{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:FYAPEOXTVUM2N2BQ3ZMBNYZFLB","short_pith_number":"pith:FYAPEOXT","schema_version":"1.0","canonical_sha256":"2e00f23af3ad19a6e830de5816e325586de26928efbf45f1cafcfa69af2f4545","source":{"kind":"arxiv","id":"1502.01690","version":2},"attestation_state":"computed","paper":{"title":"Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.QA"],"primary_cat":"cond-mat.str-el","authors_text":"Hao Zheng, Liang Kong, Xiao-Gang Wen","submitted_at":"2015-02-05T19:42:17Z","abstract_excerpt":"In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of \"bulk\" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1502.01690","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.str-el","submitted_at":"2015-02-05T19:42:17Z","cross_cats_sorted":["math.CT","math.QA"],"title_canon_sha256":"475bb3898b6d75be0264e4efcd70eeadeac1f2412622823fe3dd3833743465d2","abstract_canon_sha256":"a0aba75789c31f98835a04001d5b05949b80c453aae47da389cd9f46016ff9b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:31.826048Z","signature_b64":"jTFRqYw3jSx+aPbVPx+KTbjBxNZm8yt2eVYJXvx5a3axmyGBAHoXvw2XXjjEgR5jWlTfVSR44J3OFm8fviyPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e00f23af3ad19a6e830de5816e325586de26928efbf45f1cafcfa69af2f4545","last_reissued_at":"2026-05-18T01:29:31.825482Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:31.825482Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.QA"],"primary_cat":"cond-mat.str-el","authors_text":"Hao Zheng, Liang Kong, Xiao-Gang Wen","submitted_at":"2015-02-05T19:42:17Z","abstract_excerpt":"In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of \"bulk\" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01690","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1502.01690","created_at":"2026-05-18T01:29:31.825563+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.01690v2","created_at":"2026-05-18T01:29:31.825563+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01690","created_at":"2026-05-18T01:29:31.825563+00:00"},{"alias_kind":"pith_short_12","alias_value":"FYAPEOXTVUM2","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"FYAPEOXTVUM2N2BQ","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"FYAPEOXT","created_at":"2026-05-18T12:29:22.688609+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":17,"internal_anchor_count":13,"sample":[{"citing_arxiv_id":"2607.07786","citing_title":"Chiral Tube Algebras I: Topological Defect Lines, Twisted Modules, and Finite Gauging","ref_index":22,"is_internal_anchor":true},{"citing_arxiv_id":"2607.08583","citing_title":"Holographic Theory of Mixed-Dimensional Statistics and Conservation-Encoding Hopping-Operator Algebras","ref_index":71,"is_internal_anchor":true},{"citing_arxiv_id":"2606.03582","citing_title":"Fracton Topological Holography","ref_index":46,"is_internal_anchor":true},{"citing_arxiv_id":"2605.27672","citing_title":"Spontaneous breaking of non-invertible symmetries and duality to beyond-Landau transitions","ref_index":49,"is_internal_anchor":true},{"citing_arxiv_id":"2605.07734","citing_title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","ref_index":78,"is_internal_anchor":true},{"citing_arxiv_id":"2605.28485","citing_title":"Hilbert Space and Defect Hilbert Spaces Associated with Categorical Symmetries","ref_index":9,"is_internal_anchor":true},{"citing_arxiv_id":"2504.11449","citing_title":"SymTFT construction of gapless exotic-foliated dual models","ref_index":11,"is_internal_anchor":true},{"citing_arxiv_id":"2510.00587","citing_title":"Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle","ref_index":31,"is_internal_anchor":true},{"citing_arxiv_id":"2605.07734","citing_title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","ref_index":85,"is_internal_anchor":true},{"citing_arxiv_id":"2506.23155","citing_title":"Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders","ref_index":27,"is_internal_anchor":true},{"citing_arxiv_id":"2511.11059","citing_title":"Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls","ref_index":123,"is_internal_anchor":true},{"citing_arxiv_id":"2305.18296","citing_title":"ICTP Lectures on (Non-)Invertible Generalized Symmetries","ref_index":111,"is_internal_anchor":true},{"citing_arxiv_id":"2602.03926","citing_title":"The Line, the Strip and the Duality Defect","ref_index":11,"is_internal_anchor":true},{"citing_arxiv_id":"2604.27363","citing_title":"Constructing Bulk Topological Orders via Layered Gauging","ref_index":11,"is_internal_anchor":false},{"citing_arxiv_id":"2604.25820","citing_title":"Candidate Gaugings of Categorical Continuous Symmetry","ref_index":14,"is_internal_anchor":false},{"citing_arxiv_id":"2604.25821","citing_title":"Categorical Symmetries via Operator Algebras","ref_index":2,"is_internal_anchor":false},{"citing_arxiv_id":"2605.07734","citing_title":"Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries","ref_index":85,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB","json":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB.json","graph_json":"https://pith.science/api/pith-number/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/graph.json","events_json":"https://pith.science/api/pith-number/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/events.json","paper":"https://pith.science/paper/FYAPEOXT"},"agent_actions":{"view_html":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB","download_json":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB.json","view_paper":"https://pith.science/paper/FYAPEOXT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.01690&json=true","fetch_graph":"https://pith.science/api/pith-number/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/graph.json","fetch_events":"https://pith.science/api/pith-number/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/action/storage_attestation","attest_author":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/action/author_attestation","sign_citation":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/action/citation_signature","submit_replication":"https://pith.science/pith/FYAPEOXTVUM2N2BQ3ZMBNYZFLB/action/replication_record"}},"created_at":"2026-05-18T01:29:31.825563+00:00","updated_at":"2026-05-18T01:29:31.825563+00:00"}