{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:UNRQK5VNUHOPUOCTCQI7G67V7R","short_pith_number":"pith:UNRQK5VN","schema_version":"1.0","canonical_sha256":"a3630576ada1dcfa38531411f37bf5fc7695e946062ad8866f0430c1ef7699d6","source":{"kind":"arxiv","id":"0903.1378","version":2},"attestation_state":"computed","paper":{"title":"Mirror symmetry for P^2 and tropical geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.AG","authors_text":"Mark Gross","submitted_at":"2009-03-08T00:41:00Z","abstract_excerpt":"This paper explores the relationship between mirror symmetry for P^2, at the level of big quantum cohomology, and tropical geometry. The mirror of P^2 is typically taken to be ((C^*)^2,W), where W is a Landau-Ginzburg potential of the form x+y+1/xy. The complex moduli space of the mirror is the universal unfolding of W, and oscillatory integrals produce a Frobenius manifold structure on this universal unfolding. We show that W can be deformed by counting Maslov index two tropical disks, and the natural paramaters appearing in this deformation are then the flat coordinates on the moduli space. "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0903.1378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-03-08T00:41:00Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"89cf643c3fedc61c4a012ab5867883c786447f8924fe38e9e206af3b2ff6db1c","abstract_canon_sha256":"3c9fef996556cfd84cd9ad8f99761b7d5c0ba808a364d1bdb0170f0985da5cad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:54:28.839660Z","signature_b64":"7VQGhPDbwdaLYLP6lXANb1fJ1YtLPhUkhUvq5GTvyx9HKosvnTrvcsdbpIcxluQmPBee6tmzfbAMuoCm8ldVBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3630576ada1dcfa38531411f37bf5fc7695e946062ad8866f0430c1ef7699d6","last_reissued_at":"2026-07-04T15:54:28.839299Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:54:28.839299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mirror symmetry for P^2 and tropical geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.AG","authors_text":"Mark Gross","submitted_at":"2009-03-08T00:41:00Z","abstract_excerpt":"This paper explores the relationship between mirror symmetry for P^2, at the level of big quantum cohomology, and tropical geometry. The mirror of P^2 is typically taken to be ((C^*)^2,W), where W is a Landau-Ginzburg potential of the form x+y+1/xy. The complex moduli space of the mirror is the universal unfolding of W, and oscillatory integrals produce a Frobenius manifold structure on this universal unfolding. We show that W can be deformed by counting Maslov index two tropical disks, and the natural paramaters appearing in this deformation are then the flat coordinates on the moduli space. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1378","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/0903.1378/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0903.1378","created_at":"2026-07-04T15:54:28.839368+00:00"},{"alias_kind":"arxiv_version","alias_value":"0903.1378v2","created_at":"2026-07-04T15:54:28.839368+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.1378","created_at":"2026-07-04T15:54:28.839368+00:00"},{"alias_kind":"pith_short_12","alias_value":"UNRQK5VNUHOP","created_at":"2026-07-04T15:54:28.839368+00:00"},{"alias_kind":"pith_short_16","alias_value":"UNRQK5VNUHOPUOCT","created_at":"2026-07-04T15:54:28.839368+00:00"},{"alias_kind":"pith_short_8","alias_value":"UNRQK5VN","created_at":"2026-07-04T15:54:28.839368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R","json":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R.json","graph_json":"https://pith.science/api/pith-number/UNRQK5VNUHOPUOCTCQI7G67V7R/graph.json","events_json":"https://pith.science/api/pith-number/UNRQK5VNUHOPUOCTCQI7G67V7R/events.json","paper":"https://pith.science/paper/UNRQK5VN"},"agent_actions":{"view_html":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R","download_json":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R.json","view_paper":"https://pith.science/paper/UNRQK5VN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0903.1378&json=true","fetch_graph":"https://pith.science/api/pith-number/UNRQK5VNUHOPUOCTCQI7G67V7R/graph.json","fetch_events":"https://pith.science/api/pith-number/UNRQK5VNUHOPUOCTCQI7G67V7R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R/action/storage_attestation","attest_author":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R/action/author_attestation","sign_citation":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R/action/citation_signature","submit_replication":"https://pith.science/pith/UNRQK5VNUHOPUOCTCQI7G67V7R/action/replication_record"}},"created_at":"2026-07-04T15:54:28.839368+00:00","updated_at":"2026-07-04T15:54:28.839368+00:00"}