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\\nu_ {-} = \\int_ {S ^ {4}} \\operatorname {c h} _ {2} (\\mathcal {F}) = \\frac {- 1}{8 \\pi^ {2}} \\int_ {S ^ {4}} \\operatorname {t r} \\mathcal {F} ^ {2}. \\tag {12.90}\n$$\n", "document_sha256": "5b1f9b5886d2cd5b2354bf9b33be2a3508ddf4c1532be3fb894e08deb68450ad", "quote_sha256": "5b1f9b5886d2cd5b2354bf9b33be2a3508ddf4c1532be3fb894e08deb68450ad", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "section": "12.6.2", "pdf_page_index": 490, "node": "a-7d407e95dddbbac3a9d056a3", "source_node_sha256": "5b1f9b5886d2cd5b2354bf9b33be2a3508ddf4c1532be3fb894e08deb68450ad", "upstream": { "path": "library/concepts/source-evidence/mcp/b6d31a5bd0a4b3565ecb31ef713f665aa36c454b67732dc64abdf2abde0dc862.json", "document_sha256": "b6d31a5bd0a4b3565ecb31ef713f665aa36c454b67732dc64abdf2abde0dc862", "node_id": "a-7d407e95dddbbac3a9d056a3", "node_source_sha256": "5b1f9b5886d2cd5b2354bf9b33be2a3508ddf4c1532be3fb894e08deb68450ad" }, "conditions": "Nakahara12.6.2例12.6式12.90及后段：右端表示k∈π3(SU(2))=Z；起句指定丛仅为条件，不把整丛对象当作整数。保留同书第二陈述，不替换11.55。", "review_status": "librarian-original-source-reviewed", "source_locator": { "path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/chapter-12.yaml", "sha256": "44a17b3212fd1d77a3f77847223e64a6f1a7f32be4fff95f2b77b1d0431f1794", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "nodes": [ "a-c82022bff0d539e60a76a904", "a-7d407e95dddbbac3a9d056a3", "a-2e1e3c8562894fe274b0875e" ], "node_hashes": { "a-c82022bff0d539e60a76a904": "42fa50bf9921f4d8265866873084ec8674179e2d0f4faef5568a5080b800952a", "a-7d407e95dddbbac3a9d056a3": "5b1f9b5886d2cd5b2354bf9b33be2a3508ddf4c1532be3fb894e08deb68450ad", "a-2e1e3c8562894fe274b0875e": "b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a" }, "unit_kind": "reviewed-prose-equations" }, "definition_part": "reviewed-definition-unit", "environment_basis": { "path": "library/concepts/source-evidence/environment-basis/44a17b3212fd1d77a3f77847223e64a6f1a7f32be4fff95f2b77b1d0431f1794.yaml.txt", "document_sha256": "44a17b3212fd1d77a3f77847223e64a6f1a7f32be4fff95f2b77b1d0431f1794", "source_path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/chapter-12.yaml" }, "role": "definition", "first_in_book": "not-established", "evidence_origin": "explicit-reviewed-source-manifest", "definition_part_basis": "reviewed-source-unit-membership", "reading_links": [ { "route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-12/06-section-12-6", "anchor": "node-a-7d407e95dddbbac3a9d056a3", "lang": "en", "basis": "explicit-source-anchor" }, { "route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-12/06-section-12-6", "anchor": "node-a-7d407e95dddbbac3a9d056a3", "lang": "zh-CN", "basis": "explicit-source-anchor" } ] }, { "concept_id": "urn:uuid:70415427-4cfe-5638-9aa3-16dde8b6ef27", "definition_id": "urn:uuid:f70d2f0e-ca24-5910-b54f-4599011511c2", "definition_group": "reviewed-unit-f70d2f0e-ca24-5910-b54f-4599011511c2", "path": "library/concepts/source-evidence/mcp/b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a.txt", "range": [ 0, 439 ], "quote": "The RHS represents the instanton number $k \\in \\pi_3(\\mathrm{SU}(2)) = \\mathbb{Z}$ . Note that $k > 0$ if $\\mathcal{F} = *\\mathcal{F}$ while $k < 0$ if $\\mathcal{F} = -* \\mathcal{F}$ . It can be shown that $\\nu_{-} = 0$ ( $\\nu_{+} = 0$ ) if $k > 0$ ( $k < 0$ ), see Jackiw and Rebbi (1977). For example, let $\\mathcal{F}$ be self-dual. Suppose $\\psi^{-} \\in \\ker D^{\\dagger} = \\ker DD^{\\dagger}$ . From (12.80), we find that", "document_sha256": "b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a", "quote_sha256": "b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "scope": { "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "source_id": "nakahara-mineru-bd76b709" }, "section": "12.6.2", "pdf_page_index": 490, "node": "a-2e1e3c8562894fe274b0875e", "source_node_sha256": "b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a", "upstream": { "path": "library/concepts/source-evidence/mcp/b6d31a5bd0a4b3565ecb31ef713f665aa36c454b67732dc64abdf2abde0dc862.json", "document_sha256": "b6d31a5bd0a4b3565ecb31ef713f665aa36c454b67732dc64abdf2abde0dc862", "node_id": "a-2e1e3c8562894fe274b0875e", "node_source_sha256": "b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a" }, "conditions": "Nakahara12.6.2例12.6式12.90及后段：右端表示k∈π3(SU(2))=Z；起句指定丛仅为条件，不把整丛对象当作整数。保留同书第二陈述，不替换11.55。", "review_status": "librarian-original-source-reviewed", "source_locator": { "path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/chapter-12.yaml", "sha256": "44a17b3212fd1d77a3f77847223e64a6f1a7f32be4fff95f2b77b1d0431f1794", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "nodes": [ "a-c82022bff0d539e60a76a904", "a-7d407e95dddbbac3a9d056a3", "a-2e1e3c8562894fe274b0875e" ], "node_hashes": { "a-c82022bff0d539e60a76a904": "42fa50bf9921f4d8265866873084ec8674179e2d0f4faef5568a5080b800952a", "a-7d407e95dddbbac3a9d056a3": "5b1f9b5886d2cd5b2354bf9b33be2a3508ddf4c1532be3fb894e08deb68450ad", "a-2e1e3c8562894fe274b0875e": "b9551dbeff41639c0db9f8783c4533fd10cf1657f02cf60035525a15ebc47d9a" }, "unit_kind": "reviewed-prose-equations" }, "definition_part": "reviewed-definition-unit", "environment_basis": { "path": "library/concepts/source-evidence/environment-basis/44a17b3212fd1d77a3f77847223e64a6f1a7f32be4fff95f2b77b1d0431f1794.yaml.txt", "document_sha256": "44a17b3212fd1d77a3f77847223e64a6f1a7f32be4fff95f2b77b1d0431f1794", "source_path": "library/books/nakahara-geometry-topology-physics/second-edition-local/chapters/chapter-12.yaml" }, "role": "definition", "first_in_book": "not-established", "evidence_origin": "explicit-reviewed-source-manifest", "definition_part_basis": "reviewed-source-unit-membership", "reading_links": [ { "route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-12/06-section-12-6", "anchor": "node-a-2e1e3c8562894fe274b0875e", "lang": "en", "basis": "explicit-source-anchor" }, { "route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-12/06-section-12-6", "anchor": "node-a-2e1e3c8562894fe274b0875e", "lang": "zh-CN", "basis": "explicit-source-anchor" } ] } ], "reviewed_mentions": [ { "concept_id": "urn:uuid:70415427-4cfe-5638-9aa3-16dde8b6ef27", "path": "library/concepts/source-evidence/mcp/733d4a2752d07655b5a265a0b74e86b45136b2c567ea622af97ce62d0cda4a65.txt", "range": [ 0, 269 ], "quote": "$\\mathrm{Ch}(\\mathcal{F})$ terminates at $\\mathrm{ch}_2(\\mathcal{F})$ since $\\mathcal{F}^n = 0$ for $n \\geq 3$ . Moreover, $\\operatorname{tr}\\mathcal{F} = 0$ for $G = \\mathrm{SU}(2)$ , $n \\geq 2$ . 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"route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-11/03-section-11-3", "anchor": "node-a-5c1437f3374ad53f9801bd60", "lang": "en", "basis": "explicit-source-anchor" }, { "route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-11/03-section-11-3", "anchor": "node-a-5c1437f3374ad53f9801bd60", "lang": "zh-CN", "basis": "explicit-source-anchor" } ] }, { "concept_id": "urn:uuid:70415427-4cfe-5638-9aa3-16dde8b6ef27", "path": "library/concepts/source-evidence/mcp/a2a70718b928645c48bdc0849c3b952a5aa174c1a5f22bf015ac11c285f9095a.txt", "range": [ 0, 188 ], "quote": "\n$$\n\\frac {1}{2} \\int_ {S ^ {4}} \\operatorname {t r} \\left(\\frac {\\mathrm {i} \\mathcal {F}}{2 \\pi}\\right) ^ {2} = \\int_ {S ^ {4}} \\operatorname {c h} _ {2} (\\mathcal {F}). \\tag {11.55}\n$$\n", "document_sha256": "a2a70718b928645c48bdc0849c3b952a5aa174c1a5f22bf015ac11c285f9095a", "quote_sha256": "a2a70718b928645c48bdc0849c3b952a5aa174c1a5f22bf015ac11c285f9095a", 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"reviewed-prose-equations" }, "conditions": "用户2026-10-24经Claude交接单13授权图书管理员处理Clifford等四组概念。管理员逐字回查原书，并区分权威登记、运行临时词条和same_as误绑；首选名不能反向逼译文添加原文没有的限定。原文及裁定见reports/coordination-20261024/concept-ch12-curation/plan.json。", "review_status": "librarian-original-source-mention-reviewed", "upstream": { "path": "library/concepts/source-evidence/mcp/86d23eb1854b0054d919af72abb33c580c0d8ddc36e8a15064ba0e70db7bd269.json", "document_sha256": "86d23eb1854b0054d919af72abb33c580c0d8ddc36e8a15064ba0e70db7bd269", "node_id": "a-ac8f25a37c020bdd9c64241a", "node_source_sha256": "a2a70718b928645c48bdc0849c3b952a5aa174c1a5f22bf015ac11c285f9095a" }, "role": "mention", "first_in_book": "not-established", "evidence_origin": "explicit-reviewed-source-manifest", "reading_links": [ { "route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-11/03-section-11-3", "anchor": "node-a-ac8f25a37c020bdd9c64241a", "lang": "en", "basis": "explicit-source-anchor" }, { "route": 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Moreover, $\\operatorname{tr}\\mathcal{F} = 0$ for $G = \\mathrm{SU}(2)$ , $n \\geq 2$ . 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