{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:AZFXEZ3NTBU6ISCZIISH73S4GP","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":"ccb54872576c1b7781bb1f76062354ea553728972c81df6d6bf31c70eeb79781","cross_cats_sorted":["hep-th","math-ph","math.MP"],"license":"","primary_cat":"math.QA","submitted_at":"2006-09-20T18:06:36Z","title_canon_sha256":"b3b3e35d62c050dbf362df2afb73e18cd2a4336554cc22e0adc3532ec7dd3ac9"},"schema_version":"1.0","source":{"id":"math/0609570","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609570","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609570v2","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609570","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"pith_short_12","alias_value":"AZFXEZ3NTBU6","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"AZFXEZ3NTBU6ISCZ","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"AZFXEZ3N","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:b6379772edb2ba370c5e262bf2267c0fe19dfc2bd38ec271cef88d5700535469","target":"graph","created_at":"2026-05-18T03:06:04Z","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"},"paper":{"abstract_excerpt":"Let V^L and V^R be simple vertex operator algebras satisfying certain natural uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let F be a conformal full field algebra over the tensor product of V^L and V^R. We prove that the q_\\tau-\\bar{q_\\tau}-traces (natural traces involving q_\\tau=e^{2\\pi i\\tau} and \\bar{q_\\tau}=\\bar{e^{2\\pi i\\tau}}) of geometrically modified genus-zero correlation functions for F are convergent in suitable regions and can be extended to doubly periodic functions with periods 1 and \\tau. We obtain necessary and sufficient conditions for these func","authors_text":"Liang Kong, Yi-Zhi Huang","cross_cats":["hep-th","math-ph","math.MP"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2006-09-20T18:06:36Z","title":"Modular invariance for conformal full field algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609570","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:d4f6a1cfcd1352612babd41735d6fb725ada7f1f97a60b6cad3a946e53ffcf82","target":"record","created_at":"2026-05-18T03:06:04Z","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":"ccb54872576c1b7781bb1f76062354ea553728972c81df6d6bf31c70eeb79781","cross_cats_sorted":["hep-th","math-ph","math.MP"],"license":"","primary_cat":"math.QA","submitted_at":"2006-09-20T18:06:36Z","title_canon_sha256":"b3b3e35d62c050dbf362df2afb73e18cd2a4336554cc22e0adc3532ec7dd3ac9"},"schema_version":"1.0","source":{"id":"math/0609570","kind":"arxiv","version":2}},"canonical_sha256":"064b72676d9869e4485942247fee5c33c988ffac3fef049b27f624313294d293","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"064b72676d9869e4485942247fee5c33c988ffac3fef049b27f624313294d293","first_computed_at":"2026-05-18T03:06:04.087235Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:04.087235Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IMbUQfZx29gJT72bo6/3f0wavJTwKAGrHcdNxeayno9iMwu4HAm36Ujj6aSGopXlJava19U7Qine57TfGIpnDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:04.087742Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0609570","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4f6a1cfcd1352612babd41735d6fb725ada7f1f97a60b6cad3a946e53ffcf82","sha256:b6379772edb2ba370c5e262bf2267c0fe19dfc2bd38ec271cef88d5700535469"],"state_sha256":"1e7f72000aa3dffb055a18a4c2863f870810cefc959b31603b0212126829f864"}