1453:
78:
then showed in 1966 that, if such a manifold did in fact exist, it would carry a parallel 4-form, and that it would necessarily be Ricci-flat. The first local examples of 8-manifolds with holonomy Spin(7) were finally constructed around 1984 by
83:, and his full proof of their existence appeared in Annals of Mathematics in 1987. Next, complete (but still noncompact) 8-manifolds with holonomy Spin(7) were explicitly constructed by Bryant and Salamon in 1989. The first examples of
54:
and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, which is a calibrating form for a special class of submanifolds called Cayley cycles.
1494:
668:
136:
270:
1487:
1518:
263:
1480:
635:
221:
62:
The fact that Spin(7) might possibly arise as the holonomy group of certain
Riemannian 8-manifolds was first suggested by the 1955
865:
600:
1227:
1513:
807:
313:
256:
185:
80:
577:
1252:
308:
832:
458:
753:
405:
1092:
1047:
1272:
1192:
1007:
941:
303:
1152:
774:
748:
489:
111:
1412:
1222:
936:
779:
620:
378:
320:
1177:
918:
724:
615:
587:
410:
63:
1327:
663:
630:
494:
336:
1032:
972:
913:
880:
875:
673:
371:
366:
361:
346:
156:
1202:
1417:
415:
400:
356:
1460:
1332:
1217:
870:
769:
395:
71:
70:, and this possibility remained consistent with the simplified proof of Berger's theorem given by
39:
188:; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy",
1377:
1297:
1197:
1157:
1037:
1002:
837:
714:
610:
351:
1087:
764:
1342:
1247:
1082:
992:
962:
758:
653:
605:
499:
217:
1464:
1352:
1287:
1257:
1137:
1077:
1042:
987:
977:
957:
890:
842:
705:
698:
691:
684:
677:
595:
385:
293:
197:
84:
1452:
1432:
1387:
1337:
1322:
1312:
1207:
1172:
997:
567:
341:
1277:
1407:
1402:
1362:
1302:
1292:
1212:
1132:
1122:
1117:
1112:
1027:
1022:
1017:
982:
967:
895:
572:
435:
43:
1507:
1397:
1382:
1357:
1347:
1317:
1262:
1237:
1182:
1167:
1162:
1127:
1102:
1062:
852:
504:
420:
298:
279:
209:
88:
67:
1427:
1267:
1147:
1097:
1067:
1052:
860:
827:
719:
645:
625:
562:
425:
168:
131:
75:
74:
in 1962. Although not a single example of such a manifold had yet been discovered,
201:
1422:
1392:
1372:
1232:
1187:
1142:
1107:
1057:
822:
791:
552:
509:
100:
31:
17:
1367:
1307:
1242:
905:
885:
784:
743:
557:
51:
47:
171:(1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)",
134:(1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)",
1282:
1072:
1012:
658:
532:
453:
448:
443:
928:
817:
812:
542:
537:
474:
390:
547:
527:
484:
479:
232:
519:
248:
252:
155:
Bryant, Robert L. (1987) "Metrics with exceptional holonomy,"
1468:
950:
927:
904:
851:
736:
644:
586:
518:
467:
434:
329:
286:
1488:
264:
8:
237:Mathematical Institute, University of Oxford
87:Spin(7)-manifolds were then constructed by
1495:
1481:
271:
257:
249:
137:Comptes Rendus de l'Académie des Sciences
214:Compact Manifolds with Special Holonomy
123:
27:Eighth-dimensional Riemannian manifold
7:
1449:
1447:
669:Bogomol'nyiâPrasadâSommerfield bound
233:"Flows of G2 and Spin(7) structures"
1467:. You can help Knowledge (XXG) by
25:
1451:
866:Eleven-dimensional supergravity
1:
314:Second superstring revolution
202:10.1215/s0012-7094-89-05839-0
808:Generalized complex manifold
309:First superstring revolution
231:Karigiannis, Spiro (2009),
216:. Oxford University Press.
1535:
1446:
406:Non-critical string theory
1519:Riemannian geometry stubs
190:Duke Mathematical Journal
942:Introduction to M-theory
636:WessâZuminoâWitten model
578:HananyâWitten transition
304:History of string theory
50:. Spin(7)-manifolds are
38:is an eight-dimensional
621:Vertex operator algebra
321:String theory landscape
1463:-related article is a
919:AdS/CFT correspondence
674:Exceptional Lie groups
616:Superconformal algebra
588:Conformal field theory
459:MontonenâOlive duality
411:Non-linear sigma model
173:C. R. Acad. Sci. Paris
159:(2)126, 525–576.
64:classification theorem
914:Holographic principle
881:Type IIB supergravity
876:Type IIA supergravity
728:-form electrodynamics
347:Bosonic string theory
157:Annals of Mathematics
1514:Riemannian manifolds
833:HoĆavaâWitten theory
780:HyperkÀhler manifold
468:Particles and fields
416:Tachyon condensation
401:Matrix string theory
1461:Riemannian geometry
871:Type I supergravity
775:CalabiâYau manifold
770:Ricci-flat manifold
749:KaluzaâKlein theory
490:RamondâRamond field
396:String field theory
112:CalabiâYau manifold
40:Riemannian manifold
838:K-theory (physics)
715:ADE classification
352:Superstring theory
1476:
1475:
1441:
1440:
1223:van Nieuwenhuizen
759:Why 10 dimensions
664:ChernâSimons form
631:KacâMoody algebra
611:Conformal algebra
606:Conformal anomaly
500:Magnetic monopole
495:KalbâRamond field
337:NambuâGoto action
16:(Redirected from
1526:
1497:
1490:
1483:
1455:
1448:
951:String theorists
891:Lie superalgebra
843:Twisted K-theory
801:Spin(7)-manifold
754:Compactification
596:Virasoro algebra
379:Heterotic string
273:
266:
259:
250:
244:
227:
204:
180:
160:
153:
147:
145:
128:
46:is contained in
36:Spin(7)-manifold
21:
18:Spin(7) manifold
1534:
1533:
1529:
1528:
1527:
1525:
1524:
1523:
1504:
1503:
1502:
1501:
1444:
1442:
1437:
946:
923:
900:
847:
795:
765:KĂ€hler manifold
732:
709:
702:
695:
688:
681:
640:
601:Mirror symmetry
582:
568:Brane cosmology
514:
463:
430:
386:N=2 superstring
372:Type IIB string
367:Type IIA string
342:Polyakov action
325:
282:
277:
230:
224:
208:
184:
167:
164:
163:
154:
150:
130:
129:
125:
120:
106:
97:
60:
28:
23:
22:
15:
12:
11:
5:
1532:
1530:
1522:
1521:
1516:
1506:
1505:
1500:
1499:
1492:
1485:
1477:
1474:
1473:
1456:
1439:
1438:
1436:
1435:
1430:
1425:
1420:
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1380:
1375:
1370:
1365:
1360:
1355:
1350:
1345:
1340:
1335:
1330:
1325:
1320:
1315:
1310:
1305:
1300:
1295:
1290:
1285:
1280:
1275:
1273:Randjbar-Daemi
1270:
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1230:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1180:
1175:
1170:
1165:
1160:
1155:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1000:
995:
990:
985:
980:
975:
970:
965:
960:
954:
952:
948:
947:
945:
944:
939:
933:
931:
925:
924:
922:
921:
916:
910:
908:
902:
901:
899:
898:
896:Lie supergroup
893:
888:
883:
878:
873:
868:
863:
857:
855:
849:
848:
846:
845:
840:
835:
830:
825:
820:
815:
810:
805:
804:
803:
798:
793:
789:
788:
787:
777:
767:
762:
756:
751:
746:
740:
738:
734:
733:
731:
730:
722:
717:
712:
707:
700:
693:
686:
679:
671:
666:
661:
656:
650:
648:
642:
641:
639:
638:
633:
628:
623:
618:
613:
608:
603:
598:
592:
590:
584:
583:
581:
580:
575:
573:Quiver diagram
570:
565:
560:
555:
550:
545:
540:
535:
530:
524:
522:
516:
515:
513:
512:
507:
502:
497:
492:
487:
482:
477:
471:
469:
465:
464:
462:
461:
456:
451:
446:
440:
438:
436:String duality
432:
431:
429:
428:
423:
418:
413:
408:
403:
398:
393:
388:
383:
382:
381:
376:
375:
374:
369:
362:Type II string
359:
349:
344:
339:
333:
331:
327:
326:
324:
323:
318:
317:
316:
311:
301:
299:Cosmic strings
296:
290:
288:
284:
283:
278:
276:
275:
268:
261:
253:
247:
246:
228:
222:
206:
196:(3): 829â850,
182:
162:
161:
148:
122:
121:
119:
116:
115:
114:
109:
104:
96:
93:
59:
56:
44:holonomy group
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1531:
1520:
1517:
1515:
1512:
1511:
1509:
1498:
1493:
1491:
1486:
1484:
1479:
1478:
1472:
1470:
1466:
1462:
1457:
1454:
1450:
1445:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1418:Zamolodchikov
1416:
1414:
1413:Zamolodchikov
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
955:
953:
949:
943:
940:
938:
937:Matrix theory
935:
934:
932:
930:
926:
920:
917:
915:
912:
911:
909:
907:
903:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
858:
856:
854:
853:Supersymmetry
850:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
802:
799:
797:
790:
786:
783:
782:
781:
778:
776:
773:
772:
771:
768:
766:
763:
760:
757:
755:
752:
750:
747:
745:
742:
741:
739:
735:
729:
727:
723:
721:
718:
716:
713:
710:
703:
696:
689:
682:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
651:
649:
647:
643:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
593:
591:
589:
585:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
525:
523:
521:
517:
511:
508:
506:
505:Dual graviton
503:
501:
498:
496:
493:
491:
488:
486:
483:
481:
478:
476:
473:
472:
470:
466:
460:
457:
455:
452:
450:
447:
445:
442:
441:
439:
437:
433:
427:
424:
422:
421:RNS formalism
419:
417:
414:
412:
409:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
380:
377:
373:
370:
368:
365:
364:
363:
360:
358:
357:Type I string
355:
354:
353:
350:
348:
345:
343:
340:
338:
335:
334:
332:
328:
322:
319:
315:
312:
310:
307:
306:
305:
302:
300:
297:
295:
292:
291:
289:
285:
281:
280:String theory
274:
269:
267:
262:
260:
255:
254:
251:
242:
238:
234:
229:
225:
223:0-19-850601-5
219:
215:
211:
210:Dominic Joyce
207:
203:
199:
195:
191:
187:
183:
178:
174:
170:
166:
165:
158:
152:
149:
143:
139:
138:
133:
132:Bonan, Edmond
127:
124:
117:
113:
110:
108:
103:
99:
98:
94:
92:
90:
89:Dominic Joyce
86:
82:
81:Robert Bryant
77:
73:
69:
68:Marcel Berger
65:
57:
55:
53:
49:
45:
41:
37:
33:
19:
1469:expanding it
1458:
1443:
963:Arkani-Hamed
861:Supergravity
828:Moduli space
800:
725:
720:Dirac string
646:Gauge theory
626:Loop algebra
563:Black string
426:GS formalism
243:(4): 389â463
240:
236:
213:
193:
189:
186:Bryant, R.L.
176:
172:
151:
141:
135:
126:
101:
76:Edmond Bonan
61:
35:
29:
1323:Silverstein
823:Orientifold
558:Black holes
553:Black brane
510:Dual photon
32:mathematics
1508:Categories
1343:Strominger
1338:Steinhardt
1333:Staudacher
1248:Polchinski
1198:Nanopoulos
1158:Mandelstam
1138:Kontsevich
978:Berenstein
906:Holography
886:Superspace
785:K3 surface
744:Worldsheet
659:Instantons
287:Background
118:References
72:Jim Simons
52:Ricci-flat
1378:Veneziano
1258:Rajaraman
1153:Maldacena
1043:Gopakumar
993:Dijkgraaf
988:Curtright
654:Anomalies
533:NS5-brane
454:U-duality
449:S-duality
444:T-duality
179:: 127â129
144:: 127â129
91:in 1996.
1433:Zwiebach
1388:Verlinde
1383:Verlinde
1358:Townsend
1353:Susskind
1288:Sagnotti
1253:Polyakov
1208:Nekrasov
1173:Minwalla
1168:Martinec
1133:Knizhnik
1128:Klebanov
1123:Kapustin
1088:'t Hooft
1023:Fischler
958:AganagiÄ
929:M-theory
818:Conifold
813:Orbifold
796:manifold
737:Geometry
543:M5-brane
538:M2-brane
475:Graviton
391:F-theory
212:(2000).
169:E. Bonan
107:manifold
95:See also
1363:Trivedi
1348:Sundrum
1313:Shenker
1303:Seiberg
1298:Schwarz
1268:Randall
1228:Novikov
1218:Nielsen
1203:NÄstase
1113:Kallosh
1098:Gibbons
1038:Gliozzi
1028:Friedan
1018:Ferrara
1003:Douglas
998:Distler
548:S-brane
528:D-brane
485:Tachyon
480:Dilaton
294:Strings
85:compact
58:History
48:Spin(7)
1428:Zumino
1423:Zaslow
1408:Yoneya
1398:Witten
1318:Siegel
1293:Scherk
1263:Ramond
1238:Ooguri
1163:Marolf
1118:Kaluza
1103:Kachru
1093:HoĆava
1083:Harvey
1078:Hanson
1063:Gubser
1053:Greene
983:Bousso
968:Atiyah
520:Branes
330:Theory
220:
42:whose
1459:This
1368:Turok
1278:RoÄek
1243:Ovrut
1233:Olive
1213:Neveu
1193:Myers
1188:Mukhi
1178:Moore
1148:Linde
1143:Klein
1068:Gukov
1058:Gross
1048:Green
1033:Gates
1013:Dvali
973:Banks
1465:stub
1393:Wess
1373:Vafa
1283:Rohm
1183:Motl
1108:Kaku
1073:Guth
1008:Duff
218:ISBN
34:, a
1403:Yau
1328:SÆĄn
1308:Sen
198:doi
177:262
142:262
66:of
30:In
1510::
704:,
697:,
690:,
683:,
239:,
235:,
194:58
192:,
175:,
140:,
1496:e
1489:t
1482:v
1471:.
794:2
792:G
761:?
726:p
711:)
708:8
706:E
701:7
699:E
694:6
692:E
687:4
685:F
680:2
678:G
676:(
272:e
265:t
258:v
245:.
241:9
226:.
205:.
200::
181:.
146:.
105:2
102:G
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.