{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:NPS4W5DZFW4J3KROTCSZDRZQZN","short_pith_number":"pith:NPS4W5DZ","canonical_record":{"source":{"id":"1209.6119","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-09-27T03:37:57Z","cross_cats_sorted":["math.AG","math.DG"],"title_canon_sha256":"1fc1f04a211631e705ec8a572ef8e47fabd473733b3d9c9ba1aabe27e98d2c29","abstract_canon_sha256":"d9176923d2f4462bd18f00aa2afbb11ec1cdab012e301b1d0d7643aa0e6b5e7e"},"schema_version":"1.0"},"canonical_sha256":"6be5cb74792db89daa2e98a591c730cb4cf55dcf9d74433a3fec1242e668cb7e","source":{"kind":"arxiv","id":"1209.6119","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.6119","created_at":"2026-05-18T00:43:03Z"},{"alias_kind":"arxiv_version","alias_value":"1209.6119v4","created_at":"2026-05-18T00:43:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.6119","created_at":"2026-05-18T00:43:03Z"},{"alias_kind":"pith_short_12","alias_value":"NPS4W5DZFW4J","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NPS4W5DZFW4J3KRO","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NPS4W5DZ","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:NPS4W5DZFW4J3KROTCSZDRZQZN","target":"record","payload":{"canonical_record":{"source":{"id":"1209.6119","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-09-27T03:37:57Z","cross_cats_sorted":["math.AG","math.DG"],"title_canon_sha256":"1fc1f04a211631e705ec8a572ef8e47fabd473733b3d9c9ba1aabe27e98d2c29","abstract_canon_sha256":"d9176923d2f4462bd18f00aa2afbb11ec1cdab012e301b1d0d7643aa0e6b5e7e"},"schema_version":"1.0"},"canonical_sha256":"6be5cb74792db89daa2e98a591c730cb4cf55dcf9d74433a3fec1242e668cb7e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:03.454716Z","signature_b64":"8mO3qVdWO2RSvj6fvbuA/18Oiw7Do3q9/Xsn5Rwy4gfx+8fxrYPV5aZORrDXEQpixwI6aSPjmT0xHi1hoSN5DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6be5cb74792db89daa2e98a591c730cb4cf55dcf9d74433a3fec1242e668cb7e","last_reissued_at":"2026-05-18T00:43:03.454011Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:03.454011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.6119","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"33edzPwKr3zH6wgjvd1gOc2m25wC7T6/cnFp/JX9uRYnqpuX8NHm1gsh7upjfPdkditC/DKewSZD/IK1B8J0Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T16:25:28.784382Z"},"content_sha256":"56d1780f4e2251a9e7d32249f3b457d63f8a87926146467ecc5bada7b03c9808","schema_version":"1.0","event_id":"sha256:56d1780f4e2251a9e7d32249f3b457d63f8a87926146467ecc5bada7b03c9808"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:NPS4W5DZFW4J3KROTCSZDRZQZN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.DG"],"primary_cat":"math.SG","authors_text":"Hsian-Hua Tseng, Kwokwai Chan, Naichung Conan Leung, Siu-Cheong Lau","submitted_at":"2012-09-27T03:37:57Z","abstract_excerpt":"Let $X$ be a compact toric K\\\"ahler manifold with $-K_X$ nef. Let $L\\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6119","kind":"arxiv","version":4},"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":"2026-05-18T00:43:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hExgS8960Ru7KSX2y8lFCs5nbt2KhQExVUXDzcxGszcjjJM6x7UkCgmda4PolHepbZWSBm3MkVGubx3gnilUCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T16:25:28.784739Z"},"content_sha256":"e4248633c5d008c02d04b848c2daf9717137382d8e327e824b00fe2136abd51a","schema_version":"1.0","event_id":"sha256:e4248633c5d008c02d04b848c2daf9717137382d8e327e824b00fe2136abd51a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NPS4W5DZFW4J3KROTCSZDRZQZN/bundle.json","state_url":"https://pith.science/pith/NPS4W5DZFW4J3KROTCSZDRZQZN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NPS4W5DZFW4J3KROTCSZDRZQZN/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":"2026-06-01T16:25:28Z","links":{"resolver":"https://pith.science/pith/NPS4W5DZFW4J3KROTCSZDRZQZN","bundle":"https://pith.science/pith/NPS4W5DZFW4J3KROTCSZDRZQZN/bundle.json","state":"https://pith.science/pith/NPS4W5DZFW4J3KROTCSZDRZQZN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NPS4W5DZFW4J3KROTCSZDRZQZN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NPS4W5DZFW4J3KROTCSZDRZQZN","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":"d9176923d2f4462bd18f00aa2afbb11ec1cdab012e301b1d0d7643aa0e6b5e7e","cross_cats_sorted":["math.AG","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-09-27T03:37:57Z","title_canon_sha256":"1fc1f04a211631e705ec8a572ef8e47fabd473733b3d9c9ba1aabe27e98d2c29"},"schema_version":"1.0","source":{"id":"1209.6119","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.6119","created_at":"2026-05-18T00:43:03Z"},{"alias_kind":"arxiv_version","alias_value":"1209.6119v4","created_at":"2026-05-18T00:43:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.6119","created_at":"2026-05-18T00:43:03Z"},{"alias_kind":"pith_short_12","alias_value":"NPS4W5DZFW4J","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NPS4W5DZFW4J3KRO","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NPS4W5DZ","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:e4248633c5d008c02d04b848c2daf9717137382d8e327e824b00fe2136abd51a","target":"graph","created_at":"2026-05-18T00:43:03Z","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 $X$ be a compact toric K\\\"ahler manifold with $-K_X$ nef. Let $L\\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields ","authors_text":"Hsian-Hua Tseng, Kwokwai Chan, Naichung Conan Leung, Siu-Cheong Lau","cross_cats":["math.AG","math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-09-27T03:37:57Z","title":"Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6119","kind":"arxiv","version":4},"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":"sha256:56d1780f4e2251a9e7d32249f3b457d63f8a87926146467ecc5bada7b03c9808","target":"record","created_at":"2026-05-18T00:43:03Z","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":"d9176923d2f4462bd18f00aa2afbb11ec1cdab012e301b1d0d7643aa0e6b5e7e","cross_cats_sorted":["math.AG","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-09-27T03:37:57Z","title_canon_sha256":"1fc1f04a211631e705ec8a572ef8e47fabd473733b3d9c9ba1aabe27e98d2c29"},"schema_version":"1.0","source":{"id":"1209.6119","kind":"arxiv","version":4}},"canonical_sha256":"6be5cb74792db89daa2e98a591c730cb4cf55dcf9d74433a3fec1242e668cb7e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6be5cb74792db89daa2e98a591c730cb4cf55dcf9d74433a3fec1242e668cb7e","first_computed_at":"2026-05-18T00:43:03.454011Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:03.454011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8mO3qVdWO2RSvj6fvbuA/18Oiw7Do3q9/Xsn5Rwy4gfx+8fxrYPV5aZORrDXEQpixwI6aSPjmT0xHi1hoSN5DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:03.454716Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.6119","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:56d1780f4e2251a9e7d32249f3b457d63f8a87926146467ecc5bada7b03c9808","sha256:e4248633c5d008c02d04b848c2daf9717137382d8e327e824b00fe2136abd51a"],"state_sha256":"12ef1470c37d221f58872a4b5d9185ba5c517ee77bd419ac8689a796611abab9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4j4n+j/sHfUcpl/W4Mj/XJiG+f1uKcG0SFanljeHa4wGztsoZ/j8SHx0yZRTXe9yoJEwHhg5+Yp2QqEymwpHAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T16:25:28.786816Z","bundle_sha256":"d87ff039eab5d2abe8659364e74adc877143db66d655b73ae2601f5d43da4bf6"}}