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In this paper we study the topology of the moduli space $M(r,\\xi)^\\tau$ of stable Real vector bundles over $(\\Sigma,\\tau)$ of rank $r$ and fixed determinant $\\xi$ of degree coprime to $r$.\n  We prove that $M(r,\\xi)^{\\tau}$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $M(r,\\xi)$ so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the mod $2$ Betti numbers of $M(r,\\xi)^\\tau$ and compute mod $p$ Betti "},"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":"1703.00778","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-03-02T13:41:29Z","cross_cats_sorted":[],"title_canon_sha256":"78b2ef98a4f5037c235e8ff1bc4138a13cb8d79661cc125e070cd768609d3918","abstract_canon_sha256":"f5caeb0494807973275f97a5c303ff87dec193970c818d7d8bda53c1425808e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:37.600691Z","signature_b64":"UP86V4cppobfWVd2EkHkKeNmYanX8T+Y0DdEwZfjTknQ+kb0yjDcbqNYM2y5aWnBKnuFyX26ko6fJ44t8MD3Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d506e6afa4e1a4a8d0c9376ba62a382c46265b64a251bd4435faa2f4169b694","last_reissued_at":"2026-05-18T00:49:37.600019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:37.600019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moduli spaces of vector bundles with fixed determinant over a real curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Thomas John Baird","submitted_at":"2017-03-02T13:41:29Z","abstract_excerpt":"Let $(\\Sigma,\\tau)$ denote a Riemann surface of genus $g \\geq 2$ equipped with an anti-holomorphic involution $\\tau$. 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