fd5fec524d5ec5ad1dc5483720134a6f85fb7bf8461be94de53e3450f6","observation_id":"98021094-731a-4b21-a680-56be5f0773fa","resolution":{"observed_at":"2026-08-06T21:09:12.858969Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:12.981302Z","title":"Near-group categories","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:12.981302Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:43a452b264632e1cf86d617a7b3307b1899bc04c39c6a4511c6fc820d962a088","observation_id":"f6a94b03-d4e5-4570-870e-e6189cec90eb","resolution":{"observed_at":"2026-08-06T21:09:12.981302Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:13.120230Z","title":"Generalized near-group categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":62,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.120230Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:0c2f8cb3e9890e87c9b857a1bd9f9ba8ca9d2567fb995990a3e6a635ef959bfd","observation_id":"49c346fa-6e8f-4dd1-b1bc-68c4586cc0a7","resolution":{"observed_at":"2026-08-06T21:09:13.120230Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:13.225064Z","title":"Near-group fusion categories and their doubles","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":63,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.225064Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:2e5d9ed0a6300ab6deb7634db23bc4c5a542393644fe58d54c0446bc05788a1d","observation_id":"fb01c7a5-a666-461d-94b7-e5b75c3081ca","resolution":{"observed_at":"2026-08-06T21:09:13.225064Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1512.04288","last_updated":"2015-12-30T10:55:07Z","snapshot_observed_at":"2026-08-04T02:39:42.432984Z","submitted_at":"2015-12-14T12:52:13Z","title":"A Cuntz algebra approach to the classification of near-group categories","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1512.04288","snapshot_observed_at":"2026-08-06T21:09:13.346287Z","title":"A Cuntz algebra approach to the classification of near-group categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":64,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.346287Z"},"links":{"cited_paper":"/paper/1512.04288","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:8b45db9c5d333494a2d645b56f116e8f9fd92121c8c81b75590daaab4ae9c0ff","observation_id":"769f2d00-0777-4567-8d78-7ce9ff66ee92","resolution":{"observed_at":"2026-08-06T21:09:13.346287Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:13.500579Z","title":"Uniqueness of unitary structure for unitarizable fusion categories","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":65,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.500579Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:632357e55dfc4947986f27e5d66f1ac5b81d618b90470403dbc317efdecefa93","observation_id":"720239a7-9a91-434a-8b73-eb695ba3b0c9","resolution":{"observed_at":"2026-08-06T21:09:13.500579Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2305.03766","last_updated":"2024-02-14T17:25:16Z","snapshot_observed_at":"2026-08-02T22:15:31.311933Z","submitted_at":"2023-05-05T18:01:05Z","title":"Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2305.03766","snapshot_observed_at":"2026-08-06T21:09:13.629069Z","title":"Non-Abelian topological order and anyons on a trapped-ion processor","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":66,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.629069Z"},"links":{"cited_paper":"/paper/2305.03766","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:4dcb03d5a9400b4230c93eed9b6b7ae1114766165672b70f386b2ef29241a89b","observation_id":"b80c7d5d-c1df-44a5-834d-5f9ff3b6208c","resolution":{"observed_at":"2026-08-06T21:09:13.629069Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2411.04181","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:21.655761Z","title":"Protocols for Creating Anyons and Defects via Gauging","venue":null,"work_id":"2231375b-a119-4f27-86c0-39ab4b0b36c9","year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":67,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.749428Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:10cea24d0463009552ed385811150f23a43af7c3b8a164b72e2379f31a12d6c3","observation_id":"f5390587-e07f-41e7-81a7-9ea613f35249","resolution":{"observed_at":"2026-08-06T21:09:21.816455Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2411.04185","last_updated":"2024-11-06T19:00:02Z","snapshot_observed_at":"2026-07-06T19:46:19.931894Z","submitted_at":"2024-11-06T19:00:02Z","title":"Qutrit Toric Code and Parafermions in Trapped Ions","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2411.04185","snapshot_observed_at":"2026-08-06T21:09:13.905986Z","title":"Qutrit Toric Code and Parafermions in Trapped Ions","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":68,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:13.905986Z"},"links":{"cited_paper":"/paper/2411.04185","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:6dd991964e4e355085f11a02d99b3e5fea4f9d510c713e68bc85ae1b3a8e91b8","observation_id":"17eaea99-66f7-416e-9f79-d17c25282599","resolution":{"observed_at":"2026-08-06T21:09:13.905986Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"0905.3117","last_updated":"2009-05-19T14:50:38Z","snapshot_observed_at":"2026-08-01T02:56:16.106547Z","submitted_at":"2009-05-19T14:50:38Z","title":"Centers of graded fusion categories","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"0905.3117","snapshot_observed_at":"2026-08-06T21:09:14.003901Z","title":"Centers of graded fusion categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":69,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.003901Z"},"links":{"cited_paper":"/paper/0905.3117","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:300e8fa5e46473858e1d1983fd5e62a2b61610b1661c0f1326f17da893fe1a08","observation_id":"2b51f95e-bb2f-4656-9c78-c015840c7028","resolution":{"observed_at":"2026-08-06T21:09:14.003901Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1405.5858","last_updated":"2014-05-22T18:54:09Z","snapshot_observed_at":"2026-07-06T03:44:27.357306Z","submitted_at":"2014-05-22T18:54:09Z","title":"Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1405.5858","snapshot_observed_at":"2026-08-06T21:09:14.116611Z","title":"Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":70,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.116611Z"},"links":{"cited_paper":"/paper/1405.5858","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:d93797736f9648e348926c9eb56ebad8f90780ba94a4bf485ee5ebe5ed0a9cf4","observation_id":"88e5d204-a7ed-4b15-994f-bf094abe9925","resolution":{"observed_at":"2026-08-06T21:09:14.116611Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1103/physrevb.96.045136","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:19.499060Z","title":"Cheshire charge in (3+1)- dimensional topological phases","venue":null,"work_id":"4f003b20-824d-489b-8160-679477e01d92","year":2017},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":71,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.225364Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:37009b79ab44692bb9ca9020e3164add10b6edd724e22c46d5f5aa80c795e266","observation_id":"1caaee9e-0865-43ca-8710-054cc99e459d","resolution":{"observed_at":"2026-08-06T21:09:19.559062Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2009.06564","last_updated":"2020-09-27T15:13:38Z","snapshot_observed_at":"2026-07-06T09:55:28.543635Z","submitted_at":"2020-09-14T16:47:06Z","title":"Defects in the 3-dimensional toric code model form a braided fusion 2-category","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2009.06564","snapshot_observed_at":"2026-08-06T21:09:14.310478Z","title":"Defects in the 3-dimensional toric code model form a braided fusion 2-category","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":72,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.310478Z"},"links":{"cited_paper":"/paper/2009.06564","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:0b4a7c3feb2056c2c3da58e397b8fc944b7845e5116aa061690a13107c22806c","observation_id":"f7171a91-d950-495e-a19f-d2de1eced2fc","resolution":{"observed_at":"2026-08-06T21:09:14.310478Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2208.07361","last_updated":"2022-08-15T17:55:43Z","snapshot_observed_at":"2026-07-06T13:42:05.210724Z","submitted_at":"2022-08-15T17:55:43Z","title":"A toy model for categorical charges","version":1},"cited_work":{"arxiv_id":"2208.07361","doi":null,"metadata_source":"pith","pith_arxiv_id":"2208.07361","snapshot_observed_at":"2026-08-06T21:09:21.280726Z","title":"A toy model for categorical charges","venue":"cond-mat.str-el","work_id":"5576af49-974d-441e-b666-2509ad931412","year":2022},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":73,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.402373Z"},"links":{"cited_paper":"/paper/2208.07361","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:964c3cffbcacb12a9e79d41e2d637b8ec36cb4e22b2f72e276dd4f57e75a1bd6","observation_id":"c64821e0-2821-4de1-a71e-3a3ab3441032","resolution":{"observed_at":"2026-08-06T21:09:21.384198Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:14.535107Z","title":"Topological phase transition on the edge of two-dimensional Z2 topological order","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":74,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.535107Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:96e018d6ac461601f9b8b0e9f7faea6462dba4c725ddd5a92fb6520b95a95de1","observation_id":"ac5e19d0-d542-423a-a73e-d96c676378b3","resolution":{"observed_at":"2026-08-06T21:09:14.535107Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1004.1838","last_updated":"2010-07-28T08:42:50Z","snapshot_observed_at":"2026-07-06T02:05:52.774370Z","submitted_at":"2010-04-11T20:18:33Z","title":"Topological Order with a Twist: Ising Anyons from an Abelian Model","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1004.1838","snapshot_observed_at":"2026-08-06T21:09:14.643262Z","title":"Topological Order with a Twist: Ising Anyons from an Abelian Model","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":75,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.643262Z"},"links":{"cited_paper":"/paper/1004.1838","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:e398e633c1ad958276d48c03dcb83acb28d38cff46adf2127779550adc0c9732","observation_id":"bbfca99d-d8c6-4550-be12-e92aad04b38f","resolution":{"observed_at":"2026-08-06T21:09:14.643262Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1104.5047","last_updated":"2012-09-28T17:40:20Z","snapshot_observed_at":"2026-07-06T02:26:52.901468Z","submitted_at":"2011-04-26T23:13:57Z","title":"Models for gapped boundaries and domain walls","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1104.5047","snapshot_observed_at":"2026-08-06T21:09:14.726391Z","title":"Models for Gapped Boundaries and Domain Walls","venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":76,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.726391Z"},"links":{"cited_paper":"/paper/1104.5047","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:6f7c45bf352f6ce9ef1df613906b7975d5e8c2fec8d6129d07a2b6fc363c954e","observation_id":"c0909ef5-5b6f-4f65-93f9-523da3f1281d","resolution":{"observed_at":"2026-08-06T21:09:14.726391Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:14.809738Z","title":"Quasi hope algebras, group cohomology and orbifold models","venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":77,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.809738Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:b3ee23bf427530805b37e9e6f8443b40c7d08fe6001e3d41f7dfcf6ffffb67f0","observation_id":"a38c323a-9b5e-4550-8a86-5c449d2bb9d1","resolution":{"observed_at":"2026-08-06T21:09:14.809738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:14.895069Z","title":"The Quantum Double Model with Boundary: Condensations and Symmetries","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":78,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.895069Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:210f3981ca1da9607fe5251df53f61dceb446b491502f0b3bfd73a3c01147676","observation_id":"4de79815-5799-4dc0-9527-e9a7d0e8db59","resolution":{"observed_at":"2026-08-06T21:09:14.895069Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:14.964538Z","title":"Universal quantum computation with weakly integral anyons","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":79,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:14.964538Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:b53f4932bbe40e2751cb4250f7b4b01d0d0b08369d2c49b664f6bf7644d8bc5c","observation_id":"25377528-1f41-4f74-b9d6-9e0ea8e38990","resolution":{"observed_at":"2026-08-06T21:09:14.964538Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"math/0304194","last_updated":"2003-05-30T19:03:41Z","snapshot_observed_at":"2026-08-07T04:56:14.799440Z","submitted_at":"2003-04-15T12:23:55Z","title":"Classification of fusion categories of dimension pq","version":2},"cited_work":{"arxiv_id":"math/0304194","doi":null,"metadata_source":"pith","pith_arxiv_id":"math/0304194","snapshot_observed_at":"2026-08-06T21:09:20.875985Z","title":"Classification of fusion categories of dimension pq","venue":"math.QA","work_id":"9b602b55-a5e9-443d-94b8-235723be3ad5","year":2003},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":80,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.073280Z"},"links":{"cited_paper":"/paper/math/0304194","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:38442450d411298e2d2fb06a8e7edc01cc4527c736fdfd1356934d8a06cee4de","observation_id":"08414b96-f343-4387-9bdf-e9392e0982ae","resolution":{"observed_at":"2026-08-06T21:09:21.092177Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:15.195595Z","title":"Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":81,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.195595Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:b71be99893200a0b29f52a6c61f3efa298ba31cee1006e18e91664262b04d5c5","observation_id":"2af8f398-b370-49c9-84b3-244fcd07898e","resolution":{"observed_at":"2026-08-06T21:09:15.195595Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:15.303915Z","title":"Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups","venue":null,"work_id":null,"year":1998},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":82,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.303915Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:18e159546c8037f6224198ea229109cf34a08f115aa11cf2d15d57db79d5cb18","observation_id":"d5d23bdb-5590-4f8e-b282-3bb0b167290f","resolution":{"observed_at":"2026-08-06T21:09:15.303915Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"cond-mat/0506438","last_updated":"2008-01-01T04:29:56Z","snapshot_observed_at":"2026-08-05T12:27:26.288818Z","submitted_at":"2005-06-17T12:42:36Z","title":"Anyons in an exactly solved model and beyond","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"cond-mat/0506438","snapshot_observed_at":"2026-08-06T21:09:15.383024Z","title":"Anyons in an exactly solved model and beyond","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":83,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.383024Z"},"links":{"cited_paper":"/paper/cond-mat/0506438","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:5c4d7832d76b80e5d875c8b7fbfc8ab0a38c746e98c39698533dee676d35ec4b","observation_id":"a1419b63-6034-4291-99fd-88571f2241b3","resolution":{"observed_at":"2026-08-06T21:09:15.383024Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.08154","last_updated":"2025-06-09T19:05:15Z","snapshot_observed_at":"2026-08-07T05:16:00.874175Z","submitted_at":"2025-06-09T19:05:15Z","title":"Generalised Orbifolds and G-equivariantisation","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.08154","snapshot_observed_at":"2026-08-06T21:09:15.484777Z","title":"Generalised Orbifolds and G- equivariantisation","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":84,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.484777Z"},"links":{"cited_paper":"/paper/2506.08154","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7f87e4f4a82a4a1ef8ada54b46f2f5ac9e31cbe277188d2bf5e936faecc836c8","observation_id":"70f5535e-6565-41cf-afe7-a697ce045d4f","resolution":{"observed_at":"2026-08-06T21:09:15.484777Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.08178","last_updated":"2025-06-09T19:42:49Z","snapshot_observed_at":"2026-07-06T21:39:22.752080Z","submitted_at":"2025-06-09T19:42:49Z","title":"2-Group Symmetries of 3-dimensional Defect TQFTs and Their Gauging","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.08178","snapshot_observed_at":"2026-08-06T21:09:15.597541Z","title":"2-Group Symmetries of 3-dimensional Defect TQFTs and Their Gauging","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":85,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.597541Z"},"links":{"cited_paper":"/paper/2506.08178","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:53d8fd3ef72f644a81aa8738f8c1e645318b6b47d5fa975341a401aa3c644d24","observation_id":"49be6088-ae3b-43b1-bafd-4b35bd4be29e","resolution":{"observed_at":"2026-08-06T21:09:15.597541Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"hep-th/0607247","last_updated":"2007-01-26T19:29:31Z","snapshot_observed_at":"2026-07-06T02:11:23.670680Z","submitted_at":"2006-07-30T15:09:08Z","title":"Duality and defects in rational conformal field theory","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"hep-th/0607247","snapshot_observed_at":"2026-08-06T21:09:15.686104Z","title":"Duality and defects in rational conformal field theory","venue":null,"work_id":null,"year":2007},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":86,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.686104Z"},"links":{"cited_paper":"/paper/hep-th/0607247","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:7322d64f09169f6ad73b1682da4b192f117950bc32e0025b86057c9031ac1e66","observation_id":"29705c99-3571-4fdd-95fd-850c10935c21","resolution":{"observed_at":"2026-08-06T21:09:15.686104Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1704.02330","last_updated":"2017-10-26T16:43:09Z","snapshot_observed_at":"2026-07-31T22:06:27.209511Z","submitted_at":"2017-04-07T18:01:48Z","title":"On finite symmetries and their gauging in two dimensions","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1704.02330","snapshot_observed_at":"2026-08-06T21:09:15.823598Z","title":"On finite symmetries and their gauging in two dimen- sions","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":87,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.823598Z"},"links":{"cited_paper":"/paper/1704.02330","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:4cff685820c8bc0e816ec6c5a3872009c3e7a9a70e5269a364e28a6066f0b38d","observation_id":"0f7ae106-1d87-4129-91ff-384d731a424c","resolution":{"observed_at":"2026-08-06T21:09:15.823598Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1802.04445","last_updated":"2019-01-12T09:42:13Z","snapshot_observed_at":"2026-08-02T20:05:31.613691Z","submitted_at":"2018-02-13T02:51:54Z","title":"Topological Defect Lines and Renormalization Group Flows in Two Dimensions","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1802.04445","snapshot_observed_at":"2026-08-06T21:09:15.945043Z","title":"Topological Defect Lines and Renormalization Group Flows in Two Dimensions","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":88,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:15.945043Z"},"links":{"cited_paper":"/paper/1802.04445","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:76cf382d768377c14e5c2b2b64cb9bb23a00cbfe0e0d634777cff6afaf2a301c","observation_id":"f5a52020-e1c5-4d82-98d1-708c7822210b","resolution":{"observed_at":"2026-08-06T21:09:15.945043Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1912.02817","last_updated":"2019-12-05T18:59:58Z","snapshot_observed_at":"2026-07-06T08:42:27.599351Z","submitted_at":"2019-12-05T18:59:58Z","title":"Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1912.02817","snapshot_observed_at":"2026-08-06T21:09:16.043465Z","title":"Fusion category symmetry. Part I. Anomaly in-flow and gapped phases","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":89,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.043465Z"},"links":{"cited_paper":"/paper/1912.02817","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:71b4ef3e76ddbbf0c9b0f809d4c509af3f5a7bdf57f05067ed3b0e0643b83d51","observation_id":"14f60b5f-fede-4728-8cbd-ad0a6ca835a1","resolution":{"observed_at":"2026-08-06T21:09:16.043465Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:16.117568Z","title":"Alge- braic higher symmetry and categorical symmetry: A holographic and 90 entanglement view of symmetry","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":90,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.117568Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:2675b605205713a8e6d316cbf42a031f364ce7e8b06d79857b0b89cf22324d78","observation_id":"395d415b-34f9-4594-a3da-d6a9eaaae242","resolution":{"observed_at":"2026-08-06T21:09:16.117568Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"math/0202130","last_updated":"2006-07-10T23:55:23Z","snapshot_observed_at":"2026-07-07T05:28:59.986495Z","submitted_at":"2002-02-14T02:19:37Z","title":"Module categories over the Drinfeld double of a finite group","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"math/0202130","snapshot_observed_at":"2026-08-06T21:09:16.243023Z","title":"Module categories over the Drinfeld double of a finite group","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":91,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.243023Z"},"links":{"cited_paper":"/paper/math/0202130","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:eb042ffe7a661e355d74b6e1cce33a616378e53eb9b608a83ba666a5a0fa73c4","observation_id":"72adcff8-61cc-4229-8b6a-74eba669575a","resolution":{"observed_at":"2026-08-06T21:09:16.243023Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:16.349191Z","title":"Module categories, weak Hopf algebras and modular invariants","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":92,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.349191Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:ae1be8089794fbaee74fc24259e6c180c07ba988a125398bec99ca99fe44495c","observation_id":"d81ee159-21c8-499a-bd15-baa26df4b80d","resolution":{"observed_at":"2026-08-06T21:09:16.349191Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:16.479335Z","title":"TFT construction of RCFT cor- relators I: partition functions","venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":93,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.479335Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:663c492f4020bc7be5ef3f593b1168138ebb99eb6d93c890f31a856e07d3ae3a","observation_id":"31a5ba9a-954e-4fa8-bc00-5b5ed920970b","resolution":{"observed_at":"2026-08-06T21:09:16.479335Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"0901.4886","last_updated":"2009-01-30T13:59:28Z","snapshot_observed_at":"2026-07-06T01:53:51.165184Z","submitted_at":"2009-01-30T13:59:28Z","title":"On Frobenius algebras in rigid monoidal categories","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"0901.4886","snapshot_observed_at":"2026-08-06T21:09:16.604022Z","title":"On Frobenius algebras in rigid monoidal categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":94,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.604022Z"},"links":{"cited_paper":"/paper/0901.4886","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:1a4d7e1cf4395ddecb37b7fb0fded7d3f00492da9921a506bda785607ef0887d","observation_id":"b55d6475-4fbe-4673-95b2-4e180bd5ec3a","resolution":{"observed_at":"2026-08-06T21:09:16.604022Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1706.06904","last_updated":"2019-11-21T15:24:07Z","snapshot_observed_at":"2026-08-03T21:26:48.392743Z","submitted_at":"2017-06-21T13:47:33Z","title":"Semisimple and separable algebras in multi-fusion categories","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1706.06904","snapshot_observed_at":"2026-08-06T21:09:16.787460Z","title":"Semisimple and separable algebras in multi-fusion categories","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":95,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.787460Z"},"links":{"cited_paper":"/paper/1706.06904","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:1482e5090e80dc3d709a26de898d5ab9482be4c3cf8dcc49d94d2f107630f65b","observation_id":"288cd41c-b93d-431e-bd79-a88946c31bc0","resolution":{"observed_at":"2026-08-06T21:09:16.787460Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2112.09091","last_updated":"2023-05-11T11:07:24Z","snapshot_observed_at":"2026-07-06T12:19:40.584963Z","submitted_at":"2021-12-16T18:22:49Z","title":"Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2112.09091","snapshot_observed_at":"2026-08-06T21:09:16.913015Z","title":"Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Inter- twiners","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":96,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:16.913015Z"},"links":{"cited_paper":"/paper/2112.09091","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:772cc8d3b707e591c05c5c67ec83288876a43eccc9e739c95a61782390e473ce","observation_id":"af795c2f-c956-4756-aa2d-5c006808b91c","resolution":{"observed_at":"2026-08-06T21:09:16.913015Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2211.03777","last_updated":"2024-01-10T17:02:07Z","snapshot_observed_at":"2026-07-06T14:15:25.130495Z","submitted_at":"2022-11-07T18:54:57Z","title":"Dualities in one-dimensional quantum lattice models: topological sectors","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2211.03777","snapshot_observed_at":"2026-08-06T21:09:17.010990Z","title":"Dualities in One-Dimensional Quantum Lat- tice Models: Topological Sectors","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":97,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:17.010990Z"},"links":{"cited_paper":"/paper/2211.03777","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:e2b50c88ec5608b528bbb75b6fe806f1e6e1f3fa563862137818ffade40317ac","observation_id":"7b5acec9-c973-49d6-9404-ffb35890cb16","resolution":{"observed_at":"2026-08-06T21:09:17.010990Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2311.01439","last_updated":"2023-11-02T17:53:38Z","snapshot_observed_at":"2026-08-06T10:32:33.445013Z","submitted_at":"2023-11-02T17:53:38Z","title":"Low-depth unitary quantum circuits for dualities in one-dimensional quantum lattice models","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2311.01439","snapshot_observed_at":"2026-08-06T21:09:17.115000Z","title":"Low-Depth Unitary Quantum Circuits for Dualities in One-Dimensional Quantum Lattice Models","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":98,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:17.115000Z"},"links":{"cited_paper":"/paper/2311.01439","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:eb8935d093e3ec94e95cbe0c8ff36ec38fea1e93388eaec32ed1d4192ada68f3","observation_id":"b3aaad0f-df16-4d07-a03b-51e0996431ce","resolution":{"observed_at":"2026-08-06T21:09:17.115000Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2408.06074","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:20.437453Z","title":"Symmetry topological field theory and non-abelian Kramers- Wannier dualities of generalised Ising models","venue":null,"work_id":"d32efea0-49be-428d-94c1-7701f3c6c03e","year":null},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":99,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:17.196018Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:469ac2284e1f401f7247f39bc67c0468508437ebc5367796c0327bca2a379d41","observation_id":"58a6308a-3f2f-445e-8fff-0ddf939c4386","resolution":{"observed_at":"2026-08-06T21:09:20.512204Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T21:09:17.293258Z","title":"Traces on module categories over fu- sion categories","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":100,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:17.293258Z"},"links":{"citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:3aad54838d181d53adfa51972df87924c0710a42a405ec58c9c8363e7dbae364","observation_id":"2821a877-a811-4366-81f0-425598b5afee","resolution":{"observed_at":"2026-08-06T21:09:17.293258Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1404.7497","last_updated":"2014-09-15T11:32:41Z","snapshot_observed_at":"2026-07-06T03:42:24.656546Z","submitted_at":"2014-04-29T20:00:02Z","title":"Discrete torsion defects","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1404.7497","snapshot_observed_at":"2026-08-06T21:09:17.365266Z","title":"Discrete torsion defects","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders","version":1},"reference_index":101,"source":"pdf_text","source_observed_at":"2026-08-06T21:09:17.365266Z"},"links":{"cited_paper":"/paper/1404.7497","citing_paper":"/paper/2507.01142"},"observation_digest":"sha256:4d5bb49601e4b956e86c5347c1d0e586abbfb8e7d10831e290f7ec555e74e3a1","observation_id":"1e72a865-f9f2-4131-b162-5b2e2d260ee7","resolution":{"observed_at":"2026-08-06T21:09:17.365266Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2507.01142","last_updated":"2025-07-01T19:00:39Z","latest_version":1,"primary_category":"hep-th","snapshot_observed_at":"2026-08-07T10:37:49.533922Z","submitted_at":"2025-07-01T19:00:39Z","title":"Gauging Non-Invertible Symmetries in (2+1)d Topological Orders"},"reference_resolution":{"displayed":100,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":90,"verified_exact":10,"verified_fuzzy":0},"total_outbound_references":122},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"thesis":"As of 7 August 2026, this Paper Citation Record lists 100 of 122 outbound references and 9 inbound Pith citation observations for arXiv:2507.01142."}