{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:ZMXNZIRN4B4QKLRQYMDW4EAPPP","short_pith_number":"pith:ZMXNZIRN","canonical_record":{"source":{"id":"2002.06829","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-02-17T08:30:52Z","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"title_canon_sha256":"990e6914ed09773126bd08ea8e7b72ade38968470ba1f66927aa4c1e43440f3f","abstract_canon_sha256":"d6711b9888bb93efccab0e69d0d1eadc3c46a5dfffc6e5da08648287dfdee30d"},"schema_version":"1.0"},"canonical_sha256":"cb2edca22de079052e30c3076e100f7bdb64f106b73768f172a590f0e06bbef5","source":{"kind":"arxiv","id":"2002.06829","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2002.06829","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"arxiv_version","alias_value":"2002.06829v2","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2002.06829","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"pith_short_12","alias_value":"ZMXNZIRN4B4Q","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"pith_short_16","alias_value":"ZMXNZIRN4B4QKLRQ","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"pith_short_8","alias_value":"ZMXNZIRN","created_at":"2026-07-05T01:08:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:ZMXNZIRN4B4QKLRQYMDW4EAPPP","target":"record","payload":{"canonical_record":{"source":{"id":"2002.06829","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-02-17T08:30:52Z","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"title_canon_sha256":"990e6914ed09773126bd08ea8e7b72ade38968470ba1f66927aa4c1e43440f3f","abstract_canon_sha256":"d6711b9888bb93efccab0e69d0d1eadc3c46a5dfffc6e5da08648287dfdee30d"},"schema_version":"1.0"},"canonical_sha256":"cb2edca22de079052e30c3076e100f7bdb64f106b73768f172a590f0e06bbef5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:08:27.860643Z","signature_b64":"dTXxo0qhHJQXpxvCHrUucnksMNxoQvgGcSLAp7AmGswhKCYAjOwN52CDBjZuCVcU4D4rQohhlj2/jLJWoM4NDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb2edca22de079052e30c3076e100f7bdb64f106b73768f172a590f0e06bbef5","last_reissued_at":"2026-07-05T01:08:27.860238Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:08:27.860238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2002.06829","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":"2026-07-05T01:08:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/rKga3uae5Ihlae3Ia8/fP8IGAzortCuC+B0NhaBJvtsglIOdhn0QglJ5ztMGtao579gTa2P+3PT66zV0RplBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T01:41:01.256762Z"},"content_sha256":"d1d228850c7e0c66a5db0222e858831593edb6b8b65f264dbbab4ece96b7224e","schema_version":"1.0","event_id":"sha256:d1d228850c7e0c66a5db0222e858831593edb6b8b65f264dbbab4ece96b7224e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:ZMXNZIRN4B4QKLRQYMDW4EAPPP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exact WKB analysis and TBA equations for the Mathieu equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","quant-ph"],"primary_cat":"hep-th","authors_text":"Keita Imaizumi","submitted_at":"2020-02-17T08:30:52Z","abstract_excerpt":"We derive the Thermodynamic Bethe Ansatz (TBA) equations for the exact WKB periods of the Mathieu equation in the weak coupling region. We will use the TBA equations to calculate the quantum corrections to the WKB periods, which are regarded the quantum periods of $\\mathcal{N} = 2$ $SU(2)$ super Yang-Mills theory at strong coupling. We calculate the effective central charge of the TBA equations, which is found to be proportional to the coefficient of the one-loop beta function of the 4d theory. We also study the spectral problem for the Mathieu equation based on the TBA equations numerically."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2002.06829","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/2002.06829/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":"2026-07-05T01:08:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ct3QEMsNgiWsogsS51ZS6CMDbGa0dJBVnYpzvbh+hAq8kFu+rJc/Dfe07DG0IfVNroHqMlvQjcCKyjtydmhcBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T01:41:01.257608Z"},"content_sha256":"9f1439450f72a083093aacf139c53ca5538fe6ae32187378e41b4a9c940c88c6","schema_version":"1.0","event_id":"sha256:9f1439450f72a083093aacf139c53ca5538fe6ae32187378e41b4a9c940c88c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP/bundle.json","state_url":"https://pith.science/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP/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":"2026-08-09T01:41:01Z","links":{"resolver":"https://pith.science/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP","bundle":"https://pith.science/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP/bundle.json","state":"https://pith.science/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZMXNZIRN4B4QKLRQYMDW4EAPPP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:ZMXNZIRN4B4QKLRQYMDW4EAPPP","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":"d6711b9888bb93efccab0e69d0d1eadc3c46a5dfffc6e5da08648287dfdee30d","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-02-17T08:30:52Z","title_canon_sha256":"990e6914ed09773126bd08ea8e7b72ade38968470ba1f66927aa4c1e43440f3f"},"schema_version":"1.0","source":{"id":"2002.06829","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2002.06829","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"arxiv_version","alias_value":"2002.06829v2","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2002.06829","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"pith_short_12","alias_value":"ZMXNZIRN4B4Q","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"pith_short_16","alias_value":"ZMXNZIRN4B4QKLRQ","created_at":"2026-07-05T01:08:27Z"},{"alias_kind":"pith_short_8","alias_value":"ZMXNZIRN","created_at":"2026-07-05T01:08:27Z"}],"graph_snapshots":[{"event_id":"sha256:9f1439450f72a083093aacf139c53ca5538fe6ae32187378e41b4a9c940c88c6","target":"graph","created_at":"2026-07-05T01:08:27Z","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/2002.06829/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We derive the Thermodynamic Bethe Ansatz (TBA) equations for the exact WKB periods of the Mathieu equation in the weak coupling region. We will use the TBA equations to calculate the quantum corrections to the WKB periods, which are regarded the quantum periods of $\\mathcal{N} = 2$ $SU(2)$ super Yang-Mills theory at strong coupling. We calculate the effective central charge of the TBA equations, which is found to be proportional to the coefficient of the one-loop beta function of the 4d theory. We also study the spectral problem for the Mathieu equation based on the TBA equations numerically.","authors_text":"Keita Imaizumi","cross_cats":["math-ph","math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-02-17T08:30:52Z","title":"Exact WKB analysis and TBA equations for the Mathieu equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2002.06829","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:d1d228850c7e0c66a5db0222e858831593edb6b8b65f264dbbab4ece96b7224e","target":"record","created_at":"2026-07-05T01:08:27Z","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":"d6711b9888bb93efccab0e69d0d1eadc3c46a5dfffc6e5da08648287dfdee30d","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-02-17T08:30:52Z","title_canon_sha256":"990e6914ed09773126bd08ea8e7b72ade38968470ba1f66927aa4c1e43440f3f"},"schema_version":"1.0","source":{"id":"2002.06829","kind":"arxiv","version":2}},"canonical_sha256":"cb2edca22de079052e30c3076e100f7bdb64f106b73768f172a590f0e06bbef5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb2edca22de079052e30c3076e100f7bdb64f106b73768f172a590f0e06bbef5","first_computed_at":"2026-07-05T01:08:27.860238Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:08:27.860238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dTXxo0qhHJQXpxvCHrUucnksMNxoQvgGcSLAp7AmGswhKCYAjOwN52CDBjZuCVcU4D4rQohhlj2/jLJWoM4NDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:08:27.860643Z","signed_message":"canonical_sha256_bytes"},"source_id":"2002.06829","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d1d228850c7e0c66a5db0222e858831593edb6b8b65f264dbbab4ece96b7224e","sha256:9f1439450f72a083093aacf139c53ca5538fe6ae32187378e41b4a9c940c88c6"],"state_sha256":"5f8959c70cd0b2f281908c41cb05d6c0b494902c0060ec2509bd79df8cc5a8df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3yHeiLPY5nw7WdLgmMqy4m50CotPOBw/IRlaKM+ULrHm5aj57l/1/HZU0A2Ne/KIQWK+8GNtLTfx7r1822n6Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T01:41:01.260977Z","bundle_sha256":"ee84358918ad071d1553eb0b902b60b78d2940ce8be471fe84bd0919ed6c4e2b"}}