{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:C3MTQAUFOFQ5AKNPHTHEL7ITMW","short_pith_number":"pith:C3MTQAUF","schema_version":"1.0","canonical_sha256":"16d93802857161d029af3cce45fd1365aff5491ad131a18b3d35a6a8e7c5ee8b","source":{"kind":"arxiv","id":"1702.00673","version":3},"attestation_state":"computed","paper":{"title":"Boundary-bulk relation in topological orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"cond-mat.str-el","authors_text":"Hao Zheng, Liang Kong, Xiao-Gang Wen","submitted_at":"2017-02-02T13:35:46Z","abstract_excerpt":"In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the \"bulk\" for a given gapped boundary phase. In this paper, we show that the $n+$1D \"bulk\" phase is given by the \"center\" of the $n$D boundary phase. In other words, the geometric notion of the \"bulk\" corresponds precisely to the algebraic notion of the \""},"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":"1702.00673","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.str-el","submitted_at":"2017-02-02T13:35:46Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"bda5380c414cbbc8521e6d8fb9b6bf77ca5879e595764af809ad8459cddd90ae","abstract_canon_sha256":"51995c392b82baedc050eaf57bdf1caa7ac25c459018f76ee369f0cd667cd7b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:36.009468Z","signature_b64":"cwenT+oTvfvXP02XDnsr673gPWspUV2NUU/tqbiX5S4UsMXC+r4QTS4VSe2WK1ldYPEIoEJqlCkYCjJQJFkBDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16d93802857161d029af3cce45fd1365aff5491ad131a18b3d35a6a8e7c5ee8b","last_reissued_at":"2026-05-18T00:37:36.008783Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:36.008783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundary-bulk relation in topological orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"cond-mat.str-el","authors_text":"Hao Zheng, Liang Kong, Xiao-Gang Wen","submitted_at":"2017-02-02T13:35:46Z","abstract_excerpt":"In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the \"bulk\" for a given gapped boundary phase. In this paper, we show that the $n+$1D \"bulk\" phase is given by the \"center\" of the $n$D boundary phase. In other words, the geometric notion of the \"bulk\" corresponds precisely to the algebraic notion of the \""},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.00673","kind":"arxiv","version":3},"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":"1702.00673","created_at":"2026-05-18T00:37:36.008891+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.00673v3","created_at":"2026-05-18T00:37:36.008891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.00673","created_at":"2026-05-18T00:37:36.008891+00:00"},{"alias_kind":"pith_short_12","alias_value":"C3MTQAUFOFQ5","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"C3MTQAUFOFQ5AKNP","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"C3MTQAUF","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":6,"internal_anchor_count":4,"sample":[{"citing_arxiv_id":"2209.07471","citing_title":"Topological symmetry in quantum field theory","ref_index":66,"is_internal_anchor":true},{"citing_arxiv_id":"2508.08639","citing_title":"Extending fusion rules with finite subgroups: A general construction of $Z_{N}$ extended conformal field theories and their orbifoldings","ref_index":148,"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":29,"is_internal_anchor":true},{"citing_arxiv_id":"2603.12323","citing_title":"On the SymTFTs of Finite Non-Abelian Symmetries","ref_index":13,"is_internal_anchor":true},{"citing_arxiv_id":"2604.25820","citing_title":"Candidate Gaugings of Categorical Continuous Symmetry","ref_index":15,"is_internal_anchor":false},{"citing_arxiv_id":"2604.14275","citing_title":"Generalized Complexity Distances and Non-Invertible Symmetries","ref_index":41,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW","json":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW.json","graph_json":"https://pith.science/api/pith-number/C3MTQAUFOFQ5AKNPHTHEL7ITMW/graph.json","events_json":"https://pith.science/api/pith-number/C3MTQAUFOFQ5AKNPHTHEL7ITMW/events.json","paper":"https://pith.science/paper/C3MTQAUF"},"agent_actions":{"view_html":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW","download_json":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW.json","view_paper":"https://pith.science/paper/C3MTQAUF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.00673&json=true","fetch_graph":"https://pith.science/api/pith-number/C3MTQAUFOFQ5AKNPHTHEL7ITMW/graph.json","fetch_events":"https://pith.science/api/pith-number/C3MTQAUFOFQ5AKNPHTHEL7ITMW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW/action/storage_attestation","attest_author":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW/action/author_attestation","sign_citation":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW/action/citation_signature","submit_replication":"https://pith.science/pith/C3MTQAUFOFQ5AKNPHTHEL7ITMW/action/replication_record"}},"created_at":"2026-05-18T00:37:36.008891+00:00","updated_at":"2026-05-18T00:37:36.008891+00:00"}