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Using the formalism of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for $X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch -- Grothendieck formula applied to the universal stable map.\n  Wh"},"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":"math/0110142","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2001-10-13T22:24:02Z","cross_cats_sorted":[],"title_canon_sha256":"650ec52329f0015abcb2fa9ea5436b33ab160cb8fec2280cf5d9501170162269","abstract_canon_sha256":"58b2c3c4d7ff9e7633af27c95d92c896dd6d560df809b1b2821185f42c1f9f70"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:35:33.945051Z","signature_b64":"1+NC/ug0ABafkXBdvu9nsuwFXrlGU7+a/2SDvkzB27m9r3U8DfOW+4Qy8iMiEjEd9noN2c4Qt+HcNVKGn8roDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"60b0ebd2e9e9a209a3f1eae8b1162e243ab648638397d8a193b7172f48216ad0","last_reissued_at":"2026-07-04T14:35:33.944674Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:35:33.944674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantum Riemann - Roch, Lefschetz and Serre","license":"","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexander Givental, Tom Coates","submitted_at":"2001-10-13T22:24:02Z","abstract_excerpt":"Given a holomorphic vector bundle $E:EX X$ over a compact K\\\"ahler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: \\Sigma \\to X$ by their cap-product with a chosen multiplicative characteristic class of $H^0(\\Sigma, f^* E) - H^1(\\Sigma, f^*E)$. Using the formalism of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for $X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch -- Grothendieck formula applied to the universal stable map.\n  Wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0110142","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/0110142/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":"math/0110142","created_at":"2026-07-04T14:35:33.944736+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0110142v2","created_at":"2026-07-04T14:35:33.944736+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0110142","created_at":"2026-07-04T14:35:33.944736+00:00"},{"alias_kind":"pith_short_12","alias_value":"MCYOXUXJ5GRA","created_at":"2026-07-04T14:35:33.944736+00:00"},{"alias_kind":"pith_short_16","alias_value":"MCYOXUXJ5GRATI7R","created_at":"2026-07-04T14:35:33.944736+00:00"},{"alias_kind":"pith_short_8","alias_value":"MCYOXUXJ","created_at":"2026-07-04T14:35:33.944736+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2601.18881","citing_title":"Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials","ref_index":55,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ","json":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ.json","graph_json":"https://pith.science/api/pith-number/MCYOXUXJ5GRATI7R5LULCFROEQ/graph.json","events_json":"https://pith.science/api/pith-number/MCYOXUXJ5GRATI7R5LULCFROEQ/events.json","paper":"https://pith.science/paper/MCYOXUXJ"},"agent_actions":{"view_html":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ","download_json":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ.json","view_paper":"https://pith.science/paper/MCYOXUXJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0110142&json=true","fetch_graph":"https://pith.science/api/pith-number/MCYOXUXJ5GRATI7R5LULCFROEQ/graph.json","fetch_events":"https://pith.science/api/pith-number/MCYOXUXJ5GRATI7R5LULCFROEQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ/action/storage_attestation","attest_author":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ/action/author_attestation","sign_citation":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ/action/citation_signature","submit_replication":"https://pith.science/pith/MCYOXUXJ5GRATI7R5LULCFROEQ/action/replication_record"}},"created_at":"2026-07-04T14:35:33.944736+00:00","updated_at":"2026-07-04T14:35:33.944736+00:00"}