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When the radius of S^2 is larger than a critical radius, the scalar field develops a vacuum expectation value and creates vortices in S^2. Then the vortices break the rotational symmetry of S^2. We exactly evaluate the critical radius as r_q = \\sqrt{|q|}/\\mu, where q is the monopole number and \\mu is the imaginary m"},"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":"hep-th/0108208","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"hep-th","submitted_at":"2001-08-28T02:36:39Z","cross_cats_sorted":["cond-mat.supr-con","hep-ph"],"title_canon_sha256":"8a3f27e242fa9e4f3bfe9959c707ce339170f85168929a0fb952029d0a8311df","abstract_canon_sha256":"e153372b53a527f3743e3b7a2853f20bed484ac30a85d17225c3285e2abc76bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:35:23.820707Z","signature_b64":"NACSkogWIi5mawayO9Bz4sMeuOBU9FNoVuhOEsgIwFKPt8XDSJ+eoqBbwKeZBraVKZd3gclnR9T7YDuzSSsxCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8385d77fb0df960bb6c75aab301f4005205036d239a7853081985f62db61b31b","last_reissued_at":"2026-05-18T04:35:23.820093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:35:23.820093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spontaneous breaking of the C, P, and rotational symmetries by topological defects in two extra dimensions","license":"","headline":"","cross_cats":["cond-mat.supr-con","hep-ph"],"primary_cat":"hep-th","authors_text":"Makoto Sakamoto, Shogo Tanimura","submitted_at":"2001-08-28T02:36:39Z","abstract_excerpt":"We formulate models of complex scalar fields in the space-time that has a two-dimensional sphere as extra dimensions. 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