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We develop the all-order asymptotic expansion of the index as $q = e^{2 \\pi i \\tau}$ approaches a root of unity, i.e. as $\\widetilde \\tau \\equiv m \\tau + n \\to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\\log\\mathcal{I}(\\tau)$ has terms of the form $\\widetilde \\tau^k$, $k = -2"},"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":"2104.02051","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-th","submitted_at":"2021-04-05T17:59:12Z","cross_cats_sorted":[],"title_canon_sha256":"ea5beda23f7ad3a02dd9f68a2b17c3cbb18b89460042e3c686b91ef2f20e2d79","abstract_canon_sha256":"9e0ab5721b6711270642a1e16fd07d7d62f9cb352d9928464be98b365932a016"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:57:52.730413Z","signature_b64":"LRQJNYzgHumPLvDue4KqaUrVq1nJAu9C7LPc1wkG9RXOIXv+aoML9utrUYto2X3AAnG4SbYOw+rBByOuY9ueDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fca9360d449bf0434b461c3cbbe3a1d6457c33cf933c6ca1165e967ae44b5e0e","last_reissued_at":"2026-07-05T02:57:52.729984Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:57:52.729984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 4d superconformal index near roots of unity and 3d Chern-Simons theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Arash Arabi Ardehali, Sameer Murthy","submitted_at":"2021-04-05T17:59:12Z","abstract_excerpt":"We consider the $S^3\\times S^1$ superconformal index $\\mathcal{I}(\\tau)$ of 4d $\\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $\\tau$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 \\pi i \\tau}$ approaches a root of unity, i.e. as $\\widetilde \\tau \\equiv m \\tau + n \\to 0$, with $m,n$ relatively prime integers. 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