{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ISQ7YOSLXFPPNL6IHIRF4TFB4R","short_pith_number":"pith:ISQ7YOSL","canonical_record":{"source":{"id":"1508.02728","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2015-08-11T20:14:45Z","cross_cats_sorted":["cond-mat.str-el","math-ph","math.MP","quant-ph"],"title_canon_sha256":"596186348ea6c104abc0cc7d3014026012a0334bb20836d626d819fed9678ece","abstract_canon_sha256":"905b3937c5a5dc8b7d60ccff16376c1b41fd95bdd6fb75cfa6795372656cd35e"},"schema_version":"1.0"},"canonical_sha256":"44a1fc3a4bb95ef6afc83a225e4ca1e4435f29ee51ce357eb371ba9116606ef6","source":{"kind":"arxiv","id":"1508.02728","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02728","created_at":"2026-05-18T00:57:04Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02728v3","created_at":"2026-05-18T00:57:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02728","created_at":"2026-05-18T00:57:04Z"},{"alias_kind":"pith_short_12","alias_value":"ISQ7YOSLXFPP","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"ISQ7YOSLXFPPNL6I","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"ISQ7YOSL","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ISQ7YOSLXFPPNL6IHIRF4TFB4R","target":"record","payload":{"canonical_record":{"source":{"id":"1508.02728","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2015-08-11T20:14:45Z","cross_cats_sorted":["cond-mat.str-el","math-ph","math.MP","quant-ph"],"title_canon_sha256":"596186348ea6c104abc0cc7d3014026012a0334bb20836d626d819fed9678ece","abstract_canon_sha256":"905b3937c5a5dc8b7d60ccff16376c1b41fd95bdd6fb75cfa6795372656cd35e"},"schema_version":"1.0"},"canonical_sha256":"44a1fc3a4bb95ef6afc83a225e4ca1e4435f29ee51ce357eb371ba9116606ef6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:04.895976Z","signature_b64":"cV8G4vFv5W6rfopnad5j5/fGk/AiLtHwaaSlusgZrNLlxxNlqd8VEg735OcZ5BFDroWPFxcrGH2VFmu9WtVBBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44a1fc3a4bb95ef6afc83a225e4ca1e4435f29ee51ce357eb371ba9116606ef6","last_reissued_at":"2026-05-18T00:57:04.895418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:04.895418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.02728","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:57:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lZw4fPBiv62Q56ra29xJ+YcW7tdXc6/Ca8irH/09nhUrfS63g3UlfA+SfR9i+rwDzWUoYVE9jglP63GjOW0BBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T16:20:01.395366Z"},"content_sha256":"7423531c9c17aed3e1cdea7dafe47334572f064efacd71c8de4ad23b3271efea","schema_version":"1.0","event_id":"sha256:7423531c9c17aed3e1cdea7dafe47334572f064efacd71c8de4ad23b3271efea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ISQ7YOSLXFPPNL6IHIRF4TFB4R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular forms and a generalized Cardy formula in higher dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.str-el","math-ph","math.MP","quant-ph"],"primary_cat":"hep-th","authors_text":"Edgar Shaghoulian","submitted_at":"2015-08-11T20:14:45Z","abstract_excerpt":"We derive a formula which applies to conformal field theories on a spatial torus and gives the asymptotic density of states solely in terms of the vacuum energy on a parallel plate geometry. The formula follows immediately from global scale and Lorentz invariance, but to our knowledge has not previously been made explicit. It can also be understood from the fact that $\\log Z$ on $\\mathbb{T}^2\\times \\mathbb{R}^{d-1}$ transforms as the absolute value of a non-holomorphic modular form of weight $d-1$, which we show. The results are extended to theories which violate Lorentz invariance and hypersc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02728","kind":"arxiv","version":3},"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":"2026-05-18T00:57:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q5ZiFKM/ObgiWTcH2PEJtvgkUNlB+yi/IsYNb5Hw2dEvIk8ZNWAx8w7OhWLmI3DC6k+VFS88Zm6P3E+gyGIgBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T16:20:01.395736Z"},"content_sha256":"859ecb3432b9033cab871932acae6a6e1c86abad790aba5b78cc9f58e4609d64","schema_version":"1.0","event_id":"sha256:859ecb3432b9033cab871932acae6a6e1c86abad790aba5b78cc9f58e4609d64"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R/bundle.json","state_url":"https://pith.science/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R/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":"2026-06-28T16:20:01Z","links":{"resolver":"https://pith.science/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R","bundle":"https://pith.science/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R/bundle.json","state":"https://pith.science/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ISQ7YOSLXFPPNL6IHIRF4TFB4R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ISQ7YOSLXFPPNL6IHIRF4TFB4R","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":"905b3937c5a5dc8b7d60ccff16376c1b41fd95bdd6fb75cfa6795372656cd35e","cross_cats_sorted":["cond-mat.str-el","math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2015-08-11T20:14:45Z","title_canon_sha256":"596186348ea6c104abc0cc7d3014026012a0334bb20836d626d819fed9678ece"},"schema_version":"1.0","source":{"id":"1508.02728","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02728","created_at":"2026-05-18T00:57:04Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02728v3","created_at":"2026-05-18T00:57:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02728","created_at":"2026-05-18T00:57:04Z"},{"alias_kind":"pith_short_12","alias_value":"ISQ7YOSLXFPP","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"ISQ7YOSLXFPPNL6I","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"ISQ7YOSL","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:859ecb3432b9033cab871932acae6a6e1c86abad790aba5b78cc9f58e4609d64","target":"graph","created_at":"2026-05-18T00:57:04Z","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 derive a formula which applies to conformal field theories on a spatial torus and gives the asymptotic density of states solely in terms of the vacuum energy on a parallel plate geometry. The formula follows immediately from global scale and Lorentz invariance, but to our knowledge has not previously been made explicit. It can also be understood from the fact that $\\log Z$ on $\\mathbb{T}^2\\times \\mathbb{R}^{d-1}$ transforms as the absolute value of a non-holomorphic modular form of weight $d-1$, which we show. The results are extended to theories which violate Lorentz invariance and hypersc","authors_text":"Edgar Shaghoulian","cross_cats":["cond-mat.str-el","math-ph","math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2015-08-11T20:14:45Z","title":"Modular forms and a generalized Cardy formula in higher dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02728","kind":"arxiv","version":3},"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":"sha256:7423531c9c17aed3e1cdea7dafe47334572f064efacd71c8de4ad23b3271efea","target":"record","created_at":"2026-05-18T00:57:04Z","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":"905b3937c5a5dc8b7d60ccff16376c1b41fd95bdd6fb75cfa6795372656cd35e","cross_cats_sorted":["cond-mat.str-el","math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2015-08-11T20:14:45Z","title_canon_sha256":"596186348ea6c104abc0cc7d3014026012a0334bb20836d626d819fed9678ece"},"schema_version":"1.0","source":{"id":"1508.02728","kind":"arxiv","version":3}},"canonical_sha256":"44a1fc3a4bb95ef6afc83a225e4ca1e4435f29ee51ce357eb371ba9116606ef6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"44a1fc3a4bb95ef6afc83a225e4ca1e4435f29ee51ce357eb371ba9116606ef6","first_computed_at":"2026-05-18T00:57:04.895418Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:04.895418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cV8G4vFv5W6rfopnad5j5/fGk/AiLtHwaaSlusgZrNLlxxNlqd8VEg735OcZ5BFDroWPFxcrGH2VFmu9WtVBBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:04.895976Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.02728","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7423531c9c17aed3e1cdea7dafe47334572f064efacd71c8de4ad23b3271efea","sha256:859ecb3432b9033cab871932acae6a6e1c86abad790aba5b78cc9f58e4609d64"],"state_sha256":"64a3b6baa5af09ff898e22fa10ac99881b00c5822e4ad0a6ec43f9beb6f7fd6b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2eoY441NJNbos7oOztex6Gq+YAtuNytb/2e6auBQX59Mn0YcT6X5E0vHBDmVl6LwW79FRXwEExpWkXkvbeWKDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T16:20:01.397628Z","bundle_sha256":"51439596a30eef1a1fce924ffe7aafc8ed025e91aaefbf2f28d8b84fff7f9dc5"}}