{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:IJO26EPKA74LPZKMGEDJTZCSXS","short_pith_number":"pith:IJO26EPK","canonical_record":{"source":{"id":"0909.5582","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-30T11:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"572bd9496948d4b0308643f9c73f0a9b6a4782032f5df1cb0aac8b61302179e2","abstract_canon_sha256":"94dd2e650f3fc69884a1da7c273389f5848ce0f77ae4a2a19afb649570a940f4"},"schema_version":"1.0"},"canonical_sha256":"425daf11ea07f8b7e54c310699e452bcab281376565765a4106af9e1e4477107","source":{"kind":"arxiv","id":"0909.5582","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.5582","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"0909.5582v3","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.5582","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"IJO26EPKA74L","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"IJO26EPKA74LPZKM","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"IJO26EPK","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:IJO26EPKA74LPZKMGEDJTZCSXS","target":"record","payload":{"canonical_record":{"source":{"id":"0909.5582","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-30T11:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"572bd9496948d4b0308643f9c73f0a9b6a4782032f5df1cb0aac8b61302179e2","abstract_canon_sha256":"94dd2e650f3fc69884a1da7c273389f5848ce0f77ae4a2a19afb649570a940f4"},"schema_version":"1.0"},"canonical_sha256":"425daf11ea07f8b7e54c310699e452bcab281376565765a4106af9e1e4477107","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:23.106701Z","signature_b64":"gjlSKG3KMaTAmvdEfyrZorZlyhRo6YygEYurHlz+CXkTVb8uNq7dHF+msVLz/vwQAjhzn064KzzJSgE0QyLvBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"425daf11ea07f8b7e54c310699e452bcab281376565765a4106af9e1e4477107","last_reissued_at":"2026-05-18T04:42:23.105996Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:23.105996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.5582","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:42:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QfuscOn8rqtDoKsgbw5xJtHtZdCDVHVeYBNbxWn5kcJD/OjpGJx2Py6RD8HZ4had5od8vDg+xycYu/RNwDCWBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:32:28.965997Z"},"content_sha256":"bb1257899a6a33e92a1e0625eb07fa7d1662447912b29ab26288beca8bcd9464","schema_version":"1.0","event_id":"sha256:bb1257899a6a33e92a1e0625eb07fa7d1662447912b29ab26288beca8bcd9464"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:IJO26EPKA74LPZKMGEDJTZCSXS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Filtrations, factorizations and explicit formulae for harmonic maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"John C. Wood (Leeds), Martin Svensson (Odense)","submitted_at":"2009-09-30T11:50:36Z","abstract_excerpt":"We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact Lie groups and their inner symmetric spaces - the nonlinear sigma-model of particle physics. Our methods also give an explicit Iwasawa decomposition of the algebraic loop group."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.5582","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:42:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"78UafTfqkJjROnDK1e/O3UsTcDpDjvpIouVeMZ/iL2BFwnEYKnWRc9lFznxWMsMQhJBWn7GEB2TBdUFrxjfBBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:32:28.966823Z"},"content_sha256":"1c44bcb3f21b2932c9fdf13d2dd8839caf5a8cf04c065b5b8e51944652a5a9e4","schema_version":"1.0","event_id":"sha256:1c44bcb3f21b2932c9fdf13d2dd8839caf5a8cf04c065b5b8e51944652a5a9e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJO26EPKA74LPZKMGEDJTZCSXS/bundle.json","state_url":"https://pith.science/pith/IJO26EPKA74LPZKMGEDJTZCSXS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJO26EPKA74LPZKMGEDJTZCSXS/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":"2026-06-11T23:32:28Z","links":{"resolver":"https://pith.science/pith/IJO26EPKA74LPZKMGEDJTZCSXS","bundle":"https://pith.science/pith/IJO26EPKA74LPZKMGEDJTZCSXS/bundle.json","state":"https://pith.science/pith/IJO26EPKA74LPZKMGEDJTZCSXS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJO26EPKA74LPZKMGEDJTZCSXS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:IJO26EPKA74LPZKMGEDJTZCSXS","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":"94dd2e650f3fc69884a1da7c273389f5848ce0f77ae4a2a19afb649570a940f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-30T11:50:36Z","title_canon_sha256":"572bd9496948d4b0308643f9c73f0a9b6a4782032f5df1cb0aac8b61302179e2"},"schema_version":"1.0","source":{"id":"0909.5582","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.5582","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"0909.5582v3","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.5582","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"IJO26EPKA74L","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"IJO26EPKA74LPZKM","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"IJO26EPK","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:1c44bcb3f21b2932c9fdf13d2dd8839caf5a8cf04c065b5b8e51944652a5a9e4","target":"graph","created_at":"2026-05-18T04:42:23Z","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":"We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact Lie groups and their inner symmetric spaces - the nonlinear sigma-model of particle physics. Our methods also give an explicit Iwasawa decomposition of the algebraic loop group.","authors_text":"John C. Wood (Leeds), Martin Svensson (Odense)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-30T11:50:36Z","title":"Filtrations, factorizations and explicit formulae for harmonic maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.5582","kind":"arxiv","version":3},"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:bb1257899a6a33e92a1e0625eb07fa7d1662447912b29ab26288beca8bcd9464","target":"record","created_at":"2026-05-18T04:42:23Z","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":"94dd2e650f3fc69884a1da7c273389f5848ce0f77ae4a2a19afb649570a940f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-09-30T11:50:36Z","title_canon_sha256":"572bd9496948d4b0308643f9c73f0a9b6a4782032f5df1cb0aac8b61302179e2"},"schema_version":"1.0","source":{"id":"0909.5582","kind":"arxiv","version":3}},"canonical_sha256":"425daf11ea07f8b7e54c310699e452bcab281376565765a4106af9e1e4477107","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"425daf11ea07f8b7e54c310699e452bcab281376565765a4106af9e1e4477107","first_computed_at":"2026-05-18T04:42:23.105996Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:23.105996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gjlSKG3KMaTAmvdEfyrZorZlyhRo6YygEYurHlz+CXkTVb8uNq7dHF+msVLz/vwQAjhzn064KzzJSgE0QyLvBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:23.106701Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.5582","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb1257899a6a33e92a1e0625eb07fa7d1662447912b29ab26288beca8bcd9464","sha256:1c44bcb3f21b2932c9fdf13d2dd8839caf5a8cf04c065b5b8e51944652a5a9e4"],"state_sha256":"84260acefa09334eff44289d910071943d4f784bab8307cab89297c82d20efc9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iMJThxrS0mk+M2Ow4VkTSukSEkN5IMo5r4SgWF2sxhXrnIo4qL/9KSVoZ9x43znv3hvliFPid3jqNEcxushoAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:32:28.970514Z","bundle_sha256":"d74b3f037681075c57d49c5ddc3d5dbeffd7ad3fcd566025afd329a645d5d4a3"}}