{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:FKEX32342BAIXEZ6JDDX34S4FJ","short_pith_number":"pith:FKEX3234","canonical_record":{"source":{"id":"2101.08343","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-01-20T22:19:06Z","cross_cats_sorted":["cond-mat.str-el"],"title_canon_sha256":"938b1cca80a4de5ec43a9878f817709e7a4ab41ff9b5ba93775277fdd5e5eb89","abstract_canon_sha256":"6e5bdb8bb50afe07512194c07db3a88d5e3bb80e2ff290eb6688d474ec3cbc4d"},"schema_version":"1.0"},"canonical_sha256":"2a897deb7cd0408b933e48c77df25c2a6d3ac3b878340ecb2785ed6ff55b125e","source":{"kind":"arxiv","id":"2101.08343","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.08343","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"arxiv_version","alias_value":"2101.08343v2","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.08343","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"pith_short_12","alias_value":"FKEX32342BAI","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"pith_short_16","alias_value":"FKEX32342BAIXEZ6","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"pith_short_8","alias_value":"FKEX3234","created_at":"2026-07-05T02:47:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:FKEX32342BAIXEZ6JDDX34S4FJ","target":"record","payload":{"canonical_record":{"source":{"id":"2101.08343","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-01-20T22:19:06Z","cross_cats_sorted":["cond-mat.str-el"],"title_canon_sha256":"938b1cca80a4de5ec43a9878f817709e7a4ab41ff9b5ba93775277fdd5e5eb89","abstract_canon_sha256":"6e5bdb8bb50afe07512194c07db3a88d5e3bb80e2ff290eb6688d474ec3cbc4d"},"schema_version":"1.0"},"canonical_sha256":"2a897deb7cd0408b933e48c77df25c2a6d3ac3b878340ecb2785ed6ff55b125e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:47:30.380631Z","signature_b64":"IbRcYnWieLoJSs2n0reGPak3mPyxF/Nh0TqwyO5qfpcixs2uksxMfOEMg66W3VAxg0TaKbnFwkYhmrzK/XleDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a897deb7cd0408b933e48c77df25c2a6d3ac3b878340ecb2785ed6ff55b125e","last_reissued_at":"2026-07-05T02:47:30.380107Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:47:30.380107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2101.08343","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":"2026-07-05T02:47:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KpfgxKsr4CmTks3+dJPx50DDYs1kOm20BcyL+R5Z9C8+ifeiFXQOqGmHGWpIc4w9U7EbEOAPYGJniAAmHhoWBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T05:42:05.248496Z"},"content_sha256":"43cb5632298bdab39f040acd3f8fda479e4ae70eceb87e94c8cc600f8495c692","schema_version":"1.0","event_id":"sha256:43cb5632298bdab39f040acd3f8fda479e4ae70eceb87e94c8cc600f8495c692"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:FKEX32342BAIXEZ6JDDX34S4FJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\mathbb{Z}_N$ Symmetries, Anomalies, and the Modular Bootstrap","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.str-el"],"primary_cat":"hep-th","authors_text":"Shu-Heng Shao, Ying-Hsuan Lin","submitted_at":"2021-01-20T22:19:06Z","abstract_excerpt":"We explore constraints on (1+1)$d$ unitary conformal field theory with an internal $\\mathbb{Z}_N$ global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a $\\mathbb{Z}_N$-symmetric relevant/marginal operator if $N-1 \\le c\\le 9-N$ for $N\\leq4$, with the endpoints saturated by various WZW models that can be embedded into $(\\mathfrak{e}_8)_1$. Its existence implies that robust gapless fixed points are not possible in this range of $c$ if only a $\\mathbb{Z}_N$ symmetry is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.08343","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/2101.08343/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":"2026-07-05T02:47:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uDXJoPSRA9iPTPYOMwJjp1DqdK++vOLe8UiP/nJD0tV0JuyteXxO1ktEVa/yE3dCdsuc5W0F7trdT/ZloldMDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T05:42:05.248995Z"},"content_sha256":"ab217043a6f76521ec3b6502493b58b52a1c83e0db8c8e124e35986d046e6f8c","schema_version":"1.0","event_id":"sha256:ab217043a6f76521ec3b6502493b58b52a1c83e0db8c8e124e35986d046e6f8c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FKEX32342BAIXEZ6JDDX34S4FJ/bundle.json","state_url":"https://pith.science/pith/FKEX32342BAIXEZ6JDDX34S4FJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FKEX32342BAIXEZ6JDDX34S4FJ/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":"2026-08-06T05:42:05Z","links":{"resolver":"https://pith.science/pith/FKEX32342BAIXEZ6JDDX34S4FJ","bundle":"https://pith.science/pith/FKEX32342BAIXEZ6JDDX34S4FJ/bundle.json","state":"https://pith.science/pith/FKEX32342BAIXEZ6JDDX34S4FJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FKEX32342BAIXEZ6JDDX34S4FJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:FKEX32342BAIXEZ6JDDX34S4FJ","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":"6e5bdb8bb50afe07512194c07db3a88d5e3bb80e2ff290eb6688d474ec3cbc4d","cross_cats_sorted":["cond-mat.str-el"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-01-20T22:19:06Z","title_canon_sha256":"938b1cca80a4de5ec43a9878f817709e7a4ab41ff9b5ba93775277fdd5e5eb89"},"schema_version":"1.0","source":{"id":"2101.08343","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.08343","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"arxiv_version","alias_value":"2101.08343v2","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.08343","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"pith_short_12","alias_value":"FKEX32342BAI","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"pith_short_16","alias_value":"FKEX32342BAIXEZ6","created_at":"2026-07-05T02:47:30Z"},{"alias_kind":"pith_short_8","alias_value":"FKEX3234","created_at":"2026-07-05T02:47:30Z"}],"graph_snapshots":[{"event_id":"sha256:ab217043a6f76521ec3b6502493b58b52a1c83e0db8c8e124e35986d046e6f8c","target":"graph","created_at":"2026-07-05T02:47:30Z","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/2101.08343/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We explore constraints on (1+1)$d$ unitary conformal field theory with an internal $\\mathbb{Z}_N$ global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a $\\mathbb{Z}_N$-symmetric relevant/marginal operator if $N-1 \\le c\\le 9-N$ for $N\\leq4$, with the endpoints saturated by various WZW models that can be embedded into $(\\mathfrak{e}_8)_1$. Its existence implies that robust gapless fixed points are not possible in this range of $c$ if only a $\\mathbb{Z}_N$ symmetry is","authors_text":"Shu-Heng Shao, Ying-Hsuan Lin","cross_cats":["cond-mat.str-el"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-01-20T22:19:06Z","title":"$\\mathbb{Z}_N$ Symmetries, Anomalies, and the Modular Bootstrap"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.08343","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":"sha256:43cb5632298bdab39f040acd3f8fda479e4ae70eceb87e94c8cc600f8495c692","target":"record","created_at":"2026-07-05T02:47:30Z","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":"6e5bdb8bb50afe07512194c07db3a88d5e3bb80e2ff290eb6688d474ec3cbc4d","cross_cats_sorted":["cond-mat.str-el"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-01-20T22:19:06Z","title_canon_sha256":"938b1cca80a4de5ec43a9878f817709e7a4ab41ff9b5ba93775277fdd5e5eb89"},"schema_version":"1.0","source":{"id":"2101.08343","kind":"arxiv","version":2}},"canonical_sha256":"2a897deb7cd0408b933e48c77df25c2a6d3ac3b878340ecb2785ed6ff55b125e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2a897deb7cd0408b933e48c77df25c2a6d3ac3b878340ecb2785ed6ff55b125e","first_computed_at":"2026-07-05T02:47:30.380107Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:47:30.380107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IbRcYnWieLoJSs2n0reGPak3mPyxF/Nh0TqwyO5qfpcixs2uksxMfOEMg66W3VAxg0TaKbnFwkYhmrzK/XleDw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:47:30.380631Z","signed_message":"canonical_sha256_bytes"},"source_id":"2101.08343","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43cb5632298bdab39f040acd3f8fda479e4ae70eceb87e94c8cc600f8495c692","sha256:ab217043a6f76521ec3b6502493b58b52a1c83e0db8c8e124e35986d046e6f8c"],"state_sha256":"4ea908d38c1295c52303e6b895271ab5193b638e3b0385da77dc103913fb3617"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SYD9SKj90b5wGDG6Q8zmY3Q6aMoIvTFR8O/15k7eA75Sam498qdrpARHQcDhA60CL7x3HQQ2i8t5f9R8CkEwAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T05:42:05.252879Z","bundle_sha256":"2c4f8ae8e8a538719af0b0ce2215aba11df7c079e61dac21e8165dd611465333"}}