{"id":71104,"date":"2026-02-03T08:04:53","date_gmt":"2026-02-03T00:04:53","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/71104.html"},"modified":"2026-02-03T08:04:53","modified_gmt":"2026-02-03T00:04:53","slug":"%e3%80%90%e4%bf%a1%e6%81%af%e7%a7%91%e5%ad%a6%e4%b8%8e%e5%b7%a5%e7%a8%8b%e5%ad%a6%e3%80%91%e3%80%90%e6%95%b0%e6%8d%ae%e7%a7%91%e5%ad%a6%e3%80%91%e6%95%b0%e6%8d%ae%e7%a7%91%e5%ad%a6%e9%a2%86%e5%9f%9f","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/71104.html","title":{"rendered":"\u3010\u4fe1\u606f\u79d1\u5b66\u4e0e\u5de5\u7a0b\u5b66\u3011\u3010\u6570\u636e\u79d1\u5b66\u3011\u6570\u636e\u79d1\u5b66\u9886\u57df-\u7b2c\u4e09\u7bc7 \u6570\u5b66\u57fa\u784009 \u62d3\u6251\u5b66(2)"},"content":{"rendered":"<h2>\u56db\u7ef4\u5fae\u5206\u6d41\u5f62\u7684\u62d3\u6251\u5b66\u65b9\u7a0b&#xff1a;\u5b8c\u6574\u89e3\u91ca<\/h2>\n<h3>\u4e00\u3001\u57fa\u672c\u62d3\u6251\u4e0d\u53d8\u91cf\u4e0e\u65b9\u7a0b<\/h3>\n<h4>1. \u6b27\u62c9\u793a\u6027\u6570<\/h4>\n<p>\u5bf9\u4e8e\u95ed\u53ef\u5b9a\u5411\u7684\u56db\u7ef4\u5fae\u5206\u6d41\u5f62 M&#xff1a;<\/p>\n<p>\u65b9\u7a0b&#xff1a;<\/p>\n<p>\u03c7(M)&#061;i&#061;0\u22114\u200b(\u22121)ibi\u200b(M)&#061;2\u22122b1\u200b(M)&#043;b2\u200b(M)<\/p>\n<p>\u89e3\u91ca&#xff1a;<\/p>\n<ul>\n<li>\n<p>bi\u200b&#061;dimHi\u200b(M;R)\u662f\u8d1d\u8482\u6570<\/p>\n<\/li>\n<li>\n<p>\u7531\u5e9e\u52a0\u83b1\u5bf9\u5076&#xff1a;b0\u200b&#061;b4\u200b&#061;1&#xff0c;b1\u200b&#061;b3\u200b<\/p>\n<\/li>\n<li>\n<p>\u53ef\u5b9a\u5411\u6027\u7ed9\u51fa b0\u200b&#061;b4\u200b&#061;1<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u5355\u8fde\u901a\u6d41\u5f62&#xff1a;b1\u200b&#061;b3\u200b&#061;0&#xff0c;\u6545 \u03c7&#061;2&#043;b2\u200b<\/p>\n<\/li>\n<\/ul>\n<p>\u51e0\u4f55\u610f\u4e49&#xff1a;\u6b27\u62c9\u793a\u6027\u6570\u662f\u6d41\u5f62\u62d3\u6251\u590d\u6742\u6027\u7684\u5ea6\u91cf&#xff0c;\u7b49\u4e8e\u66f2\u7387\u7684\u79ef\u5206&#xff08;Gauss-Bonnet\u63a8\u5e7f&#xff09;\u3002<\/p>\n<h4>2. \u7b26\u53f7\u5dee\u4e0e\u76f8\u4ea4\u5f62\u5f0f<\/h4>\n<p>\u76f8\u4ea4\u5f62\u5f0f\u65b9\u7a0b&#xff1a;<\/p>\n<p>QM\u200b:H2(M;Z)\u00d7H2(M;Z)\u2192Z<\/p>\n<p>QM\u200b(\u03b1,\u03b2)&#061;\u27e8\u03b1\u222a\u03b2,[M]\u27e9<\/p>\n<p>\u7b26\u53f7\u5dee\u65b9\u7a0b&#xff1a;<\/p>\n<p>\u8bbe b2&#043;\u200b\u548c b2\u2212\u200b\u5206\u522b\u4e3a QM\u200b\u5728 H2(M;R)\u4e0a\u7684\u6b63\u3001\u8d1f\u60ef\u6027\u6307\u6570<\/p>\n<p>b2\u200b&#061;b2&#043;\u200b&#043;b2\u2212\u200b<\/p>\n<p>\u03c4(M)&#061;b2&#043;\u200b\u2212b2\u2212\u200b<\/p>\n<p>\u5206\u7c7b\u5b9a\u7406&#xff08;Freedman, 1982&#xff09;&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5355\u8fde\u901a\u95ed\u56db\u7ef4\u62d3\u6251\u6d41\u5f62\u7684\u540c\u80da\u578b\u7531\u76f8\u4ea4\u5f62\u5f0f QM\u200b\u7684<\/p>\n<ul>\n<li>\n<p>\u7c7b\u578b&#xff08;\u5076\/\u5947&#xff09;<\/p>\n<\/li>\n<li>\n<p>Kirby\u2013Siebenmann\u4e0d\u53d8\u91cf w(M)\u2208Z2\u200b<\/p>\n<p>\u5b8c\u5168\u51b3\u5b9a<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u7f57\u8d6b\u6797\u5b9a\u7406&#xff1a;<\/p>\n<p>\u82e5 M\u5149\u6ed1\u4e14 QM\u200b\u4e3a\u5076\u5f62\u5f0f&#xff0c;\u5219<\/p>\n<p>\u03c4(M)\u22610(mod16)<\/p>\n<h3>\u4e8c\u3001\u590d\u66f2\u9762&#xff08;\u590d\u4e8c\u7ef4\u6d41\u5f62&#xff09;\u7684\u9648\u7c7b\u65b9\u7a0b<\/h3>\n<h4>1. \u9648\u7c7b\u5b9a\u4e49<\/h4>\n<p>\u5bf9\u590d\u66f2\u9762 M&#xff08;\u590d\u4e8c\u7ef4\u590d\u6d41\u5f62&#xff09;&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u7b2c\u4e00\u9648\u7c7b&#xff1a;c1\u200b(M)&#061;c1\u200b(TM)\u2208H2(M;Z)<\/p>\n<\/li>\n<li>\n<p>\u7b2c\u4e8c\u9648\u7c7b&#xff1a;c2\u200b(M)&#061;c2\u200b(TM)\u2208H4(M;Z)\u2245Z<\/p>\n<\/li>\n<\/ul>\n<h4>2. \u9648\u6570\u65b9\u7a0b<\/h4>\n<p>\u6b27\u62c9\u793a\u6027\u6570\u4e0e\u9648\u6570&#xff1a;<\/p>\n<p>\u03c7(M)&#061;\u222bM\u200bc2\u200b(M)<\/p>\n<p>\u7b26\u53f7\u5dee\u4e0e\u9648\u6570&#xff08;Hirzebruch\u7b7e\u540d\u5b9a\u7406&#xff09;&#xff1a;<\/p>\n<p>\u03c4(M)&#061;31\u200b\u222bM\u200b(c12\u200b(M)\u22122c2\u200b(M))<\/p>\n<p>\u8bfa\u7279\u516c\u5f0f&#xff08;\u590d\u66f2\u9762&#xff09;&#xff1a;<\/p>\n<p>c12\u200b(M)&#061;2\u03c7(M)&#043;3\u03c4(M)<\/p>\n<p>\u9648\u6570\u5173\u7cfb&#xff1a;<\/p>\n<p>c12\u200b(M)&#043;c2\u200b(M)&#061;3\u03c3(M)&#043;2\u03c7(M)<\/p>\n<h4>3. \u5178\u5219\u7c7b\u65b9\u7a0b<\/h4>\n<p>\u5bf9\u4ee3\u6570\u66f2\u9762&#xff0c;\u5178\u5219\u7c7b K&#061;\u2212c1\u200b(TM)&#xff0c;\u6709&#xff1a;<\/p>\n<p>K2&#061;c12\u200b(M)&#061;2\u03c7(M)&#043;3\u03c4(M)<\/p>\n<p>\u03c7(OM\u200b)&#061;121\u200b(K2&#043;c2\u200b(M))<\/p>\n<h3>\u4e09\u3001\u6307\u6807\u5b9a\u7406\u5728\u56db\u7ef4\u7684\u5e94\u7528<\/h3>\n<h4>1. \u7b26\u53f7\u7b97\u5b50\u7684\u6307\u6807\u5b9a\u7406<\/h4>\n<p>Hirzebruch\u7b26\u53f7\u5b9a\u7406&#xff1a;<\/p>\n<p>\u03c4(M)&#061;31\u200b\u222bM\u200bL1\u200b(p1\u200b)<\/p>\n<p>\u5176\u4e2d L1\u200b\u662fHirzebruch L-\u591a\u9879\u5f0f&#xff0c;p1\u200b\u662f\u7b2c\u4e00\u4e2a\u5e9e\u7279\u91cc\u4e9a\u91d1\u7c7b<\/p>\n<p>\u56db\u7ef4\u5177\u4f53\u5f62\u5f0f&#xff1a;<\/p>\n<p>\u03c4(M)&#061;31\u200bp1\u200b(M)[M]&#061;\u221224\u03c021\u200b\u222bM\u200btr(R\u2227R)<\/p>\n<h4>2. Dirac\u7b97\u5b50\u7684\u6307\u6807<\/h4>\n<p>\u5bf9\u5e26\u81ea\u65cb\u7ed3\u6784\u7684\u56db\u7ef4\u6d41\u5f62&#xff0c;Dirac\u7b97\u5b50 D\u7684\u6307\u6807&#xff1a;<\/p>\n<p>ind(D)&#061;81\u200b\u03c4(M)<\/p>\n<p>\u5bf9Spin^c\u7ed3\u6784&#xff1a;<\/p>\n<p>ind(DA\u200b)&#061;81\u200b(c1\u200b(L)2\u2212\u03c4(M))<\/p>\n<p>\u5176\u4e2d L\u662fSpin^c\u7ed3\u6784\u7684\u786e\u5b9a\u7ebf\u4e1b<\/p>\n<h4>3. Dolbeault\u7b97\u5b50\u7684\u6307\u6807<\/h4>\n<p>\u5bf9\u590d\u66f2\u9762&#xff0c;\u2202\u02c9\u7b97\u5b50\u7684\u6307\u6807&#xff1a;<\/p>\n<p>ind(\u2202\u02c9)&#061;121\u200b\u222bM\u200b(c12\u200b&#043;c2\u200b)&#061;121\u200b(c12\u200b&#043;\u03c7)<\/p>\n<h3>\u56db\u3001\u51e0\u4f55\u4e0e\u62d3\u6251\u7ea6\u675f\u65b9\u7a0b<\/h3>\n<h4>1. Hitchin-Thorpe\u4e0d\u7b49\u5f0f<\/h4>\n<p>\u7231\u56e0\u65af\u5766\u6d41\u5f62\u7ea6\u675f&#xff1a;<\/p>\n<p>\u5bf9\u7d27\u81f4\u53ef\u5b9a\u5411\u7684\u56db\u7ef4\u7231\u56e0\u65af\u5766\u6d41\u5f62&#xff1a;<\/p>\n<p>2\u03c7(M)\u22653\u2223\u03c4(M)\u2223<\/p>\n<p>\u7b49\u53f7\u60c5\u5f62&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5f53 2\u03c7&#061;3\u03c4\u65f6&#xff0c;\u6d41\u5f62\u5c40\u90e8\u8d85\u51ef\u52d2\u6216\u5e73\u5766<\/p>\n<\/li>\n<li>\n<p>\u5f53 2\u03c7&#061;\u22123\u03c4\u65f6&#xff0c;\u7c7b\u4f3c\u7ed3\u8bba<\/p>\n<\/li>\n<\/ul>\n<p>\u63a8\u5e7f\u5f62\u5f0f&#xff1a;<\/p>\n<p>2\u03c7(M)\u22653\u2223\u03c4(M)\u2223&#043;2\u03c021\u200b\u222bM\u200b\u2223W\u2212\u22232dV<\/p>\n<p>\u5176\u4e2d W\u2212\u662f\u81ea\u53cd\u5bf9\u5076\u97e6\u5c14\u5f20\u91cf<\/p>\n<h4>2. Bogomolov\u2013Miyaoka\u2013Yau\u4e0d\u7b49\u5f0f<\/h4>\n<p>\u590d\u66f2\u9762\u7ea6\u675f&#xff1a;<\/p>\n<p>\u5bf9\u5177\u6709\u4e00\u822c\u578b&#xff08;general type&#xff09;\u7684\u590d\u66f2\u9762&#xff1a;<\/p>\n<p>c12\u200b(M)\u22643c2\u200b(M)<\/p>\n<p>\u5373<\/p>\n<p>K2\u22643\u03c7(OM\u200b)<\/p>\n<p>\u7b49\u53f7\u60c5\u5f62&#xff1a;\u8fbe\u5230\u4e0a\u754c\u65f6&#xff0c;\u6d41\u5f62\u662f\u5546\u4e8e\u5355\u4f4d\u7403 B2\u2282C2<\/p>\n<h4>3. \u6807\u91cf\u66f2\u7387\u7ea6\u675f<\/h4>\n<p>Lichnerowicz\u5b9a\u7406&#xff1a;\u82e5\u56db\u7ef4\u95ed\u81ea\u65cb\u6d41\u5f62\u6709\u6b63\u6807\u91cf\u66f2\u7387&#xff0c;\u5219<\/p>\n<p>A^(M)&#061;\u221281\u200b\u03c4(M)&#061;0<\/p>\n<p>\u56db\u7ef4\u5f62\u5f0f&#xff1a;\u5bf9\u81ea\u65cb\u6d41\u5f62&#xff0c;<\/p>\n<p>\u03c4(M)\u22610(mod16)\u4e14\u03c4(M)&#061;0\u00a0\u82e5\u5b58\u5728\u6b63\u6807\u91cf\u66f2\u7387<\/p>\n<h3>\u4e94\u3001\u89c4\u8303\u7406\u8bba\u4e0d\u53d8\u91cf\u65b9\u7a0b<\/h3>\n<h4>1. Donaldson\u4e0d\u53d8\u91cf<\/h4>\n<p>\u77ac\u5b50\u6a21\u7a7a\u95f4\u7ef4\u5ea6\u516c\u5f0f&#xff1a;<\/p>\n<p>\u5bf9\u4e3b SU(2)\u4e1b P\u2192M\u5177\u6709\u7b2c\u4e8c\u9648\u7c7b k&#061;c2\u200b(P)&#xff0c;\u77ac\u5b50\u6a21\u7a7a\u95f4&#xff08;\u82e5\u6b63\u5219&#xff09;\u7684\u865a\u62df\u7ef4\u5ea6&#xff1a;<\/p>\n<p>dimMk\u200b&#061;8k\u22123(1\u2212b1\u200b&#043;b2&#043;\u200b)<\/p>\n<p>\u7b80\u5355\u8fde\u901a\u60c5\u5f62&#xff1a;b1\u200b&#061;0&#xff0c;\u5219<\/p>\n<p>dimMk\u200b&#061;8k\u22123(1&#043;b2&#043;\u200b)<\/p>\n<h4>2. Seiberg-Witten\u65b9\u7a0b<\/h4>\n<p>\u57fa\u672c\u65b9\u7a0b&#xff1a;<\/p>\n<p>\u5bf9Spin^c\u7ed3\u6784 (W&#043;,W\u2212)\u548c\u8fde\u63a5 A&#xff0c;\u65cb\u91cf\u573a \u03c8\u2208\u0393(W&#043;)&#xff1a;<\/p>\n<p>DA\u200b\u03c8FA&#043;\u200b\u200b&#061;0&#061;iq(\u03c8)&#061;i(\u03c8\u2297\u03c8\u2217)0\u200b\u200b<\/p>\n<p>\u5176\u4e2d FA&#043;\u200b\u662f\u81ea\u5bf9\u5076\u66f2\u7387\u90e8\u5206&#xff0c;q(\u03c8)\u662f\u4e8c\u6b21\u5f62\u5f0f<\/p>\n<p>\u6a21\u7a7a\u95f4\u7ef4\u5ea6&#xff1a;<\/p>\n<p>dimMSW\u200b&#061;41\u200b(c1\u200b(L)2\u2212(2\u03c7&#043;3\u03c4))<\/p>\n<p>\u5176\u4e2d L&#061;det(W&#043;)<\/p>\n<p>Seiberg-Witten\u4e0d\u53d8\u91cf\u65b9\u7a0b&#xff1a;<\/p>\n<p>\u8bbe SWM\u200b:Spinc(M)\u2192Z\u4e3a\u4e0d\u53d8\u91cf&#xff0c;\u5219<\/p>\n<ul>\n<li>\n<p>\u5bf9\u5177\u6709\u6b63\u6807\u91cf\u66f2\u7387\u7684\u6d41\u5f62&#xff0c;SWM\u200b\u22610<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u8f9b\u6d41\u5f62&#xff0c;\u57fa\u672c\u7c7b\u6ee1\u8db3 \u2223SWM\u200b(\u00b1K)\u2223&#061;1<\/p>\n<\/li>\n<\/ul>\n<h4>3. \u7b80\u5355\u578b\u731c\u60f3<\/h4>\n<p>Witten\u516c\u5f0f&#xff1a;Donaldson\u4e0d\u53d8\u91cf\u4e0eSeiberg-Witten\u4e0d\u53d8\u91cf\u7684\u5173\u7cfb&#xff1a;<\/p>\n<p>DM(p,s)&#061;22\u221247\u03c7&#043;11\u03c3\u200bx\u2208H2(M;Z)\u2211\u200bSWM\u200b(x)e2(x\u22c5s&#043;x2)<\/p>\n<p>\u5176\u4e2d p,s\u662f\u4e0a\u540c\u8c03\u63d2\u5165<\/p>\n<h3>\u516d\u3001\u66f2\u7387\u79ef\u5206\u65b9\u7a0b<\/h3>\n<h4>1. Gauss-Bonnet-Chern\u5b9a\u7406<\/h4>\n<p>\u56db\u7ef4\u5f62\u5f0f&#xff1a;<\/p>\n<p>\u03c7(M)&#061;32\u03c021\u200b\u222bM\u200b(\u2223R\u22232\u22124\u2223Ric\u22232&#043;R2)dV<\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\n<p>R\u662f\u9ece\u66fc\u66f2\u7387\u5f20\u91cf<\/p>\n<\/li>\n<li>\n<p>Ric\u662fRicci\u66f2\u7387\u5f20\u91cf<\/p>\n<\/li>\n<li>\n<p>R\u662f\u6807\u91cf\u66f2\u7387<\/p>\n<\/li>\n<\/ul>\n<p>\u5206\u91cf\u5f62\u5f0f&#xff1a;<\/p>\n<p>\u03c7(M)&#061;8\u03c021\u200b\u222bM\u200b(Rijkl\u200bRijkl\u22124Rij\u200bRij&#043;R2)dV<\/p>\n<h4>2. \u7b26\u53f7\u5b9a\u7406\u7684\u66f2\u7387\u5f62\u5f0f<\/h4>\n<p>\u03c4(M)&#061;12\u03c021\u200b\u222bM\u200b(\u2223W&#043;\u22232\u2212\u2223W\u2212\u22232)dV<\/p>\n<p>\u5176\u4e2d W&#043;,W\u2212\u662f\u97e6\u5c14\u5f20\u91cf\u7684\u81ea\u5bf9\u5076\u548c\u53cd\u81ea\u5bf9\u5076\u90e8\u5206<\/p>\n<h4>3. \u9648-\u9ad8\u65af-\u535a\u5185\u7279\u5f62\u5f0f<\/h4>\n<p>\u7528\u66f2\u7387\u5f62\u5f0f\u8868\u793a\u9648\u7c7b&#xff1a;<\/p>\n<p>c1\u200b(M)&#061;2\u03c0i\u200btr(\u0398)<\/p>\n<p>c2\u200b(M)&#061;8\u03c021\u200b[tr(\u0398\u2227\u0398)\u2212tr(\u0398)\u2227tr(\u0398)]<\/p>\n<p>\u5176\u4e2d \u0398\u662f\u66f2\u73872-\u5f62\u5f0f<\/p>\n<h3>\u4e03\u3001\u62d3\u6251\u91cf\u5b50\u573a\u8bba\u65b9\u7a0b<\/h3>\n<h4>1. Donaldson-Witten\u7406\u8bba<\/h4>\n<p>\u914d\u5206\u51fd\u6570&#xff1a;<\/p>\n<p>ZDW\u200b(M)&#061;k\u2211\u200bqk\u222bMk\u200b\u200b1<\/p>\n<p>\u5176\u4e2d q&#061;e2\u03c0i\u03c4&#xff0c;\u03c4\u662f\u8026\u5408\u5e38\u6570<\/p>\n<p>\u7b80\u5355\u578b\u6d41\u5f62&#xff1a;<\/p>\n<p>ZDW\u200b(M)&#061;21&#043;47\u03c7&#043;11\u03c3\u200bx\u2211\u200bSWM\u200b(x)e2i\u03c0(\u03bb0\u200b\u22c5x&#043;x2)<\/p>\n<h4>2. Seiberg-Witten\u7406\u8bba<\/h4>\n<p>\u4f4e\u80fd\u6709\u6548\u4f5c\u7528\u91cf&#xff1a;<\/p>\n<p>Leff\u200b&#061;Im\u03c4(a)\u2223da\u22232&#043;Re\u03c4(a)F\u2227F<\/p>\n<p>\u5176\u4e2d a\u662f\u6807\u91cf\u573a&#xff0c;\u03c4(a)\u662f\u6709\u6548\u8026\u5408\u5e38\u6570<\/p>\n<p>\u6a21\u7a7a\u95f4\u5947\u70b9&#xff1a;<\/p>\n<p>\u5f53 u&#061;\u27e8tr\u03d52\u27e9&#061;\u00b1\u039b2\u65f6&#xff0c;\u51fa\u73b0\u5947\u70b9<\/p>\n<h3>\u516b\u3001\u6d41\u5f62\u5206\u89e3\u65b9\u7a0b<\/h3>\n<h4>1. \u8fde\u901a\u548c\u516c\u5f0f<\/h4>\n<p>\u6b27\u62c9\u793a\u6027\u6570&#xff1a;<\/p>\n<p>\u03c7(M#N)&#061;\u03c7(M)&#043;\u03c7(N)\u22122<\/p>\n<p>\u7b26\u53f7\u5dee&#xff1a;<\/p>\n<p>\u03c4(M#N)&#061;\u03c4(M)&#043;\u03c4(N)<\/p>\n<p>\u8d1d\u8482\u6570&#xff1a;<\/p>\n<p>b1\u200b(M#N)&#061;b1\u200b(M)&#043;b1\u200b(N)<\/p>\n<p>b2&#043;\u200b(M#N)&#061;b2&#043;\u200b(M)&#043;b2&#043;\u200b(N)<\/p>\n<p>b2\u2212\u200b(M#N)&#061;b2\u2212\u200b(M)&#043;b2\u2212\u200b(N)<\/p>\n<h4>2. \u7ea4\u7ef4\u4e1b\u516c\u5f0f<\/h4>\n<p>Leray-Serre\u8c31\u5e8f\u5217&#xff1a;<\/p>\n<p>\u5bf9\u7ea4\u7ef4\u4e1b F\u2192E\u2192B&#xff0c;\u6709<\/p>\n<p>\u03c7(E)&#061;\u03c7(F)\u03c7(B)<\/p>\n<p>\u9648\u7c7b\u516c\u5f0f&#xff1a;\u5bf9\u590d\u5411\u91cf\u4e1b E\u2192M&#xff0c;<\/p>\n<p>c(TE)&#061;c(TB)\u222ac(F)<\/p>\n<h3>\u4e5d\u3001\u5f17\u7f57\u8d1d\u5c3c\u4e4c\u65af\u6d41\u5f62\u65b9\u7a0b&#xff08;\u91cf\u5b50\u4e0a\u540c\u8c03&#xff09;<\/h3>\n<h4>1. WDVV\u65b9\u7a0b<\/h4>\n<p>\u5bf9\u5f17\u7f57\u8d1d\u5c3c\u4e4c\u65af\u6d41\u5f62&#xff0c;\u7ed3\u5408\u5f8b\u65b9\u7a0b&#xff1a;<\/p>\n<p>\u03a6ijk\u200b\u03b7kl\u03a6lmn\u200b&#061;\u03a6ijn\u200b\u03b7kl\u03a6lmk\u200b<\/p>\n<p>\u5176\u4e2d \u03a6ijk\u200b&#061;\u2202i\u200b\u2202j\u200b\u2202k\u200bF&#xff0c;F\u662f\u81ea\u7531\u80fd<\/p>\n<h4>2. \u91cf\u5b50\u4e0a\u540c\u8c03\u73af<\/h4>\n<p>\u7ed3\u6784\u5e38\u6570&#xff1a;<\/p>\n<p>\u03a6\u03b1\u200b\u2217\u03a6\u03b2\u200b&#061;\u03b3,A\u2211\u200bC\u03b1\u03b2\u03b3,A\u200b\u03a6\u03b3\u200bqA<\/p>\n<p>\u5176\u4e2d qA&#061;e2\u03c0i\u222bA\u200b\u03c9&#xff0c;A\u2208H2\u200b(M)<\/p>\n<h3>\u5341\u3001\u7279\u6b8a\u56db\u7ef4\u6d41\u5f62\u7684\u65b9\u7a0b<\/h3>\n<h4>1. K3\u66f2\u9762<\/h4>\n<p>\u62d3\u6251\u4e0d\u53d8\u91cf&#xff1a;<\/p>\n<p>\u03c7(K3)&#061;24,\u03c4(K3)&#061;\u221216,b2&#043;\u200b&#061;3,b2\u2212\u200b&#061;19<\/p>\n<p>c1\u200b&#061;0,c2\u200b&#061;24<\/p>\n<p>\u9648\u6570\u65b9\u7a0b&#xff1a;<\/p>\n<p>c12\u200b&#061;0,2\u03c7&#043;3\u03c4&#061;2\u22c524\u22123\u22c516&#061;48\u221248&#061;0<\/p>\n<h4>2. \u590d\u5c04\u5f71\u5e73\u9762 CP2<\/h4>\n<p>\u62d3\u6251\u4e0d\u53d8\u91cf&#xff1a;<\/p>\n<p>\u03c7(CP2)&#061;3,\u03c4(CP2)&#061;1,b2&#043;\u200b&#061;1,b2\u2212\u200b&#061;0<\/p>\n<p>c1\u200b&#061;3H,c2\u200b&#061;3H2<\/p>\n<p>\u5176\u4e2d H\u662f\u8d85\u5e73\u9762\u7c7b<\/p>\n<p>\u9648\u6570\u9a8c\u8bc1&#xff1a;<\/p>\n<p>c12\u200b&#061;9,2\u03c7&#043;3\u03c4&#061;2\u22c53&#043;3\u22c51&#061;9<\/p>\n<h4>3. \u4e58\u79ef\u66f2\u9762 \u03a3g\u200b\u00d7\u03a3h\u200b<\/h4>\n<p>\u6b27\u62c9\u793a\u6027\u6570&#xff1a;<\/p>\n<p>\u03c7(\u03a3g\u200b\u00d7\u03a3h\u200b)&#061;4(1\u2212g)(1\u2212h)<\/p>\n<p>\u7b26\u53f7\u5dee&#xff1a;<\/p>\n<p>\u03c4(\u03a3g\u200b\u00d7\u03a3h\u200b)&#061;0<\/p>\n<p>\u8d1d\u8482\u6570&#xff1a;<\/p>\n<p>b1\u200b&#061;2g&#043;2h,b2&#043;\u200b&#061;2gh&#043;1,b2\u2212\u200b&#061;2gh&#043;1<\/p>\n<h3>\u603b\u7ed3<\/h3>\n<p>\u56db\u7ef4\u5fae\u5206\u6d41\u5f62\u7684\u62d3\u6251\u5b66\u65b9\u7a0b\u6784\u6210\u4e86\u4e00\u4e2a\u4e30\u5bcc\u800c\u6df1\u523b\u7684\u6570\u5b66\u6846\u67b6&#xff0c;\u5176\u6838\u5fc3\u7279\u5f81\u5305\u62ec&#xff1a;<\/p>\n<li>\n<p>\u76f8\u4ea4\u5f62\u5f0f\u7684\u4e3b\u5bfc\u4f5c\u7528&#xff1a;Freedman\u5b9a\u7406\u8868\u660e&#xff0c;\u5355\u8fde\u901a\u56db\u7ef4\u62d3\u6251\u6d41\u5f62\u7684\u540c\u80da\u578b\u51e0\u4e4e\u5b8c\u5168\u7531\u76f8\u4ea4\u5f62\u5f0f\u51b3\u5b9a\u3002<\/p>\n<\/li>\n<li>\n<p>\u6307\u6807\u5b9a\u7406\u7684\u6838\u5fc3\u5730\u4f4d&#xff1a;Atiyah-Singer\u6307\u6807\u5b9a\u7406\u5c06\u62d3\u6251\u4e0d\u53d8\u91cf&#xff08;\u9648\u6570\u3001\u7b26\u53f7\u5dee&#xff09;\u4e0e\u5206\u6790\u4e0d\u53d8\u91cf&#xff08;\u7b97\u5b50\u6307\u6807&#xff09;\u8054\u7cfb\u8d77\u6765\u3002<\/p>\n<\/li>\n<li>\n<p>\u51e0\u4f55\u5206\u6790\u7684\u521a\u6027&#xff1a;Hitchin-Thorpe\u3001Bogomolov\u2013Miyaoka\u2013Yau\u7b49\u4e0d\u7b49\u5f0f\u7ed9\u51fa\u51e0\u4f55\u7ed3\u6784\u4e0e\u62d3\u6251\u4e0d\u53d8\u91cf\u4e4b\u95f4\u7684\u5f3a\u7ea6\u675f\u3002<\/p>\n<\/li>\n<li>\n<p>\u89c4\u8303\u7406\u8bba\u7684\u9769\u547d\u6027\u5f71\u54cd&#xff1a;Donaldson\u548cSeiberg-Witten\u7406\u8bba\u901a\u8fc7\u89c4\u8303\u7406\u8bba\u65b9\u7a0b\u6784\u9020\u4e86\u5fae\u5206\u62d3\u6251\u4e0d\u53d8\u91cf&#xff0c;\u63ed\u793a\u4e86\u56db\u7ef4\u5149\u6ed1\u7ed3\u6784\u7684\u4e30\u5bcc\u6027\u3002<\/p>\n<\/li>\n<li>\n<p>\u4f8b\u5916\u7ef4\u5ea6&#xff1a;\u56db\u7ef4\u662f\u552f\u4e00\u5b58\u5728&#034;\u602a\u5f02&#034;\u5fae\u5206\u7ed3\u6784\u7684\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u7ef4\u5ea6&#xff0c;\u4e5f\u662f\u552f\u4e00\u65e2\u6709\u4e30\u5bcc\u62d3\u6251\u7ed3\u6784\u53c8\u9002\u7528\u4e8e\u7269\u7406\u91cf\u5b50\u573a\u8bba\u7684\u7ef4\u5ea6\u3002<\/p>\n<\/li>\n<p>\u8fd9\u4e9b\u65b9\u7a0b\u4e0d\u4ec5\u662f\u7eaf\u6570\u5b66\u7684\u6838\u5fc3&#xff0c;\u4e5f\u4e0e\u7406\u8bba\u7269\u7406&#xff08;\u7279\u522b\u662f\u5f26\u8bba\u548cM-\u7406\u8bba&#xff09;\u6709\u7740\u6df1\u523b\u8054\u7cfb&#xff0c;\u4f53\u73b0\u4e86\u6570\u5b66\u4e0e\u7269\u7406\u5728\u56db\u7ef4\u51e0\u4f55\u62d3\u6251\u4e2d\u7684\u7f8e\u5999\u7edf\u4e00\u3002<\/p>\n<p>\u5173\u4e8e\u56db\u7ef4\u5fae\u5206\u6d41\u5f62\u7684\u62d3\u6251\u5b66\u65b9\u7a0b\u7684\u6df1\u5165\u3001\u7cfb\u7edf\u6027\u7684\u9610\u8ff0\u3002\u56db\u7ef4\u62d3\u6251\u662f\u73b0\u4ee3\u51e0\u4f55\u4e0e\u62d3\u6251\u5b66\u7684\u6838\u5fc3\u524d\u6cbf&#xff0c;\u4ee5\u5176\u72ec\u7279\u7684\u590d\u6742\u6027\u548c\u4e30\u5bcc\u6027\u800c\u95fb\u540d&#xff08;\u4f8b\u5982\u201c\u56db\u7ef4\u662f\u7279\u6b8a\u7684\u201d\u8fd9\u4e00\u8457\u540d\u8bba\u65ad&#xff09;\u3002<\/p>\n<hr \/>\n<h4>\u6838\u5fc3\u601d\u60f3&#xff1a;\u8fde\u63a5\u51e0\u4f55\u3001\u5206\u6790\u4e0e\u62d3\u6251\u7684\u6865\u6881<\/h4>\n<p>\u5728\u56db\u7ef4\u5fae\u5206\u6d41\u5f62\u7684\u7814\u7a76\u4e2d&#xff0c;\u62d3\u6251\u5b66\u65b9\u7a0b\u901a\u5e38\u6307\u90a3\u4e9b\u6e90\u4e8e\u51e0\u4f55\u6216\u7269\u7406\u3001\u5176\u89e3\u7a7a\u95f4&#xff08;\u6a21\u7a7a\u95f4&#xff09;\u7684\u62d3\u6251\u6027\u8d28\u80fd\u63ed\u793a\u6d41\u5f62\u5e95\u5c42\u62d3\u6251\u4fe1\u606f\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u5176\u57fa\u672c\u8303\u5f0f\u662f&#xff1a;<\/p>\n<li>\n<p>\u5728\u5149\u6ed1\u56db\u7ef4\u6d41\u5f62 M\u4e0a\u5b9a\u4e49\u4e00\u4e2a\u975e\u7ebf\u6027\u504f\u5fae\u5206\u65b9\u7a0b\u3002<\/p>\n<\/li>\n<li>\n<p>\u8be5\u65b9\u7a0b\u7684\u89e3\u6784\u6210\u4e00\u4e2a\u6709\u9650\u7684\u3001\u53ef\u5b9a\u5411\u7684\u3001\u7d27\u7684\u201c\u6a21\u7a7a\u95f4\u201d M\u3002<\/p>\n<\/li>\n<li>\n<p>\u8ba1\u7b97\u8fd9\u4e2a\u6a21\u7a7a\u95f4 M\u7684\u4ee3\u6570\u62d3\u6251\u4e0d\u53d8\u91cf&#xff08;\u5982\u8ba1\u6570\u5176\u5e26\u7b26\u53f7\u7684\u70b9\u6570&#xff0c;\u5373\u201c\u6a21\u7a7a\u95f4\u7ef4\u6570\u4e3a\u96f6\u201d\u65f6\u7684\u60c5\u5f62&#xff09;\u3002<\/p>\n<\/li>\n<li>\n<p>\u8fd9\u4e9b\u4e0d\u53d8\u91cf\u6210\u4e3a\u539f\u59cb\u6d41\u5f62 M\u7684\u5fae\u5206\u62d3\u6251\u4e0d\u53d8\u91cf&#xff0c;\u4ece\u800c\u533a\u5206\u4e0d\u540c\u7684\u5149\u6ed1\u7ed3\u6784\u3002<\/p>\n<\/li>\n<hr \/>\n<h4>\u5173\u952e\u65b9\u7a0b\u4f53\u7cfb&#xff1a;\u4ece\u7ecf\u5178\u5230\u73b0\u4ee3<\/h4>\n<p>\u4ee5\u4e0b\u662f\u56db\u7ef4\u62d3\u6251\u4e2d\u6700\u91cd\u8981\u7684\u51e0\u7c7b\u65b9\u7a0b&#xff0c;\u5b83\u4eec\u5171\u540c\u6784\u6210\u4e86\u8be5\u9886\u57df\u7684\u652f\u67f1\u3002<\/p>\n<h5>1. \u9ece\u66fc\u51e0\u4f55\u4e0e\u7231\u56e0\u65af\u5766\u65b9\u7a0b<\/h5>\n<p>\u8fd9\u662f\u7406\u89e3\u6d41\u5f62\u5185\u8574\u51e0\u4f55\u7684\u57fa\u7840&#xff0c;\u4e0d\u9650\u4e8e\u56db\u7ef4&#xff0c;\u4f46\u5728\u56db\u7ef4\u65f6\u4e0e\u7269\u7406\u6df1\u523b\u5173\u8054\u3002<\/p>\n<ul>\n<li>\n<p>\u65b9\u7a0b&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u7231\u56e0\u65af\u5766\u65b9\u7a0b&#xff08;\u771f\u7a7a&#xff09;&#xff1a;<\/p>\n<p>Ric(g) &#061; \u03bb g<\/p>\n<p>\u5176\u4e2d g\u662f\u6d1b\u4f26\u5179\u5ea6\u89c4&#xff0c;Ric\u662f\u91cc\u5947\u66f2\u7387\u5f20\u91cf&#xff0c;\u03bb\u662f\u5b87\u5b99\u5e38\u6570\u3002\u5728\u9ece\u66fc\u51e0\u4f55\u80cc\u666f\u4e0b&#xff0c;\u6211\u4eec\u5e38\u8003\u8651\u7231\u56e0\u65af\u5766\u6d41\u5f62\u7684\u7248\u672c&#xff0c;\u5176\u4e2d g\u662f\u6b63\u5b9a\u5ea6\u89c4\u3002<\/p>\n<\/li>\n<li>\n<p>\u7b49\u53c2\u6570\u91cf\u66f2\u7387\u65b9\u7a0b&#xff1a; \u4e00\u4e2a\u4e0e\u4e4b\u76f8\u5173\u7684\u6807\u91cf\u65b9\u7a0b&#xff0c;\u7814\u7a76\u5728\u5171\u5f62\u7c7b\u4e2d\u5bfb\u627e\u5e38\u6570\u6807\u91cf\u66f2\u7387\u5ea6\u91cf\u7684\u95ee\u9898\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u62d3\u6251\u610f\u4e49&#xff1a; \u89e3\u7684\u5b58\u5728\u6027&#xff08;\u5373\u7231\u56e0\u65af\u5766\u5ea6\u91cf\u7684\u5b58\u5728\u6027&#xff09;\u5f3a\u70c8\u5236\u7ea6\u7740\u6d41\u5f62\u7684\u5fae\u5206\u62d3\u6251\u3002\u4f8b\u5982&#xff0c;\u8bb8\u591a\u56db\u7ef4\u6d41\u5f62&#xff08;\u5982 K3\u66f2\u9762&#xff09;\u5141\u8bb8\u7231\u56e0\u65af\u5766\u5ea6\u91cf&#xff0c;\u8fd9\u4e0e\u5176\u4e30\u5bcc\u7684\u51e0\u4f55\u7ed3\u6784\u76f8\u5173\u3002\u6807\u91cf\u66f2\u7387\u7684\u7b26\u53f7\u200b \u4e0e\u6d41\u5f62\u7684\u62d3\u6251\u6709\u6df1\u523b\u8054\u7cfb&#xff08;\u5982Gromov-Lawson\u5b9a\u7406&#xff09;\u3002<\/p>\n<\/li>\n<\/ul>\n<h5>2. \u89c4\u8303\u7406\u8bba\u4e0e\u6768-\u7c73\u5c14\u65af\u65b9\u7a0b<\/h5>\n<p>\u8fd9\u662f\u56db\u7ef4\u5fae\u5206\u62d3\u6251\u9769\u547d\u7684\u8d77\u6e90&#xff0c;\u7531\u897f\u8499\u00b7\u5510\u7eb3\u68ee\u7b49\u4eba\u5f00\u521b\u3002\u5b83\u5c06\u7269\u7406\u4e2d\u7684\u89c4\u8303\u573a\u8bba\u8f6c\u5316\u4e3a\u5f3a\u6709\u529b\u7684\u6570\u5b66\u5de5\u5177\u3002<\/p>\n<ul>\n<li>\n<p>\u6570\u5b66\u5bf9\u8c61&#xff1a; \u5728\u4e00\u4e2a\u4e3bG-\u4e1b P\u4e0a\u7684\u8054\u7edc&#xff08;\u89c4\u8303\u573a&#xff09;\u200b A&#xff0c;\u5176\u66f2\u7387\u4e3a F_A\u3002<\/p>\n<\/li>\n<li>\n<p>\u65b9\u7a0b&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u6768-\u7c73\u5c14\u65af\u65b9\u7a0b&#xff1a;<\/p>\n<p>d_A * F_A &#061; 0<\/p>\n<p>\u5176\u4e2d d_A\u662f\u534f\u53d8\u5916\u5fae\u5206&#xff0c;*\u662f\u970d\u5947\u661f\u7b97\u5b50\u3002\u8fd9\u662f\u53d8\u5206\u539f\u7406&#xff08;\u6768-\u7c73\u5c14\u65af\u4f5c\u7528\u91cf\u6781\u5c0f&#xff09;\u200b \u7684\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u3002<\/p>\n<\/li>\n<li>\n<p>\u81ea\u5bf9\u5076\/\u53cd\u81ea\u5bf9\u5076\u65b9\u7a0b&#xff08;ASD\u65b9\u7a0b&#xff09;&#xff1a;<\/p>\n<p>F_A &#061; \u00b1 * F_A<\/p>\n<p>\u8fd9\u662f\u6768-\u7c73\u5c14\u65af\u65b9\u7a0b\u5728\u56db\u7ef4\u6d41\u5f62\u4e0a\u7684\u4e00\u7c7b\u7279\u6b8a\u89e3&#xff08;\u77ac\u5b50&#xff09;&#xff0c;\u7531\u4e8e\u56db\u7ef4\u7a7a\u95f4\u970d\u661f\u7b97\u5b50\u6ee1\u8db3 *\u00b2 &#061; 1\u3002\u8fd9\u4e2a\u65b9\u7a0b\u662f\u5510\u7eb3\u68ee\u7406\u8bba\u7684\u6838\u5fc3\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u62d3\u6251\u610f\u4e49&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5510\u7eb3\u68ee\u4e0d\u53d8\u91cf&#xff1a; \u901a\u8fc7\u8ba1\u7b97\u81ea\u5bf9\u5076\u77ac\u5b50\u7684\u6a21\u7a7a\u95f4\u7684\u62d3\u6251\u6027\u8d28&#xff08;\u8ba1\u6570&#xff09;\u6765\u5b9a\u4e49\u3002\u5b83\u4eec\u662f\u533a\u5206\u56db\u7ef4\u6d41\u5f62\u5149\u6ed1\u7ed3\u6784\u7684\u5212\u65f6\u4ee3\u5de5\u5177\u3002\u4f8b\u5982&#xff0c;\u5510\u7eb3\u68ee\u7406\u8bba\u8bc1\u660e\u4e86\u5728 R\u2074\u4e0a\u5b58\u5728\u4e0d\u53ef\u6570\u65e0\u7a77\u591a\u79cd\u4e92\u4e0d\u76f8\u540c\u7684\u5fae\u5206\u7ed3\u6784&#xff08;\u201c\u602a\u5f02 R\u2074\u201d&#xff09;\u3002<\/p>\n<\/li>\n<li>\n<p>\u6a21\u7a7a\u95f4\u7684\u62d3\u6251&#xff1a; \u77ac\u5b50\u6a21\u7a7a\u95f4\u672c\u8eab\u7684\u62d3\u6251&#xff08;\u5982\u5b83\u7684\u76f8\u4ea4\u7406\u8bba\u3001\u4e0a\u540c\u8c03\u73af&#xff09;\u7f16\u7801\u4e86\u539f\u59cb\u6d41\u5f62 M\u6df1\u523b\u7684\u62d3\u6251\u4fe1\u606f\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h5>3. \u585e\u4f2f\u683c-\u5a01\u6ed5\u65b9\u7a0b<\/h5>\n<p>\u8fd9\u662f\u4e00\u7ec4\u66f4\u4e3a\u4f18\u96c5\u7684\u65b9\u7a0b&#xff0c;\u572820\u4e16\u7eaa90\u5e74\u4ee3\u7531\u7231\u5fb7\u534e\u00b7\u5a01\u6ed5\u7b49\u4eba\u4ece\u7269\u7406\u4e2d\u7684\u8d85\u5bf9\u79f0\u89c4\u8303\u7406\u8bba\u5bfc\u51fa&#xff0c;\u6781\u5927\u5730\u7b80\u5316\u548c\u5f3a\u5316\u4e86\u56db\u7ef4\u62d3\u6251\u7406\u8bba\u3002<\/p>\n<ul>\n<li>\n<p>\u6570\u5b66\u5bf9\u8c61&#xff1a;<\/p>\n<ul>\n<li>\n<p>Spin^c\u7ed3\u6784&#xff1a; \u5728\u56db\u7ef4\u6d41\u5f62\u4e0a&#xff0c;\u8fd9\u662f\u4e00\u4e2a\u6bd4\u65cb\u91cf\u7ed3\u6784\u66f4\u5f31\u3001\u603b\u662f\u5b58\u5728\u7684\u7ed3\u6784\u3002\u5b83\u7531\u4e00\u4e2a U(1)-\u4e1b L\u53ca\u5176\u8054\u7edc A\u63cf\u8ff0\u3002<\/p>\n<\/li>\n<li>\n<p>\u65cb\u91cf\u573a&#xff1a; \u4e00\u5bf9\u590d\u65cb\u91cf\u573a (\u03c6, \u03c8)\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u65b9\u7a0b&#xff1a;<\/p>\n<p>&#096;{ D_A \u03c6 &#061; 0<\/p>\n<p>F_A^&#043; &#061; i \u03c3(\u03c6) }&#096;<\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\n<p>D_A\u662f\u72c4\u62c9\u514b\u7b97\u5b50\u3002<\/p>\n<\/li>\n<li>\n<p>F_A^&#043;\u662f\u8054\u7edc A\u7684\u66f2\u7387\u7684\u81ea\u5bf9\u5076\u90e8\u5206\u3002<\/p>\n<\/li>\n<li>\n<p>\u03c3(\u03c6)\u662f\u4e00\u4e2a\u7531\u65cb\u91cf\u573a \u03c6\u7684\u4e8c\u6b21\u578b\u6784\u9020\u7684&#xff08;2,0)-\u5f62\u5f0f\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u62d3\u6251\u610f\u4e49&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u585e\u4f2f\u683c-\u5a01\u6ed5\u4e0d\u53d8\u91cf&#xff1a; \u540c\u6837\u901a\u8fc7\u8ba1\u6570\u8fd9\u4e2a\u65b9\u7a0b\u7684\u89e3&#xff08;\u6a21\u6389\u89c4\u8303\u5bf9\u79f0\u6027&#xff09;\u6765\u5b9a\u4e49\u3002\u5b83\u4eec\u6bd4\u5510\u7eb3\u68ee\u4e0d\u53d8\u91cf\u8ba1\u7b97\u4e0a\u66f4\u6613\u5904\u7406&#xff0c;\u4f46\u5728\u63ed\u793a\u5149\u6ed1\u7ed3\u6784\u65b9\u9762\u540c\u6837\u5f3a\u5927\u3002<\/p>\n<\/li>\n<li>\n<p>\u4e0e\u5510\u7eb3\u68ee\u4e0d\u53d8\u91cf\u7684\u5173\u7cfb&#xff1a; \u4e24\u8005\u901a\u8fc7\u201c\u7b80\u5355\u578b\u731c\u60f3\u201d\u200b &#xff08;\u5df2\u88ab\u8bc1\u660e\u4e3a\u5b9a\u7406&#xff09;\u6df1\u523b\u8054\u7cfb&#xff0c;\u5171\u540c\u6784\u6210\u4e86\u5bf9\u56db\u7ef4\u6d41\u5f62\u5149\u6ed1\u7ed3\u6784\u7684\u5f3a\u5927\u7ea6\u675f\u3002SW\u65b9\u7a0b\u80fd\u63a8\u51fa\u8bb8\u591a\u5510\u7eb3\u68ee\u7406\u8bba\u7684\u7ed3\u679c&#xff0c;\u4e14\u8bc1\u660e\u66f4\u7b80\u6d01\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h5>4. \u683c\u7f57\u83ab\u592b-\u5a01\u6ed5\u7406\u8bba\u4e0e\u91cf\u5b50\u4e0a\u540c\u8c03<\/h5>\n<p>\u8fd9\u662f\u4ece\u8f9b\u51e0\u4f55\u548c\u4ee3\u6570\u51e0\u4f55\u89d2\u5ea6\u7814\u7a76\u56db\u7ef4&#xff08;\u53ca\u66f4\u9ad8\u7ef4&#xff09;\u6d41\u5f62\u7684\u7406\u8bba&#xff0c;\u6838\u5fc3\u662f\u8ba1\u6570\u6d41\u5f62\u4e2d\u7684\u4f2a\u5168\u7eaf\u66f2\u7ebf\u3002<\/p>\n<ul>\n<li>\n<p>\u65b9\u7a0b&#xff1a; \u5168\u7eaf\u66f2\u7ebf\u65b9\u7a0b&#xff08;\u975e\u7ebf\u6027\u67ef\u897f-\u9ece\u66fc\u65b9\u7a0b&#xff09;<\/p>\n<p>\u2202\u0304_J u &#061; 0<\/p>\n<p>\u5176\u4e2d u\u662f\u4ece\u9ece\u66fc\u9762&#xff08;\u5982\u7403\u9762&#xff09;\u6620\u5c04\u5230\u76ee\u6807\u8f9b\u6d41\u5f62 M\u7684\u6620\u5c04&#xff0c;\u2202\u0304_J\u662f\u5173\u4e8e M\u4e0a\u6b86\u590d\u7ed3\u6784 J\u7684\u67ef\u897f-\u9ece\u66fc\u7b97\u5b50\u3002<\/p>\n<\/li>\n<li>\n<p>\u62d3\u6251\u610f\u4e49&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u683c\u7f57\u83ab\u592b-\u5a01\u6ed5\u4e0d\u53d8\u91cf&#xff1a; \u901a\u8fc7\u8ba1\u6570\u6ee1\u8db3\u7279\u5b9a\u540c\u8c03\u6761\u4ef6\u7684\u4f2a\u5168\u7eaf\u66f2\u7ebf\u7684\u6570\u76ee&#xff08;\u6a21\u53bb\u53c2\u6570\u5316&#xff09;\u6765\u5b9a\u4e49\u3002\u5b83\u662f\u8f9b\u6d41\u5f62\u7684\u5fae\u5206\u4e0d\u53d8\u91cf\u3002<\/p>\n<\/li>\n<li>\n<p>\u5728\u56db\u7ef4\u7684\u5e94\u7528&#xff1a; \u5f53\u56db\u7ef4\u6d41\u5f62\u662f\u8f9b\u6d41\u5f62\u65f6&#xff08;\u5982\u4ee3\u6570\u66f2\u9762&#xff09;&#xff0c;GW\u4e0d\u53d8\u91cf\u63d0\u4f9b\u4e86\u5f3a\u5927\u7684\u5206\u7c7b\u5de5\u5177\u3002\u5b83\u4eec\u4e0eSW\u4e0d\u53d8\u91cf\u6709\u6df1\u523b\u7684\u8054\u7cfb&#xff08;Taubes\u5b9a\u7406&#xff09;\u3002<\/p>\n<\/li>\n<li>\n<p>\u91cf\u5b50\u4e0a\u540c\u8c03\u73af&#xff1a; GW\u4e0d\u53d8\u91cf\u5b9a\u4e49\u4e86\u6d41\u5f62\u4e0a\u540c\u8c03\u73af\u7684\u4e00\u4e2a\u91cf\u5b50\u5f62\u53d8&#xff0c;\u7f16\u7801\u4e86\u6d41\u5f62\u7684\u201c\u91cf\u5b50\u201d\u62d3\u6251\u4fe1\u606f\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr \/>\n<h4 style=\"background-color:transparent\">\u603b\u7ed3\u5bf9\u6bd4\u4e0e\u6846\u67b6<\/h4>\n<p>\u4e0b\u8868\u603b\u7ed3\u4e86\u56db\u7ef4\u5fae\u5206\u6d41\u5f62\u62d3\u6251\u5b66\u4e2d\u7684\u6838\u5fc3\u65b9\u7a0b&#xff1a;<\/p>\n<table>\n<tr>\n<p>\u65b9\u7a0b\u4f53\u7cfb<\/p>\n<p>\u8d77\u6e90\u9886\u57df<\/p>\n<p>\u6838\u5fc3\u65b9\u7a0b<\/p>\n<p>\u6838\u5fc3\u6570\u5b66\u5bf9\u8c61<\/p>\n<p>\u5b9a\u4e49\u7684\u4e0d\u53d8\u91cf<\/p>\n<p>\u4e3b\u8981\u62d3\u6251\u610f\u4e49<\/p>\n<\/tr>\n<tbody>\n<tr>\n<td>\n<p>\u7231\u56e0\u65af\u5766\u65b9\u7a0b\u200b<\/p>\n<\/td>\n<td>\n<p>\u5e7f\u4e49\u76f8\u5bf9\u8bba\/\u9ece\u66fc\u51e0\u4f55<\/p>\n<\/td>\n<td>\n<p>Ric(g) &#061; \u03bbg<\/p>\n<\/td>\n<td>\n<p>\u9ece\u66fc\/\u6d1b\u4f26\u5179\u5ea6\u89c4 g<\/p>\n<\/td>\n<td>\n<p>\u7231\u56e0\u65af\u5766\u5ea6\u91cf\u7684\u5b58\u5728\u6027<\/p>\n<\/td>\n<td>\n<p>\u8054\u7cfb\u51e0\u4f55\u4e0e\u62d3\u6251&#xff0c;\u5236\u7ea6\u5fae\u5206\u7ed3\u6784<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u6768-\u7c73\u5c14\u65af\u65b9\u7a0b (ASD)\u200b<\/p>\n<\/td>\n<td>\n<p>\u89c4\u8303\u573a\u8bba&#xff08;\u7269\u7406\/\u6570\u5b66&#xff09;<\/p>\n<\/td>\n<td>\n<p>F_A &#061; \u00b1*F_A<\/p>\n<\/td>\n<td>\n<p>\u4e3b\u4e1b\u8054\u7edc A&#xff0c; \u66f2\u7387 F_A<\/p>\n<\/td>\n<td>\n<p>\u5510\u7eb3\u68ee\u4e0d\u53d8\u91cf\u200b<\/p>\n<\/td>\n<td>\n<p>\u9769\u547d\u6027\u7a81\u7834&#xff0c;\u533a\u5206\u5149\u6ed1\u7ed3\u6784&#xff0c;\u53d1\u73b0\u602a\u5f02R\u2074<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u585e\u4f2f\u683c-\u5a01\u6ed5\u65b9\u7a0b\u200b<\/p>\n<\/td>\n<td>\n<p>\u8d85\u5bf9\u79f0\u89c4\u8303\u573a\u8bba<\/p>\n<\/td>\n<td>\n<p>D_A \u03c6&#061;0, F_A^&#043;&#061;i\u03c3(\u03c6)<\/p>\n<\/td>\n<td>\n<p>Spin^c\u8054\u7edc A, \u65cb\u91cf\u573a \u03c6<\/p>\n<\/td>\n<td>\n<p>\u585e\u4f2f\u683c-\u5a01\u6ed5\u4e0d\u53d8\u91cf\u200b<\/p>\n<\/td>\n<td>\n<p>\u4e0e\u5510\u7eb3\u68ee\u4e0d\u53d8\u91cf\u7b49\u4ef7\u4f46\u66f4\u6613\u8ba1\u7b97&#xff0c;\u5206\u7c7b\u5149\u6ed1\u7ed3\u6784<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u5168\u7eaf\u66f2\u7ebf\u65b9\u7a0b\u200b<\/p>\n<\/td>\n<td>\n<p>\u8f9b\u51e0\u4f55\/\u590d\u51e0\u4f55<\/p>\n<\/td>\n<td>\n<p>\u2202\u0304_J u &#061; 0<\/p>\n<\/td>\n<td>\n<p>\u4f2a\u5168\u7eaf\u66f2\u7ebf u<\/p>\n<\/td>\n<td>\n<p>\u683c\u7f57\u83ab\u592b-\u5a01\u6ed5\u4e0d\u53d8\u91cf\u200b<\/p>\n<\/td>\n<td>\n<p>\u8f9b\u6d41\u5f62\u7684\u4e0d\u53d8\u91cf&#xff0c;\u4e0eSW\u4e0d\u53d8\u91cf\u76f8\u8fde&#xff0c;\u7814\u7a76\u91cf\u5b50\u4e0a\u540c\u8c03<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u903b\u8f91\u6846\u67b6\u56fe<\/h4>\n<p>\u56db\u7ef4\u5149\u6ed1\u6d41\u5f62 M<br \/>\n    \u251c\u2500\u2500 \u9ece\u66fc\u51e0\u4f55\u89c2\u70b9<br \/>\n    \u2502       \u251c\u2500\u2500 \u5bfb\u627e\u7279\u6b8a\u5ea6\u91cf&#xff08;\u7231\u56e0\u65af\u5766\u65b9\u7a0b&#xff09;<br \/>\n    \u2502       \u2514\u2500\u2500 \u62d3\u6251\u7ea6\u675f&#xff08;\u5982\u6807\u91cf\u66f2\u7387\u7b26\u53f7\u5b9a\u7406&#xff09;<br \/>\n    \u2502<br \/>\n    \u251c\u2500\u2500 \u89c4\u8303\u7406\u8bba\u89c2\u70b9 (\u4e3b\u8981\u5de5\u5177)<br \/>\n    \u2502       \u251c\u2500\u2500 \u6784\u9020\u4e3b\u4e1b\u53ca\u5176\u8054\u7edc<br \/>\n    \u2502       \u2502       \u251c\u2500\u2500 \u89e3 ASD \u65b9\u7a0b \u2192 \u5510\u7eb3\u68ee\u7406\u8bba<br \/>\n    \u2502       \u2502       \u2514\u2500\u2500 \u89e3 SW \u65b9\u7a0b \u2192 \u585e\u4f2f\u683c-\u5a01\u6ed5\u7406\u8bba<br \/>\n    \u2502       \u2514\u2500\u2500 \u7814\u7a76\u89e3\u7a7a\u95f4&#xff08;\u6a21\u7a7a\u95f4&#xff09;\u7684\u62d3\u6251<br \/>\n    \u2502               \u2514\u2500\u2500 \u63d0\u53d6 M \u7684\u5fae\u5206\u62d3\u6251\u4e0d\u53d8\u91cf<br \/>\n    \u2502<br \/>\n    \u2514\u2500\u2500 \u8f9b\u51e0\u4f55\/\u4ee3\u6570\u51e0\u4f55\u89c2\u70b9 (\u82e5 M \u5177\u6709\u76f8\u5e94\u7ed3\u6784)<br \/>\n            \u2514\u2500\u2500 \u89e3\u4f2a\u5168\u7eaf\u66f2\u7ebf\u65b9\u7a0b<br \/>\n                    \u2514\u2192 \u683c\u7f57\u83ab\u592b-\u5a01\u6ed5\u7406\u8bba \u2192 \u91cf\u5b50\u4e0a\u540c\u8c03<\/p>\n<h4 style=\"background-color:transparent\">\u524d\u6cbf\u4e0e\u542f\u793a<\/h4>\n<li>\n<p>\u56db\u7ef4\u7684\u72ec\u7279\u6027&#xff1a; \u56db\u7ef4\u4e4b\u6240\u4ee5\u7279\u6b8a&#xff0c;\u662f\u56e0\u4e3a SO(4)\u4e0d\u662f\u5355\u674e\u7fa4&#xff0c;\u8fd9\u5bfc\u81f4\u4e86\u81ea\u5bf9\u5076\/\u53cd\u81ea\u5bf9\u5076\u5f62\u5f0f\u7684\u7279\u6b8a\u5730\u4f4d&#xff0c;\u4ece\u800c\u50ac\u751f\u4e86ASD\u65b9\u7a0b\u548cSW\u65b9\u7a0b\u3002\u9ad8\u7ef4\u6ca1\u6709\u5982\u6b64\u4e30\u5bcc\u7684\u81ea\u5bf9\u5076\u7406\u8bba\u3002<\/p>\n<\/li>\n<li>\n<p>\u201c\u602a\u5f02\u201d\u73b0\u8c61&#xff1a; \u8fd9\u4e9b\u65b9\u7a0b\u63ed\u793a\u4e86\u5728\u56db\u7ef4\u4e2d&#xff0c;\u62d3\u6251\u8303\u7574\u3001PL\u8303\u7574\u548c\u5fae\u5206\u8303\u7574\u622a\u7136\u4e0d\u540c&#xff08;\u5b58\u5728\u540c\u80da\u4f46\u4e0d\u5fae\u5206\u540c\u80da\u7684\u6d41\u5f62&#xff0c;\u5982\u602a\u5f02 R\u2074&#xff09;\u3002<\/p>\n<\/li>\n<li>\n<p>\u7269\u7406\u4e0e\u6570\u5b66\u7684\u5171\u751f&#xff1a; \u8fd9\u4e2a\u9886\u57df\u662f\u6570\u5b66\u4e0e\u7406\u8bba\u7269\u7406&#xff08;\u91cf\u5b50\u573a\u8bba\u3001\u5f26\u8bba&#xff09;\u6df1\u5ea6\u4ea4\u53c9\u7684\u5178\u8303\u3002\u5510\u7eb3\u68ee\u7406\u8bba\u6e90\u4e8e\u6768-\u7c73\u5c14\u65af\u7406\u8bba&#xff0c;SW\u7406\u8bba\u6e90\u4e8e\u8d85\u5bf9\u79f0&#xff0c;\u800c\u8fd1\u5e74\u6765\u66f4\u6709Kapustin-Witten\u65b9\u7a0b\u7b49\u6e90\u4e8e\u51e0\u4f55\u6717\u5170\u5179\u7eb2\u9886\u7684\u65b9\u7a0b\u5728\u56db\u7ef4\u62d3\u6251\u4e2d\u626e\u6f14\u65b0\u89d2\u8272\u3002<\/p>\n<\/li>\n<li>\n<p>\u672a\u89e3\u4e4b\u8c1c&#xff1a; \u6700\u8457\u540d\u7684\u5149\u6ed1\u56db\u7ef4\u5e9e\u52a0\u83b1\u731c\u60f3&#xff08;\u4e00\u4e2a\u540c\u4f26\u56db\u7ef4\u7403\u9762\u662f\u5426\u4e00\u5b9a\u5fae\u5206\u540c\u80da\u4e8e\u6807\u51c6\u56db\u7ef4\u7403\u9762&#xff1f;&#xff09;\u4ecd\u672a\u89e3\u51b3\u3002\u8fd9\u4e9b\u65b9\u7a0b\u662f\u653b\u51fb\u6b64\u7c7b\u96be\u9898\u7684\u4e3b\u8981\u6b66\u5668\u3002<\/p>\n<\/li>\n<p>\u603b\u800c\u8a00\u4e4b&#xff0c;\u56db\u7ef4\u5fae\u5206\u6d41\u5f62\u7684\u62d3\u6251\u5b66\u65b9\u7a0b\u5e76\u975e\u5b64\u7acb\u7684\u516c\u5f0f&#xff0c;\u800c\u662f\u4e00\u4e2a\u4ee5\u975e\u7ebf\u6027\u51e0\u4f55\u5206\u6790\u65b9\u7a0b&#xff08;ASD, SW, \u2202\u0304_J&#xff09;\u4e3a\u5de5\u5177&#xff0c;\u4ee5\u5176\u89e3\u7a7a\u95f4\u7684\u62d3\u6251\u4e3a\u6865\u6881&#xff0c;\u6765\u63a2\u6d4b\u56db\u7ef4\u6d41\u5f62\u672c\u8eab\u6700\u6df1\u5965\u7684\u5fae\u5206\u62d3\u6251\u7ed3\u6784\u7684\u5b8f\u4f1f\u7406\u8bba\u4f53\u7cfb\u3002\u5b83\u662f20\u4e16\u7eaa\u4e0b\u534a\u53f6\u4ee5\u6765\u6570\u5b66\u53f2\u4e0a\u6700\u6fc0\u52a8\u4eba\u5fc3\u7684\u7bc7\u7ae0\u4e4b\u4e00\u3002<\/p>\n<\/p>\n<h2>\u5f26\u8bba\u4e0eM-\u7406\u8bba&#xff1a;\u7edf\u4e00\u6846\u67b6<\/h2>\n<h3>\u4e00\u3001\u57fa\u672c\u5b9e\u4f53\u4e0e\u7ef4\u5ea6<\/h3>\n<table>\n<tr>\n<p>\u7406\u8bba<\/p>\n<p>\u7ef4\u5ea6<\/p>\n<p>\u57fa\u672c\u5b9e\u4f53<\/p>\n<p>\u8d85\u5bf9\u79f0<\/p>\n<p>\u4f4e\u80fd\u6781\u9650<\/p>\n<p>\u89c4\u8303\u7fa4<\/p>\n<\/tr>\n<tbody>\n<tr>\n<td>\n<p>\u73bb\u8272\u5f26\u200b<\/p>\n<\/td>\n<td>\n<p>26\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u95ed\u5f26\u3001\u5f00\u5f26<\/p>\n<\/td>\n<td>\n<p>\u65e0<\/p>\n<\/td>\n<td>\n<p>\u65e0<\/p>\n<\/td>\n<td>\n<p>\u53ef\u5177\u6709\u89c4\u8303\u5bf9\u79f0\u6027<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u2160\u578b\u8d85\u5f26\u200b<\/p>\n<\/td>\n<td>\n<p>10\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u5f00\u5f26&#043;\u95ed\u5f26<\/p>\n<\/td>\n<td>\n<p>N&#061;1<\/p>\n<\/td>\n<td>\n<p>10\u7ef4N&#061;1\u8d85\u5f15\u529b&#043;SO(32)<\/p>\n<\/td>\n<td>\n<p>SO(32)<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u2161A\u578b\u8d85\u5f26\u200b<\/p>\n<\/td>\n<td>\n<p>10\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u95ed\u5f26<\/p>\n<\/td>\n<td>\n<p>N&#061;2 (\u975e\u624b\u5f81)<\/p>\n<\/td>\n<td>\n<p>10\u7ef4N&#061;2A\u8d85\u5f15\u529b<\/p>\n<\/td>\n<td>\n<p>U(1)<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u2161B\u578b\u8d85\u5f26\u200b<\/p>\n<\/td>\n<td>\n<p>10\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u95ed\u5f26<\/p>\n<\/td>\n<td>\n<p>N&#061;2 (\u624b\u5f81)<\/p>\n<\/td>\n<td>\n<p>10\u7ef4N&#061;2B\u8d85\u5f15\u529b<\/p>\n<\/td>\n<td>\n<p>&#8211;<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u6742\u5316\u5f26SO(32)\u200b<\/p>\n<\/td>\n<td>\n<p>10\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u95ed\u5f26<\/p>\n<\/td>\n<td>\n<p>N&#061;1<\/p>\n<\/td>\n<td>\n<p>10\u7ef4N&#061;1\u8d85\u5f15\u529b&#043;SO(32)<\/p>\n<\/td>\n<td>\n<p>SO(32)<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>\u6742\u5316\u5f26E8\u00d7E8\u200b<\/p>\n<\/td>\n<td>\n<p>10\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u95ed\u5f26<\/p>\n<\/td>\n<td>\n<p>N&#061;1<\/p>\n<\/td>\n<td>\n<p>10\u7ef4N&#061;1\u8d85\u5f15\u529b&#043;E8\u00d7E8<\/p>\n<\/td>\n<td>\n<p>E8\u00d7E8<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>M-\u7406\u8bba\u200b<\/p>\n<\/td>\n<td>\n<p>11\u7ef4<\/p>\n<\/td>\n<td>\n<p>M2\u819c\u3001M5\u819c<\/p>\n<\/td>\n<td>\n<p>N&#061;1<\/p>\n<\/td>\n<td>\n<p>11\u7ef4\u8d85\u5f15\u529b<\/p>\n<\/td>\n<td>\n<p>&#8211;<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>F-\u7406\u8bba\u200b<\/p>\n<\/td>\n<td>\n<p>12\u7ef4<\/p>\n<\/td>\n<td>\n<p>\u692d\u5706\u7ea4\u7ef4\u5316<\/p>\n<\/td>\n<td>\n<p>&#8211;<\/p>\n<\/td>\n<td>\n<p>\u2161B\u5f26\u8bba\u7684\u975e\u5fae\u6270\u5b8c\u6210<\/p>\n<\/td>\n<td>\n<p>SL(2,Z)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>\u4e8c\u3001\u5f26\u8bba\u57fa\u672c\u65b9\u7a0b<\/h3>\n<h4>1. \u4e16\u754c\u9762\u4f5c\u7528\u91cf<\/h4>\n<h5>Polyakov\u4f5c\u7528\u91cf&#xff08;\u73bb\u8272\u5f26&#xff09;<\/h5>\n<p>SP\u200b&#061;\u22124\u03c0\u03b1\u20321\u200b\u222bd2\u03c3\u2212h\u200bh\u03b1\u03b2\u2202\u03b1\u200bX\u03bc\u2202\u03b2\u200bX\u03bd\u03b7\u03bc\u03bd\u200b<\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\n<p>X\u03bc(\u03c3,\u03c4): \u5f26\u5728D\u7ef4\u65f6\u7a7a\u4e2d\u7684\u5d4c\u5165<\/p>\n<\/li>\n<li>\n<p>h\u03b1\u03b2\u200b: \u4e16\u754c\u9762\u5ea6\u89c4 (\u03b1,\u03b2&#061;0,1)<\/p>\n<\/li>\n<li>\n<p>\u03b1\u2032&#061;ls2\u200b: \u5f26\u5f20\u529b\u53c2\u6570 (ls\u200b\u4e3a\u5f26\u5c3a\u5ea6)<\/p>\n<\/li>\n<\/ul>\n<h5>\u5171\u5f62\u89c4\u8303\u4e0b\u7684\u8fd0\u52a8\u65b9\u7a0b<\/h5>\n<p>\u5728\u5171\u5f62\u89c4\u8303 h\u03b1\u03b2\u200b&#061;e2\u03d5(\u03c3,\u03c4)\u03b7\u03b1\u03b2\u200b\u4e0b&#xff1a;<\/p>\n<p>(\u2202\u03c42\u200b\u2212\u2202\u03c32\u200b)X\u03bc&#061;0<\/p>\n<p>\u8fb9\u754c\u6761\u4ef6&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5f00\u5f26&#xff1a;\u8bfa\u4f0a\u66fc \u2202\u03c3\u200bX\u03bc\u2223\u03c3&#061;0,\u03c0\u200b&#061;0\u6216\u72c4\u5229\u514b\u96f7 X\u03bc\u2223\u03c3&#061;0,\u03c0\u200b&#061;\u5e38\u6570<\/p>\n<\/li>\n<li>\n<p>\u95ed\u5f26&#xff1a;\u5468\u671f\u6027 X\u03bc(\u03c3&#043;\u03c0,\u03c4)&#061;X\u03bc(\u03c3,\u03c4)<\/p>\n<\/li>\n<\/ul>\n<h4>2. \u8d85\u5f26\u7684Green-Schwarz\u4f5c\u7528\u91cf<\/h4>\n<p>S&#061;\u22124\u03c0\u03b1\u20321\u200b\u222bd2\u03c3(\u2212h\u200bh\u03b1\u03b2\u03a0\u03b1\u200b\u22c5\u03a0\u03b2\u200b&#043;2i\u03f5\u03b1\u03b2\u2202\u03b1\u200bX\u03bc(\u03b8\u02c91\u0393\u03bc\u200b\u2202\u03b2\u200b\u03b81\u2212\u03b8\u02c92\u0393\u03bc\u200b\u2202\u03b2\u200b\u03b82)\u22122\u03f5\u03b1\u03b2\u03b8\u02c91\u0393\u03bc\u2202\u03b1\u200b\u03b81\u03b8\u02c92\u0393\u03bc\u200b\u2202\u03b2\u200b\u03b82)<\/p>\n<p>\u5176\u4e2d \u03a0\u03b1\u03bc\u200b&#061;\u2202\u03b1\u200bX\u03bc\u2212i\u03b8\u02c9A\u0393\u03bc\u2202\u03b1\u200b\u03b8A&#xff0c;A&#061;1,2\u3002<\/p>\n<h4>3. \u91cf\u5b50\u5316\u4e0e\u632f\u52a8\u6a21\u5f0f<\/h4>\n<h5>\u95ed\u5f26\u6a21\u5f0f\u5c55\u5f00<\/h5>\n<p>X\u03bc(\u03c3,\u03c4)&#061;x\u03bc&#043;\u03b1\u2032p\u03bc\u03c4&#043;i2\u03b1\u2032\u200b\u200bn\ue020&#061;0\u2211\u200bn1\u200b(\u03b1n\u03bc\u200be\u22122in(\u03c4\u2212\u03c3)&#043;\u03b1~n\u03bc\u200be\u22122in(\u03c4&#043;\u03c3))<\/p>\n<p>\u5bf9\u6613\u5173\u7cfb&#xff1a;<\/p>\n<p>[\u03b1m\u03bc\u200b,\u03b1n\u03bd\u200b]&#061;m\u03b4m&#043;n,0\u200b\u03b7\u03bc\u03bd,[\u03b1~m\u03bc\u200b,\u03b1~n\u03bd\u200b]&#061;m\u03b4m&#043;n,0\u200b\u03b7\u03bc\u03bd,[\u03b1m\u03bc\u200b,\u03b1~n\u03bd\u200b]&#061;0<\/p>\n<p>\u8d28\u91cf\u58f3\u6761\u4ef6&#xff1a;<\/p>\n<p>M2&#061;\u03b1\u20322\u200b(N&#043;N~\u22122)(\u73bb\u8272\u5f26)<\/p>\n<p>\u6c34\u5e73\u5339\u914d\u6761\u4ef6&#xff1a;N&#061;N~<\/p>\n<h4>4. \u8d85\u5f26\u7684GSO\u6295\u5f71<\/h4>\n<p>\u4e3a\u4fdd\u8bc1\u65f6\u7a7a\u8d85\u5bf9\u79f0\u548c\u6d88\u9664\u5feb\u5b50&#xff0c;\u5f15\u5165Gliozzi-Scherk-Olive\u6295\u5f71&#xff1a;<\/p>\n<p>PGSO\u200b&#061;21\u200b(1\u2212(\u22121)F)<\/p>\n<p>\u5176\u4e2dF\u662f\u4e16\u754c\u9762\u8d39\u7c73\u5b50\u6570\u3002\u8fd9\u4fdd\u7559\u4e86\u65f6\u7a7a\u8d85\u5bf9\u79f0\u8868\u793a\u3002<\/p>\n<h3>\u4e09\u3001\u4f4e\u80fd\u6709\u6548\u7406\u8bba&#xff1a;\u8d85\u5f15\u529b<\/h3>\n<h4>1. \u2161A\u578b\u8d85\u5f15\u529b&#xff08;10\u7ef4&#xff0c;\u975e\u624b\u5f81&#xff09;<\/h4>\n<p>\u4f5c\u7528\u91cf&#xff1a;<\/p>\n<p>SIIA\u200b&#061;2\u03ba102\u200b1\u200b\u222bd10x\u2212G\u200be\u22122\u03d5(R&#043;4(\u2202\u03d5)2\u221221\u200b\u2223H3\u200b\u22232)\u22124\u03ba102\u200b1\u200b\u222bd10x\u2212G\u200b(\u2223F2\u200b\u22232&#043;\u2223F~4\u200b\u22232)\u22124\u03ba102\u200b1\u200b\u222bB2\u200b\u2227F4\u200b\u2227F4\u200b<\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\n<p>H3\u200b&#061;dB2\u200b(NS-NS 3-\u5f62\u5f0f\u573a\u5f3a)<\/p>\n<\/li>\n<li>\n<p>F2\u200b&#061;dC1\u200b, F4\u200b&#061;dC3\u200b(R-R \u573a\u5f3a)<\/p>\n<\/li>\n<li>\n<p>F~4\u200b&#061;F4\u200b\u2212C1\u200b\u2227H3\u200b<\/p>\n<\/li>\n<li>\n<p>\u03d5: \u4f38\u7f29\u5b50&#xff0c;\u5f26\u8026\u5408\u5e38\u6570 gs\u200b&#061;e\u27e8\u03d5\u27e9<\/p>\n<\/li>\n<\/ul>\n<h4>2. \u2161B\u578b\u8d85\u5f15\u529b&#xff08;10\u7ef4&#xff0c;\u624b\u5f81&#xff09;<\/h4>\n<p>SIIB\u200b&#061;2\u03ba102\u200b1\u200b\u222bd10x\u2212G\u200be\u22122\u03d5(R&#043;4(\u2202\u03d5)2\u221221\u200b\u2223H3\u200b\u22232)\u22124\u03ba102\u200b1\u200b\u222bd10x\u2212G\u200b(\u2223F1\u200b\u22232&#043;\u2223F~3\u200b\u22232&#043;21\u200b\u2223F~5\u200b\u22232)&#043;4\u03ba102\u200b1\u200b\u222bC4\u200b\u2227H3\u200b\u2227F3\u200b<\/p>\n<p>\u9644\u52a0\u81ea\u5bf9\u5076\u6761\u4ef6&#xff1a;F~5\u200b&#061;\u22c6F~5\u200b<\/p>\n<p>\u573a\u5f3a&#xff1a;<\/p>\n<ul>\n<li>\n<p>F1\u200b&#061;dC0\u200b(\u8f74\u5b50\u573a\u5f3a)<\/p>\n<\/li>\n<li>\n<p>F3\u200b&#061;dC2\u200b<\/p>\n<\/li>\n<li>\n<p>F5\u200b&#061;dC4\u200b<\/p>\n<\/li>\n<li>\n<p>F~3\u200b&#061;F3\u200b\u2212C0\u200bH3\u200b<\/p>\n<\/li>\n<li>\n<p>F~5\u200b&#061;F5\u200b\u221221\u200bC2\u200b\u2227H3\u200b&#043;21\u200bB2\u200b\u2227F3\u200b<\/p>\n<\/li>\n<\/ul>\n<h4>3. 11\u7ef4\u8d85\u5f15\u529b&#xff08;M-\u7406\u8bba\u4f4e\u80fd\u6781\u9650&#xff09;<\/h4>\n<p>S11\u200b&#061;2\u03ba112\u200b1\u200b\u222bd11x\u2212G\u200b(R\u221221\u200b\u2223F4\u200b\u22232)\u221212\u03ba112\u200b1\u200b\u222bC3\u200b\u2227F4\u200b\u2227F4\u200b<\/p>\n<p>\u5176\u4e2d F4\u200b&#061;dC3\u200b\u3002<\/p>\n<h3>\u56db\u3001\u5bf9\u5076\u6027\u7f51\u7edc<\/h3>\n<h4>1. T-\u5bf9\u5076<\/h4>\n<p>\u5728\u7d27\u81f4\u7ef4\u5ea6 X\u223cX&#043;2\u03c0R\u4e0a&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u534a\u5f84\u53d8\u6362&#xff1a;R\u2194\u03b1\u2032\/R<\/p>\n<\/li>\n<li>\n<p>\u52a8\u91cf\u6a21\u5f0f\u4e0e\u7f20\u7ed5\u6a21\u5f0f\u4ea4\u6362&#xff1a;n\u2194w<\/p>\n<\/li>\n<li>\n<p>\u5f26\u8026\u5408\u5e38\u6570\u53d8\u6362&#xff1a;gs\u200b\u2194gs\u200b\u03b1\u2032\u200b\/R<\/p>\n<\/li>\n<\/ul>\n<p>T-\u5bf9\u5076\u5bf9&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u2161A\u5728\u534a\u5f84R\u7684\u5706\u4e0a \u2194 \u2161B\u5728\u534a\u5f84\u03b1\u2032\/R\u7684\u5706\u4e0a<\/p>\n<\/li>\n<li>\n<p>\u6742\u5316E8\u00d7E8\u5728\u534a\u5f84R\u7684\u5706\u4e0a \u2194 \u6742\u5316SO(32)\u5728\u534a\u5f84\u03b1\u2032\/R\u7684\u5706\u4e0a<\/p>\n<\/li>\n<\/ul>\n<h4>2. S-\u5bf9\u5076<\/h4>\n<p>\u5f3a\u5f31\u8026\u5408\u5bf9\u5076&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u2161B\u81ea\u5bf9\u5076&#xff1a;gs\u200b\u21941\/gs\u200b&#xff0c;\u57fa\u672c\u5f26 \u2194 D1-\u5f26<\/p>\n<\/li>\n<li>\n<p>\u2160\u578b\u4e0e\u6742\u5316SO(32)&#xff1a;gs(I)\u200b\u21941\/gs(het)\u200b<\/p>\n<\/li>\n<li>\n<p>\u5f26\u8026\u5408\u5e38\u6570&#xff1a;\u03c4&#061;C0\u200b&#043;ie\u2212\u03d5\u2192\u22121\/\u03c4\u5728SL(2,Z)\u4e0b<\/p>\n<\/li>\n<\/ul>\n<h4>3. U-\u5bf9\u5076<\/h4>\n<p>T-\u5bf9\u5076\u548cS-\u5bf9\u5076\u7684\u7ed3\u5408&#xff0c;\u5f62\u6210\u66f4\u5927\u7684\u5bf9\u5076\u7fa4\u3002\u5728M-\u7406\u8bba\u4e2d&#xff0c;U-\u5bf9\u5076\u662fE_n(\u2124)\u79bb\u6563\u5bf9\u79f0\u7fa4\u3002<\/p>\n<h3>\u4e94\u3001D-\u819c\u52a8\u529b\u5b66<\/h3>\n<h4>1. Dp-\u819c\u4f5c\u7528\u91cf<\/h4>\n<p>Dirac-Born-Infeld\u4f5c\u7528\u91cf&#xff1a;<\/p>\n<p>SDBI\u200b&#061;\u2212Tp\u200b\u222bdp&#043;1\u03bee\u2212\u03d5\u2212det(G\u03b1\u03b2\u200b&#043;B\u03b1\u03b2\u200b&#043;2\u03c0\u03b1\u2032F\u03b1\u03b2\u200b)\u200b<\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\n<p>Tp\u200b&#061;(2\u03c0)pgs\u200b(\u03b1\u2032)(p&#043;1)\/21\u200b\u662fDp-\u819c\u5f20\u529b<\/p>\n<\/li>\n<li>\n<p>G\u03b1\u03b2\u200b&#061;\u2202\u03b1\u200bX\u03bc\u2202\u03b2\u200bX\u03bdG\u03bc\u03bd\u200b\u662f\u8bf1\u5bfc\u5ea6\u89c4<\/p>\n<\/li>\n<li>\n<p>B\u03b1\u03b2\u200b&#061;\u2202\u03b1\u200bX\u03bc\u2202\u03b2\u200bX\u03bdB\u03bc\u03bd\u200b<\/p>\n<\/li>\n<li>\n<p>F\u03b1\u03b2\u200b&#061;\u2202\u03b1\u200bA\u03b2\u200b\u2212\u2202\u03b2\u200bA\u03b1\u200b\u662f\u89c4\u8303\u573a\u573a\u5f3a<\/p>\n<\/li>\n<\/ul>\n<p>Chern-Simons\u4f5c\u7528\u91cf&#xff1a;<\/p>\n<p>SCS\u200b&#061;\u03bcp\u200b\u222bWp&#043;1\u200b\u200bq\u2211\u200bCq\u200b\u2227eB&#043;2\u03c0\u03b1\u2032F<\/p>\n<h4>2. \u5f00\u5f26\u6fc0\u53d1<\/h4>\n<p>Dp-\u819c\u4e0a\u7684\u5f00\u5f26\u7aef\u70b9\u6ee1\u8db3\u8fb9\u754c\u6761\u4ef6&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u8bfa\u4f0a\u66fc\u65b9\u5411&#xff1a;X\u03bc\u81ea\u7531\u53d8\u5316<\/p>\n<\/li>\n<li>\n<p>\u72c4\u5229\u514b\u96f7\u65b9\u5411&#xff1a;X\u03bc\u56fa\u5b9a\u5728D-\u819c\u4e0a<\/p>\n<\/li>\n<\/ul>\n<p>\u8d28\u91cf\u8c31&#xff1a;<\/p>\n<p>M2&#061;\u03b1\u20321\u200b(N\u22121&#043;21\u200bi&#061;p&#043;1\u22119\u200b(xi\u200b\u2212yi\u200b)2)<\/p>\n<p>\u5176\u4e2d xi\u200b,yi\u200b\u662f\u4e24\u4e2aD-\u819c\u5728\u6a2a\u5411\u65b9\u5411\u7684\u4f4d\u7f6e\u3002<\/p>\n<h3>\u516d\u3001\u7d27\u5316\u4e0e\u5361\u62c9\u6bd4-\u4e18\u6d41\u5f62<\/h3>\n<h4>1. \u5361\u62c9\u6bd4-\u4e18\u6761\u4ef6<\/h4>\n<p>6\u7ef4\u7d27\u81f4\u6d41\u5f62M\u4e3a\u5361\u62c9\u6bd4-\u4e18\u6d41\u5f62\u9700\u6ee1\u8db3&#xff1a;<\/p>\n<li>\n<p>\u51ef\u52d2&#xff1a;\u5b58\u5728\u95ed\u7684\u51ef\u52d2\u5f62\u5f0f J&#061;igij\u02c9\u200b\u200bdzi\u2227dz\u02c9j\u02c9\u200b<\/p>\n<\/li>\n<li>\n<p>\u91cc\u5947\u5e73\u5766&#xff1a;Rij\u02c9\u200b\u200b&#061;0<\/p>\n<\/li>\n<li>\n<p>\u548c\u4e50\u7fa4\u4e3aSU(3)&#xff1a;\u5b58\u5728\u5168\u5c40\u65cb\u91cf\u573a<\/p>\n<\/li>\n<p>\u62d3\u6251\u4e0d\u53d8\u91cf&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u970d\u5947\u6570&#xff1a;hp,q&#061;dimHp,q(M)<\/p>\n<\/li>\n<li>\n<p>\u6b27\u62c9\u793a\u6027\u6570&#xff1a;\u03c7&#061;\u2211p,q\u200b(\u22121)p&#043;qhp,q&#061;2(h1,1\u2212h2,1)<\/p>\n<\/li>\n<li>\n<p>\u975e\u96f6\u970d\u5947\u6570&#xff1a;h0,0&#061;h3,0&#061;h0,3&#061;h3,3&#061;1<\/p>\n<\/li>\n<li>\n<p>\u91cd\u8981\u970d\u5947\u6570&#xff1a;h1,1(\u51ef\u52d2\u6a21\u6570), h2,1(\u590d\u7ed3\u6784\u6a21\u6570)<\/p>\n<\/li>\n<\/ul>\n<h4>2. \u7d27\u5316\u540e\u76844\u7ef4\u7406\u8bba<\/h4>\n<p>10\u7ef4\u2161\u578b\u5f26\u5728\u5361\u62c9\u6bd4-\u4e18\u6d41\u5f62M\u4e0a\u7d27\u5316\u5f97\u52304\u7ef4N&#061;2\u8d85\u5f15\u529b&#xff08;\u2161A&#xff09;\u6216N&#061;1\u8d85\u5f15\u529b&#xff08;\u2161B&#xff09;\u3002<\/p>\n<p>\u6a21\u7a7a\u95f4&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u51ef\u52d2\u6a21\u7a7a\u95f4&#xff1a;\u590d\u7ef4\u6570 h1,1&#xff0c;\u53c2\u6570\u5316\u51ef\u52d2\u5f62\u5f0f<\/p>\n<\/li>\n<li>\n<p>\u590d\u7ed3\u6784\u6a21\u7a7a\u95f4&#xff1a;\u590d\u7ef4\u6570 h2,1&#xff0c;\u53c2\u6570\u5316\u590d\u7ed3\u6784<\/p>\n<\/li>\n<li>\n<p>\u8f74-\u4f38\u7f29\u5b50&#xff1a;\u590d\u6807\u91cf\u573a S&#061;a&#043;ie\u2212\u03d5<\/p>\n<\/li>\n<\/ul>\n<h4>3. \u955c\u9762\u5bf9\u79f0<\/h4>\n<p>\u4ea4\u6362\u4e00\u5bf9\u5361\u62c9\u6bd4-\u4e18\u6d41\u5f62M\u548cW&#xff0c;\u6ee1\u8db3&#xff1a;<\/p>\n<p>h1,1(M)&#061;h2,1(W),h2,1(M)&#061;h1,1(W)<\/p>\n<p>\u7269\u7406\u4e0a\u4ea4\u6362\u4e86\u2161A\u548c\u2161B\u5728\u955c\u50cf\u5bf9\u4e0a\u7684\u7d27\u5316\u3002<\/p>\n<h3>\u4e03\u3001M-\u7406\u8bba\u4e0e\u7edf\u4e00<\/h3>\n<h4>1. 11\u7ef4\u8d85\u5f15\u529b\u4e0eM-\u7406\u8bba<\/h4>\n<p>M-\u7406\u8bba\u5728\u4f4e\u80fd\u4e0b\u753111\u7ef4\u8d85\u5f15\u529b\u63cf\u8ff0&#xff0c;\u57fa\u672c\u6fc0\u53d1\u5305\u62ec&#xff1a;<\/p>\n<ul>\n<li>\n<p>M2-\u819c&#xff1a;2\u7ef4\u819c&#xff0c;\u4e16\u754c\u4f53\u4f5c\u7528\u91cf\u7c7b\u4f3c\u4e8e\u5f26<\/p>\n<\/li>\n<li>\n<p>M5-\u819c&#xff1a;5\u7ef4\u819c&#xff0c;\u4e16\u754c\u4f53\u4f5c\u7528\u91cf\u5305\u542b\u81ea\u5bf9\u50762-\u5f62\u5f0f\u573a<\/p>\n<\/li>\n<\/ul>\n<p>\u5173\u7cfb&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5c06M-\u7406\u8bba\u7d27\u5316\u5728S1\u4e0a\u5f97\u5230\u2161A\u578b\u5f26\u8bba&#xff0c;\u534a\u5f84 R11\u200b&#061;gs\u200bls\u200b<\/p>\n<\/li>\n<li>\n<p>\u5c06M-\u7406\u8bba\u7d27\u5316\u5728S1\/Z2\u200b\u8f68\u5f62\u5f97\u5230\u6742\u5316E8\u00d7E8\u5f26\u8bba<\/p>\n<\/li>\n<\/ul>\n<h4>2. 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