{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2002:TDPU5YM7YXMSCM35HOWQIEGXVE","short_pith_number":"pith:TDPU5YM7","canonical_record":{"source":{"id":"math/0210017","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2002-10-02T01:16:17Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"3624e4c7edb2b3ea26ab388e85e19188db56117516a4e0821741f8e2a2afcf02","abstract_canon_sha256":"86161d18606e605ffd1df9491597948e8a181fe5ff1b72e30c2c679de8d82577"},"schema_version":"1.0"},"canonical_sha256":"98df4ee19fc5d921337d3bad0410d7a93773ea78769bc7ca3e6b8c8cebc7a834","source":{"kind":"arxiv","id":"math/0210017","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0210017","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"arxiv_version","alias_value":"math/0210017v2","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0210017","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"pith_short_12","alias_value":"TDPU5YM7YXMS","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"pith_short_16","alias_value":"TDPU5YM7YXMSCM35","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"pith_short_8","alias_value":"TDPU5YM7","created_at":"2026-07-05T04:03:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2002:TDPU5YM7YXMSCM35HOWQIEGXVE","target":"record","payload":{"canonical_record":{"source":{"id":"math/0210017","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2002-10-02T01:16:17Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"3624e4c7edb2b3ea26ab388e85e19188db56117516a4e0821741f8e2a2afcf02","abstract_canon_sha256":"86161d18606e605ffd1df9491597948e8a181fe5ff1b72e30c2c679de8d82577"},"schema_version":"1.0"},"canonical_sha256":"98df4ee19fc5d921337d3bad0410d7a93773ea78769bc7ca3e6b8c8cebc7a834","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:03:07.928790Z","signature_b64":"UaIAtuZXT5QnZEcObf4vZNjc3QQN39QfZ775FYP8Wzm5CVniEJmBrNyEBpmfiiEaDn4Zqz+jHIbq21YjfJSZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98df4ee19fc5d921337d3bad0410d7a93773ea78769bc7ca3e6b8c8cebc7a834","last_reissued_at":"2026-07-05T04:03:07.928313Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:03:07.928313Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0210017","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":"2026-07-05T04:03:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NGZg1QWWsnRFo4GkSKbBMbtbmGRv0fTdVHXIy0psHl651uLq3ZppyGXUEtIxzbo/x2uzstDLEDrm3r0QF8MZDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T09:00:46.227169Z"},"content_sha256":"2a6272d3e9e03e3f12da9479cceeb510eed55b53516ba8594eba8858f266d2d5","schema_version":"1.0","event_id":"sha256:2a6272d3e9e03e3f12da9479cceeb510eed55b53516ba8594eba8858f266d2d5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2002:TDPU5YM7YXMSCM35HOWQIEGXVE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Holomorphic line bundles on the loop space of the Riemann sphere","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Ning Zhang","submitted_at":"2002-10-02T01:16:17Z","abstract_excerpt":"The loop space $L\\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\\mathbb{C})$ acts on $L\\mathbb{P}_1$ . We prove that the group of $LPGL(2,\\mathbb{C})$ invariant holomorphic line bundles on $L\\mathbb{P}_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0210017","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/math/0210017/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":"2026-07-05T04:03:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sOT84yEsvBeTZRZEOr1TDady3HFz55du09qDExzmFOrgSveXK2v+ZkqnGyln9wj5KUCKd9gpFYUPnxU3jG7jDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T09:00:46.227537Z"},"content_sha256":"91ad26f562a9e3d9f6d3750891e2bc4e1ad36c0c71f4f7e78a252cf4c4713335","schema_version":"1.0","event_id":"sha256:91ad26f562a9e3d9f6d3750891e2bc4e1ad36c0c71f4f7e78a252cf4c4713335"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TDPU5YM7YXMSCM35HOWQIEGXVE/bundle.json","state_url":"https://pith.science/pith/TDPU5YM7YXMSCM35HOWQIEGXVE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TDPU5YM7YXMSCM35HOWQIEGXVE/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":"2026-07-07T09:00:46Z","links":{"resolver":"https://pith.science/pith/TDPU5YM7YXMSCM35HOWQIEGXVE","bundle":"https://pith.science/pith/TDPU5YM7YXMSCM35HOWQIEGXVE/bundle.json","state":"https://pith.science/pith/TDPU5YM7YXMSCM35HOWQIEGXVE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TDPU5YM7YXMSCM35HOWQIEGXVE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2002:TDPU5YM7YXMSCM35HOWQIEGXVE","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":"86161d18606e605ffd1df9491597948e8a181fe5ff1b72e30c2c679de8d82577","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2002-10-02T01:16:17Z","title_canon_sha256":"3624e4c7edb2b3ea26ab388e85e19188db56117516a4e0821741f8e2a2afcf02"},"schema_version":"1.0","source":{"id":"math/0210017","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0210017","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"arxiv_version","alias_value":"math/0210017v2","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0210017","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"pith_short_12","alias_value":"TDPU5YM7YXMS","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"pith_short_16","alias_value":"TDPU5YM7YXMSCM35","created_at":"2026-07-05T04:03:07Z"},{"alias_kind":"pith_short_8","alias_value":"TDPU5YM7","created_at":"2026-07-05T04:03:07Z"}],"graph_snapshots":[{"event_id":"sha256:91ad26f562a9e3d9f6d3750891e2bc4e1ad36c0c71f4f7e78a252cf4c4713335","target":"graph","created_at":"2026-07-05T04:03:07Z","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/math/0210017/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The loop space $L\\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\\mathbb{C})$ acts on $L\\mathbb{P}_1$ . We prove that the group of $LPGL(2,\\mathbb{C})$ invariant holomorphic line bundles on $L\\mathbb{P}_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle.","authors_text":"Ning Zhang","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2002-10-02T01:16:17Z","title":"Holomorphic line bundles on the loop space of the Riemann sphere"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0210017","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":"sha256:2a6272d3e9e03e3f12da9479cceeb510eed55b53516ba8594eba8858f266d2d5","target":"record","created_at":"2026-07-05T04:03:07Z","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":"86161d18606e605ffd1df9491597948e8a181fe5ff1b72e30c2c679de8d82577","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2002-10-02T01:16:17Z","title_canon_sha256":"3624e4c7edb2b3ea26ab388e85e19188db56117516a4e0821741f8e2a2afcf02"},"schema_version":"1.0","source":{"id":"math/0210017","kind":"arxiv","version":2}},"canonical_sha256":"98df4ee19fc5d921337d3bad0410d7a93773ea78769bc7ca3e6b8c8cebc7a834","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98df4ee19fc5d921337d3bad0410d7a93773ea78769bc7ca3e6b8c8cebc7a834","first_computed_at":"2026-07-05T04:03:07.928313Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:03:07.928313Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UaIAtuZXT5QnZEcObf4vZNjc3QQN39QfZ775FYP8Wzm5CVniEJmBrNyEBpmfiiEaDn4Zqz+jHIbq21YjfJSZBw==","signature_status":"signed_v1","signed_at":"2026-07-05T04:03:07.928790Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0210017","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a6272d3e9e03e3f12da9479cceeb510eed55b53516ba8594eba8858f266d2d5","sha256:91ad26f562a9e3d9f6d3750891e2bc4e1ad36c0c71f4f7e78a252cf4c4713335"],"state_sha256":"824eeeb6c5f55a33adaa0a507a23ece4ae2cce2f409758a1efebc30c10264897"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BJYPwofbCWuuIn77HFrtJ04MZTSa1H9ey5asOMHaCspulgqIvTniM3Q446RKwTDpYpbuExXcQThCbuuaeJCWDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T09:00:46.229394Z","bundle_sha256":"ad199f95503673df697d9c14390b7dc355fbb36a837eb71cd628291cd6728fde"}}