{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:PL7CWBWVMZCI7JW3UGEMXHSKJE","short_pith_number":"pith:PL7CWBWV","canonical_record":{"source":{"id":"1012.2048","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-12-09T16:51:23Z","cross_cats_sorted":["hep-th"],"title_canon_sha256":"0abc7efda9816e5a12fdecb5fef29f57a487a9cfc0cc62580c8337230a12f2a7","abstract_canon_sha256":"1fce85eee136c10b4123af8ce8ce72a9ee871c48d493b1e90fbcbb475df4adad"},"schema_version":"1.0"},"canonical_sha256":"7afe2b06d566448fa6dba188cb9e4a491e1878967e75660df5cd8a8125ea45b8","source":{"kind":"arxiv","id":"1012.2048","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2048","created_at":"2026-05-18T02:42:54Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2048v2","created_at":"2026-05-18T02:42:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2048","created_at":"2026-05-18T02:42:54Z"},{"alias_kind":"pith_short_12","alias_value":"PL7CWBWVMZCI","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PL7CWBWVMZCI7JW3","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PL7CWBWV","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:PL7CWBWVMZCI7JW3UGEMXHSKJE","target":"record","payload":{"canonical_record":{"source":{"id":"1012.2048","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-12-09T16:51:23Z","cross_cats_sorted":["hep-th"],"title_canon_sha256":"0abc7efda9816e5a12fdecb5fef29f57a487a9cfc0cc62580c8337230a12f2a7","abstract_canon_sha256":"1fce85eee136c10b4123af8ce8ce72a9ee871c48d493b1e90fbcbb475df4adad"},"schema_version":"1.0"},"canonical_sha256":"7afe2b06d566448fa6dba188cb9e4a491e1878967e75660df5cd8a8125ea45b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:54.481048Z","signature_b64":"fdr8ZVtJW8eFqNcol1MN7SGM6zJt9VP/R+iTlCa5nlJW9EvmhJd0pzouYz4mzuEqxnH4YP1BpQr4gMGmNq5KCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7afe2b06d566448fa6dba188cb9e4a491e1878967e75660df5cd8a8125ea45b8","last_reissued_at":"2026-05-18T02:42:54.480595Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:54.480595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.2048","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":"2026-05-18T02:42:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z1ikYfVZaEAseOeAuYJ7tQzXMXw6RAUBJjXbcrrhwNogWZUAo9ekUaqJAZCGHw1NDb2mrsu9vCg12rbrNtyYAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T16:16:39.657367Z"},"content_sha256":"17eaf09b8fd3071f057f8084d755e147c4e61589e31fb0592583358b1096279e","schema_version":"1.0","event_id":"sha256:17eaf09b8fd3071f057f8084d755e147c4e61589e31fb0592583358b1096279e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:PL7CWBWVMZCI7JW3UGEMXHSKJE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multi-moment maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th"],"primary_cat":"math.DG","authors_text":"Andrew Swann, Thomas Bruun Madsen","submitted_at":"2010-12-09T16:51:23Z","abstract_excerpt":"We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2048","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":""},"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":"2026-05-18T02:42:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jqWgArf27HZ7eBHcpGE5GJOWt7kZcFt1SREo0+/WvCHFLdMvMHfxKUuhhZxYTj64Bjyx5IwVg+hNhpCZIkHuBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T16:16:39.657719Z"},"content_sha256":"53c66046c4afa26788f4f677579f9685597dbf57cc7ae5d3df9343e9dd46a154","schema_version":"1.0","event_id":"sha256:53c66046c4afa26788f4f677579f9685597dbf57cc7ae5d3df9343e9dd46a154"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE/bundle.json","state_url":"https://pith.science/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE/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":"2026-06-25T16:16:39Z","links":{"resolver":"https://pith.science/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE","bundle":"https://pith.science/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE/bundle.json","state":"https://pith.science/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PL7CWBWVMZCI7JW3UGEMXHSKJE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:PL7CWBWVMZCI7JW3UGEMXHSKJE","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":"1fce85eee136c10b4123af8ce8ce72a9ee871c48d493b1e90fbcbb475df4adad","cross_cats_sorted":["hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-12-09T16:51:23Z","title_canon_sha256":"0abc7efda9816e5a12fdecb5fef29f57a487a9cfc0cc62580c8337230a12f2a7"},"schema_version":"1.0","source":{"id":"1012.2048","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2048","created_at":"2026-05-18T02:42:54Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2048v2","created_at":"2026-05-18T02:42:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2048","created_at":"2026-05-18T02:42:54Z"},{"alias_kind":"pith_short_12","alias_value":"PL7CWBWVMZCI","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PL7CWBWVMZCI7JW3","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PL7CWBWV","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:53c66046c4afa26788f4f677579f9685597dbf57cc7ae5d3df9343e9dd46a154","target":"graph","created_at":"2026-05-18T02:42:54Z","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 introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by","authors_text":"Andrew Swann, Thomas Bruun Madsen","cross_cats":["hep-th"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-12-09T16:51:23Z","title":"Multi-moment maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2048","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":"sha256:17eaf09b8fd3071f057f8084d755e147c4e61589e31fb0592583358b1096279e","target":"record","created_at":"2026-05-18T02:42:54Z","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":"1fce85eee136c10b4123af8ce8ce72a9ee871c48d493b1e90fbcbb475df4adad","cross_cats_sorted":["hep-th"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-12-09T16:51:23Z","title_canon_sha256":"0abc7efda9816e5a12fdecb5fef29f57a487a9cfc0cc62580c8337230a12f2a7"},"schema_version":"1.0","source":{"id":"1012.2048","kind":"arxiv","version":2}},"canonical_sha256":"7afe2b06d566448fa6dba188cb9e4a491e1878967e75660df5cd8a8125ea45b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7afe2b06d566448fa6dba188cb9e4a491e1878967e75660df5cd8a8125ea45b8","first_computed_at":"2026-05-18T02:42:54.480595Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:54.480595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fdr8ZVtJW8eFqNcol1MN7SGM6zJt9VP/R+iTlCa5nlJW9EvmhJd0pzouYz4mzuEqxnH4YP1BpQr4gMGmNq5KCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:54.481048Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.2048","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17eaf09b8fd3071f057f8084d755e147c4e61589e31fb0592583358b1096279e","sha256:53c66046c4afa26788f4f677579f9685597dbf57cc7ae5d3df9343e9dd46a154"],"state_sha256":"e5d43a1d3f32ea4c3e1d831cb664fb3016c8279f3b7c82b492c73f53b7e2b101"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TGmXf15hQeA/nv3VPSdY/cgNgRdunxSlL7wzz4f4DqEpvKf68rcIm8WIMdLg32nrylUILu93RhoR+zwascKtAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T16:16:39.659667Z","bundle_sha256":"e9142b457c85dc9d8a6d83882247cf5da2023b26a63a77ec6b267d092d730813"}}