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A unitary connection $A_0$ on the Hermitian line bundle $F\\times_{\\det U(2)}\\mathbb{C}$ determines a self-adjoint Dirac operator $D_0$ on the $\\mathbb{C}^2$-bundle $F\\times_{U(2)}\\mathbb{C}^2$.\n  The contact form $a$ can be used to perturb the connection $A_0$ by $A_0-ira$. This associates a one parameter family of Dirac operators $D_r"},"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":"1307.4604","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-07-17T12:55:55Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"426ca216385a274edf86731a78779ad55783b3b5e50e1daafa47725ea148fb82","abstract_canon_sha256":"691fe30a15a42e3a6ae4888bf398e94498f4df6b9c3375f12c9f32cef0ba8d97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:17.567658Z","signature_b64":"SJBjqC9/pxymTHeGA51EzKr9eHXkTm18ptOPwgPvzAtSZenJqSrmqMyE/MlhuDVnIXjljTDgaUHAs23MG34FAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b2fba5bba6d928c0fb77d83913de30bdc1de8dfa8821352011266dc42c9610b","last_reissued_at":"2026-05-18T03:18:17.566989Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:17.566989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dirac spectral flow on contact three manifolds I: eigensection estimates and spectral asymmetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.DG","authors_text":"Chung-Jun Tsai","submitted_at":"2013-07-17T12:55:55Z","abstract_excerpt":"Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and a metric $ds^2$. 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