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We focus, as an example, on the $D=3$, $\\mathcal{N}=2$ Wess-Zumino model with cubic superpotential $W=XYZ+\\frac{\\tau}{6}(X^3+Y^3+Z^3)$, and compute $\\Delta(Q_R,\\tau)$ using the $\\epsilon$-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be \\begin{equation*}\n  \\Delta(Q_R,\\tau)=\n  \\begin{cases}\n  \\left(\\text{BPS bound}\\right)\\left"},"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":"2008.01106","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2020-08-03T18:01:04Z","cross_cats_sorted":[],"title_canon_sha256":"2799c8ab546ce47ab81c65b9034f7293b7bc3cead63bb577d0ad49627de38fb3","abstract_canon_sha256":"d92201185f3dab15f67f08785a2e855342d69d0950210f3908d15336f9dca82a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:11:57.814001Z","signature_b64":"mX/HuD17obEjX7bSQ9P5v2T0H9L9xYOxn/BYPgcA8sCUTgYzfun8kMR3TH0P0Ux6ZYM0YpTwD5rSMN6BnhfMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7a991e3b45e50bcf64e96e27d57b31df94aee0040e302ed83b81cc42925c0f2","last_reissued_at":"2026-07-05T02:11:57.813521Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:11:57.813521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transition of Large $R$-Charge Operators on a Conformal Manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Adar Sharon, Masataka Watanabe","submitted_at":"2020-08-03T18:01:04Z","abstract_excerpt":"We study the transition between phases at large $R$-charge on a conformal manifold. These phases are characterized by the behaviour of the lowest operator dimension $\\Delta(Q_R)$ for fixed and large $R$-charge $Q_R$. We focus, as an example, on the $D=3$, $\\mathcal{N}=2$ Wess-Zumino model with cubic superpotential $W=XYZ+\\frac{\\tau}{6}(X^3+Y^3+Z^3)$, and compute $\\Delta(Q_R,\\tau)$ using the $\\epsilon$-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be \\begin{equation*}\n  \\Delta(Q_R,\\tau)=\n  \\begin{cases}\n  \\left(\\text{BPS bound}\\right)\\left"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.01106","kind":"arxiv","version":1},"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/2008.01106/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":"2008.01106","created_at":"2026-07-05T02:11:57.813579+00:00"},{"alias_kind":"arxiv_version","alias_value":"2008.01106v1","created_at":"2026-07-05T02:11:57.813579+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.01106","created_at":"2026-07-05T02:11:57.813579+00:00"},{"alias_kind":"pith_short_12","alias_value":"W6UZDY5ULZIL","created_at":"2026-07-05T02:11:57.813579+00:00"},{"alias_kind":"pith_short_16","alias_value":"W6UZDY5ULZILZ5SO","created_at":"2026-07-05T02:11:57.813579+00:00"},{"alias_kind":"pith_short_8","alias_value":"W6UZDY5U","created_at":"2026-07-05T02:11:57.813579+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2406.19441","citing_title":"Moduli Spaces in CFT: Large Charge Operators","ref_index":45,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX","json":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX.json","graph_json":"https://pith.science/api/pith-number/W6UZDY5ULZILZ5SOS3RH2V5TDX/graph.json","events_json":"https://pith.science/api/pith-number/W6UZDY5ULZILZ5SOS3RH2V5TDX/events.json","paper":"https://pith.science/paper/W6UZDY5U"},"agent_actions":{"view_html":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX","download_json":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX.json","view_paper":"https://pith.science/paper/W6UZDY5U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2008.01106&json=true","fetch_graph":"https://pith.science/api/pith-number/W6UZDY5ULZILZ5SOS3RH2V5TDX/graph.json","fetch_events":"https://pith.science/api/pith-number/W6UZDY5ULZILZ5SOS3RH2V5TDX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX/action/storage_attestation","attest_author":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX/action/author_attestation","sign_citation":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX/action/citation_signature","submit_replication":"https://pith.science/pith/W6UZDY5ULZILZ5SOS3RH2V5TDX/action/replication_record"}},"created_at":"2026-07-05T02:11:57.813579+00:00","updated_at":"2026-07-05T02:11:57.813579+00:00"}