{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:F2V6NSS4D6O6UHCZKW6JS3U4YS","short_pith_number":"pith:F2V6NSS4","canonical_record":{"source":{"id":"2107.05718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2021-07-12T20:14:42Z","cross_cats_sorted":["hep-th","math-ph","math.CT","math.MP","math.RT"],"title_canon_sha256":"5acfef7bd4c8aedafa958f717563949680b18ca10add13aea7935465f6e62d95","abstract_canon_sha256":"94ab68d7b357889acce012fbfdabe5c6f35cfebe8f9bfa9b1ef5caf9329f2d78"},"schema_version":"1.0"},"canonical_sha256":"2eabe6ca5c1f9dea1c5955bc996e9cc49227d25273d7710359d4969a63282af5","source":{"kind":"arxiv","id":"2107.05718","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.05718","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"arxiv_version","alias_value":"2107.05718v2","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.05718","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"pith_short_12","alias_value":"F2V6NSS4D6O6","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"pith_short_16","alias_value":"F2V6NSS4D6O6UHCZ","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"pith_short_8","alias_value":"F2V6NSS4","created_at":"2026-07-05T03:12:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:F2V6NSS4D6O6UHCZKW6JS3U4YS","target":"record","payload":{"canonical_record":{"source":{"id":"2107.05718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2021-07-12T20:14:42Z","cross_cats_sorted":["hep-th","math-ph","math.CT","math.MP","math.RT"],"title_canon_sha256":"5acfef7bd4c8aedafa958f717563949680b18ca10add13aea7935465f6e62d95","abstract_canon_sha256":"94ab68d7b357889acce012fbfdabe5c6f35cfebe8f9bfa9b1ef5caf9329f2d78"},"schema_version":"1.0"},"canonical_sha256":"2eabe6ca5c1f9dea1c5955bc996e9cc49227d25273d7710359d4969a63282af5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:12:10.516965Z","signature_b64":"PxiaZJdTw3OHAHJLOWSNIixbpd9u1imN3alDigG7IHhvr9fnQtI79FTBUDnnMqUgPA/iWxshqnU9OzHbCN4BAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2eabe6ca5c1f9dea1c5955bc996e9cc49227d25273d7710359d4969a63282af5","last_reissued_at":"2026-07-05T03:12:10.516607Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:12:10.516607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2107.05718","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:12:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gX0rphl5EUb8hR+jSQ80Va3gXpM1pAzg9O87YDHhG5uorNM232A9SxcOxLSSYVzdpj8aN93skNw9woMx27H5AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T11:27:03.697384Z"},"content_sha256":"50b14ccd4ddcd8ae73943a48967eb0dd2c668e3683b87a57a19d758d468042bc","schema_version":"1.0","event_id":"sha256:50b14ccd4ddcd8ae73943a48967eb0dd2c668e3683b87a57a19d758d468042bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:F2V6NSS4D6O6UHCZKW6JS3U4YS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.CT","math.MP","math.RT"],"primary_cat":"math.QA","authors_text":"Christoph Schweigert, Robert Allen, Simon Lentner, Simon Wood","submitted_at":"2021-07-12T20:14:42Z","abstract_excerpt":"Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.05718","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2107.05718/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:12:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"evc0O44qcv+qL0Z8aauUeaUOFygqxVwL2ZZBkQn+yK4sbghy0M7UQRl58wk5NKJwi2dAnKHW/KVhuBuN7AGLAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T11:27:03.697987Z"},"content_sha256":"ad813425ba8334d58e399ad4bab2413784e58cbff2289d83fd3ded600d3fa8cc","schema_version":"1.0","event_id":"sha256:ad813425ba8334d58e399ad4bab2413784e58cbff2289d83fd3ded600d3fa8cc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS/bundle.json","state_url":"https://pith.science/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-06T11:27:03Z","links":{"resolver":"https://pith.science/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS","bundle":"https://pith.science/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS/bundle.json","state":"https://pith.science/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F2V6NSS4D6O6UHCZKW6JS3U4YS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:F2V6NSS4D6O6UHCZKW6JS3U4YS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"94ab68d7b357889acce012fbfdabe5c6f35cfebe8f9bfa9b1ef5caf9329f2d78","cross_cats_sorted":["hep-th","math-ph","math.CT","math.MP","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2021-07-12T20:14:42Z","title_canon_sha256":"5acfef7bd4c8aedafa958f717563949680b18ca10add13aea7935465f6e62d95"},"schema_version":"1.0","source":{"id":"2107.05718","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.05718","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"arxiv_version","alias_value":"2107.05718v2","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.05718","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"pith_short_12","alias_value":"F2V6NSS4D6O6","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"pith_short_16","alias_value":"F2V6NSS4D6O6UHCZ","created_at":"2026-07-05T03:12:10Z"},{"alias_kind":"pith_short_8","alias_value":"F2V6NSS4","created_at":"2026-07-05T03:12:10Z"}],"graph_snapshots":[{"event_id":"sha256:ad813425ba8334d58e399ad4bab2413784e58cbff2289d83fd3ded600d3fa8cc","target":"graph","created_at":"2026-07-05T03:12:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2107.05718/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fac","authors_text":"Christoph Schweigert, Robert Allen, Simon Lentner, Simon Wood","cross_cats":["hep-th","math-ph","math.CT","math.MP","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2021-07-12T20:14:42Z","title":"Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.05718","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:50b14ccd4ddcd8ae73943a48967eb0dd2c668e3683b87a57a19d758d468042bc","target":"record","created_at":"2026-07-05T03:12:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"94ab68d7b357889acce012fbfdabe5c6f35cfebe8f9bfa9b1ef5caf9329f2d78","cross_cats_sorted":["hep-th","math-ph","math.CT","math.MP","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2021-07-12T20:14:42Z","title_canon_sha256":"5acfef7bd4c8aedafa958f717563949680b18ca10add13aea7935465f6e62d95"},"schema_version":"1.0","source":{"id":"2107.05718","kind":"arxiv","version":2}},"canonical_sha256":"2eabe6ca5c1f9dea1c5955bc996e9cc49227d25273d7710359d4969a63282af5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2eabe6ca5c1f9dea1c5955bc996e9cc49227d25273d7710359d4969a63282af5","first_computed_at":"2026-07-05T03:12:10.516607Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:12:10.516607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PxiaZJdTw3OHAHJLOWSNIixbpd9u1imN3alDigG7IHhvr9fnQtI79FTBUDnnMqUgPA/iWxshqnU9OzHbCN4BAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:12:10.516965Z","signed_message":"canonical_sha256_bytes"},"source_id":"2107.05718","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50b14ccd4ddcd8ae73943a48967eb0dd2c668e3683b87a57a19d758d468042bc","sha256:ad813425ba8334d58e399ad4bab2413784e58cbff2289d83fd3ded600d3fa8cc"],"state_sha256":"79aa684c7be948ceef8e9857f8e3e98c05afca483619c9c21df262efcd664bae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UWQ7ohYxvjp3xMPhR51QhWFciSbagufUPobJ5TyA6VT5DKi7HrG9z9kdR9ZBGKgWoH4tXt6Ysit77syfS9Z7BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T11:27:03.703259Z","bundle_sha256":"633950f9158efcd55dd71b60defe812c26548dcfa4e1ae9a3c267cd2c3724a04"}}