{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:CKAVHPSHEM7WBDFSXM7ZCATTZS","short_pith_number":"pith:CKAVHPSH","canonical_record":{"source":{"id":"2009.10099","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-09-21T18:00:32Z","cross_cats_sorted":[],"title_canon_sha256":"7f10eb8211a4e17abac698e710dcbe84799e548d81e75a0487c3bee50314263a","abstract_canon_sha256":"2245a10d4b670b9b5014444f357ffbdbbcc483a506ab71a1fbe8e37c352e1788"},"schema_version":"1.0"},"canonical_sha256":"128153be47233f608cb2bb3f910273cc9d66d077da351add0f8e7c92c5d2e6d2","source":{"kind":"arxiv","id":"2009.10099","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.10099","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"arxiv_version","alias_value":"2009.10099v2","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.10099","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"pith_short_12","alias_value":"CKAVHPSHEM7W","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"pith_short_16","alias_value":"CKAVHPSHEM7WBDFS","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"pith_short_8","alias_value":"CKAVHPSH","created_at":"2026-07-05T02:02:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:CKAVHPSHEM7WBDFSXM7ZCATTZS","target":"record","payload":{"canonical_record":{"source":{"id":"2009.10099","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-09-21T18:00:32Z","cross_cats_sorted":[],"title_canon_sha256":"7f10eb8211a4e17abac698e710dcbe84799e548d81e75a0487c3bee50314263a","abstract_canon_sha256":"2245a10d4b670b9b5014444f357ffbdbbcc483a506ab71a1fbe8e37c352e1788"},"schema_version":"1.0"},"canonical_sha256":"128153be47233f608cb2bb3f910273cc9d66d077da351add0f8e7c92c5d2e6d2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:02:45.171786Z","signature_b64":"23jIicC0E7sH082miy7NgEDws1p0CA+OWYDZuy65GLqrmpi48Ts/BWt1eMV7CTU1WgeYg7/TVNgD5/DS53joDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"128153be47233f608cb2bb3f910273cc9d66d077da351add0f8e7c92c5d2e6d2","last_reissued_at":"2026-07-05T02:02:45.171202Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:02:45.171202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.10099","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":"2026-07-05T02:02:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9aXYsX9CoSaGa3S3tmuIbgRqfKe4v9VmHc/zknsP0qglU+bStzqbFzoJrJrcx4SJFcPyCIzd6+Idk0EFwt3wDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:47:47.350434Z"},"content_sha256":"0cbc83499cc8c1d97653cbdb9682d47db7a9f03a97eb2a39538c43b6d0649af2","schema_version":"1.0","event_id":"sha256:0cbc83499cc8c1d97653cbdb9682d47db7a9f03a97eb2a39538c43b6d0649af2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:CKAVHPSHEM7WBDFSXM7ZCATTZS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$SL(2,\\mathbb{Z})$ action on QFTs with $\\mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Lakshya Bhardwaj, Yasunori Lee, Yuji Tachikawa","submitted_at":"2020-09-21T18:00:32Z","abstract_excerpt":"We consider an analogue of Witten's $SL(2,\\mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $\\mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,\\mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,\\mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $\\mathbb{Z}_2$ gauge theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.10099","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/2009.10099/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":"2026-07-05T02:02:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gtv5Zy1RodyeK0W08/f51tlkmnif9h5++rTvG8jcGtxomsNhZBmeNdN48HoA+bjnYHAB9algpSycKnD0uIHTDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:47:47.350964Z"},"content_sha256":"57516531cf87d41ae204ed7d9f95e88426360f4324ed0434518976dd90312992","schema_version":"1.0","event_id":"sha256:57516531cf87d41ae204ed7d9f95e88426360f4324ed0434518976dd90312992"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS/bundle.json","state_url":"https://pith.science/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS/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":"2026-08-13T11:47:47Z","links":{"resolver":"https://pith.science/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS","bundle":"https://pith.science/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS/bundle.json","state":"https://pith.science/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CKAVHPSHEM7WBDFSXM7ZCATTZS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:CKAVHPSHEM7WBDFSXM7ZCATTZS","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":"2245a10d4b670b9b5014444f357ffbdbbcc483a506ab71a1fbe8e37c352e1788","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-09-21T18:00:32Z","title_canon_sha256":"7f10eb8211a4e17abac698e710dcbe84799e548d81e75a0487c3bee50314263a"},"schema_version":"1.0","source":{"id":"2009.10099","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.10099","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"arxiv_version","alias_value":"2009.10099v2","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.10099","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"pith_short_12","alias_value":"CKAVHPSHEM7W","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"pith_short_16","alias_value":"CKAVHPSHEM7WBDFS","created_at":"2026-07-05T02:02:45Z"},{"alias_kind":"pith_short_8","alias_value":"CKAVHPSH","created_at":"2026-07-05T02:02:45Z"}],"graph_snapshots":[{"event_id":"sha256:57516531cf87d41ae204ed7d9f95e88426360f4324ed0434518976dd90312992","target":"graph","created_at":"2026-07-05T02:02:45Z","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/2009.10099/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider an analogue of Witten's $SL(2,\\mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $\\mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,\\mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,\\mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $\\mathbb{Z}_2$ gauge theory.","authors_text":"Lakshya Bhardwaj, Yasunori Lee, Yuji Tachikawa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-09-21T18:00:32Z","title":"$SL(2,\\mathbb{Z})$ action on QFTs with $\\mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.10099","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":"sha256:0cbc83499cc8c1d97653cbdb9682d47db7a9f03a97eb2a39538c43b6d0649af2","target":"record","created_at":"2026-07-05T02:02:45Z","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":"2245a10d4b670b9b5014444f357ffbdbbcc483a506ab71a1fbe8e37c352e1788","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-09-21T18:00:32Z","title_canon_sha256":"7f10eb8211a4e17abac698e710dcbe84799e548d81e75a0487c3bee50314263a"},"schema_version":"1.0","source":{"id":"2009.10099","kind":"arxiv","version":2}},"canonical_sha256":"128153be47233f608cb2bb3f910273cc9d66d077da351add0f8e7c92c5d2e6d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"128153be47233f608cb2bb3f910273cc9d66d077da351add0f8e7c92c5d2e6d2","first_computed_at":"2026-07-05T02:02:45.171202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:02:45.171202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"23jIicC0E7sH082miy7NgEDws1p0CA+OWYDZuy65GLqrmpi48Ts/BWt1eMV7CTU1WgeYg7/TVNgD5/DS53joDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:02:45.171786Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.10099","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0cbc83499cc8c1d97653cbdb9682d47db7a9f03a97eb2a39538c43b6d0649af2","sha256:57516531cf87d41ae204ed7d9f95e88426360f4324ed0434518976dd90312992"],"state_sha256":"a9d8bec48bfcb3d51f20a2f14bae1cd15b8546b74b48faf9827319a83164af56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4t0AaboXDenlaMFNAUHDLbTRipU6shMMic7vlciTpbgG2VtTWjFkZVJLF2ZXkU9vzQTOd3cWECPYFolOBlcGBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T11:47:47.357766Z","bundle_sha256":"e1d4a027d5c15dd8354c07ffe854ec451ab521977b9b5c1991ee94f87de862cc"}}