1493:
949:
588:
gauge theory with an equal number of colors and flavors, as was demonstrated in
Vortices, instantons and branes. The generalization to greater numbers of flavors appeared in Solitons in the Higgs phase: The Moduli matrix approach. In both cases the
742:
764:
1092:
1201:
277:
1366:
1000:
356:
1296:
431:
944:{\displaystyle \Delta ={\begin{pmatrix}I&B_{2}+z_{2}&B_{1}+z_{1}\\J^{\dagger }&-B_{1}^{\dagger }-{\bar {z_{1}}}&B_{2}^{\dagger }+{\bar {z_{2}}}\end{pmatrix}}.}
626:
551:
1008:
1131:
1476:
1534:
1563:
1568:
1558:
146:
1315:
957:
590:
1377:
1118:
557:. In this case instantons exist even when the gauge group is U(1). The noncommutative instantons were discovered by
1527:
480:
24:
288:
1302:
1553:
521:
1258:
468:
737:{\displaystyle x_{ij}={\begin{pmatrix}z_{2}&z_{1}\\-{\bar {z_{1}}}&{\bar {z_{2}}}\end{pmatrix}}.}
390:
1520:
438:
1438:
1426:
597:
487:
20:
553:
is set equal to the self-dual projection of the noncommutativity matrix of the spacetime times the
527:
1087:{\displaystyle \Delta \Delta ^{\dagger }={\begin{pmatrix}f^{-1}&0\\0&f^{-1}\end{pmatrix}}}
1500:
1454:
1418:
44:
1504:
1446:
1110:
1466:
1407:
1462:
1403:
1196:{\displaystyle P=\Delta ^{\dagger }{\begin{pmatrix}f&0\\0&f\end{pmatrix}}\Delta .}
613:
558:
554:
459:
can be obtained in this way and are in one-to-one correspondence with solutions up to a U(
1442:
1414:
1395:
1382:
562:
282:
40:
1547:
1472:
1450:
1422:
585:
52:
48:
491:
445:
68:
138:
1252:
A regularity condition on the rank of Δ guarantees the completeness condition
620:
456:
141:
1458:
361:
Then the ADHM construction claims that, given certain regularity conditions,
1207:
616:
584:
to zero, one obtains the classical moduli space of nonabelian vortices in a
434:
36:
600:, plays the role of the noncommutativity parameter in the real moment map.
1492:
594:
524:
gauge theory, the ADHM construction is identical but the moment map
1222:. The basis vectors for this null-space can be assembled into an (
494:
of instantons is that inherited from the flat metric on
1508:
272:{\displaystyle \mu _{r}=++II^{\dagger }-J^{\dagger }J,}
1156:
1033:
779:
651:
1318:
1261:
1134:
1011:
961:
960:
767:
629:
530:
393:
292:
291:
149:
1360:
1290:
1195:
1086:
994:
943:
736:
545:
425:
350:
271:
1361:{\displaystyle A_{m}=U^{\dagger }\partial _{m}U.}
995:{\displaystyle \displaystyle \mu _{r}=\mu _{c}=0}
1002:are equivalent to the factorization condition
63:The ADHM construction uses the following data:
1528:
55:in their paper "Construction of Instantons."
8:
351:{\displaystyle \displaystyle \mu _{c}=+IJ.}
1535:
1521:
16:Method of constructing instanton solutions
1346:
1336:
1323:
1317:
1287:
1275:
1260:
1151:
1145:
1133:
1067:
1040:
1028:
1019:
1010:
979:
966:
959:
918:
912:
911:
902:
897:
879:
873:
872:
863:
858:
843:
829:
816:
804:
791:
774:
766:
711:
705:
704:
690:
684:
683:
670:
658:
646:
634:
628:
532:
531:
529:
411:
398:
392:
326:
313:
297:
290:
257:
244:
225:
220:
207:
188:
183:
170:
154:
148:
1402:, Scuola Normale Superiore Pisa, Pisa,
1429:(1978), "Construction of instantons",
7:
1489:
1487:
1291:{\displaystyle P+UU^{\dagger }=1.\,}
1242:) with orthonormalization condition
483:and antifundamental representations
426:{\displaystyle \mu _{r}=\mu _{c}=0}
39:using methods of linear algebra by
1507:. You can help Knowledge (XXG) by
1477:"On the Construction of Monopoles"
1343:
1187:
1142:
1016:
1012:
768:
14:
1491:
924:
885:
717:
696:
537:
463:) rotation which acts on each
332:
306:
231:
200:
194:
163:
1:
1400:Geometry of Yang-Mills fields
546:{\displaystyle {\vec {\mu }}}
1451:10.1016/0375-9601(78)90141-X
1378:Monad (homological algebra)
35:is the construction of all
1585:
1486:
1305:is then constructed from
516:Noncommutative instantons
604:The construction formula
1415:Atiyah, Michael Francis
1396:Atiyah, Michael Francis
1125:can be constructed as
619:coordinates written in
1564:Quantum chromodynamics
1503:-related article is a
1362:
1292:
1197:
1088:
996:
945:
738:
547:
469:adjoint representation
448:with instanton number
427:
352:
273:
1569:Quantum physics stubs
1559:Differential geometry
1427:Manin, Yuri Ivanovich
1363:
1293:
1198:
1089:
997:
946:
739:
612:be the 4-dimensional
593:, which determines a
591:Fayet–Iliopoulos term
548:
428:
353:
274:
1316:
1259:
1132:
1009:
958:
954:Then the conditions
765:
627:
528:
433:, an anti-self-dual
391:
289:
147:
130:complex matrix
21:mathematical physics
1481:Commun. Math. Phys.
1443:1978PhLA...65..185A
1230:) ×
907:
868:
750: × (
455:All anti-self-dual
452:can be constructed,
230:
193:
1358:
1301:The anti-selfdual
1288:
1214:) is of dimension
1193:
1181:
1084:
1078:
992:
991:
941:
932:
893:
854:
734:
725:
543:
423:
348:
347:
269:
216:
179:
126: ×
114: ×
92: ×
33:monad construction
1516:
1515:
1501:quantum mechanics
1431:Physics Letters A
1117:Then a hermitian
927:
888:
720:
699:
540:
96:complex matrices
45:Vladimir Drinfeld
29:ADHM construction
1576:
1537:
1530:
1523:
1495:
1488:
1469:
1410:
1367:
1365:
1364:
1359:
1351:
1350:
1341:
1340:
1328:
1327:
1309:by the formula
1297:
1295:
1294:
1289:
1280:
1279:
1249: = 1.
1202:
1200:
1199:
1194:
1186:
1185:
1150:
1149:
1111:Hermitian matrix
1093:
1091:
1090:
1085:
1083:
1082:
1075:
1074:
1048:
1047:
1024:
1023:
1001:
999:
998:
993:
984:
983:
971:
970:
950:
948:
947:
942:
937:
936:
929:
928:
923:
922:
913:
906:
901:
890:
889:
884:
883:
874:
867:
862:
848:
847:
834:
833:
821:
820:
809:
808:
796:
795:
743:
741:
740:
735:
730:
729:
722:
721:
716:
715:
706:
701:
700:
695:
694:
685:
675:
674:
663:
662:
642:
641:
552:
550:
549:
544:
542:
541:
533:
432:
430:
429:
424:
416:
415:
403:
402:
357:
355:
354:
349:
331:
330:
318:
317:
302:
301:
278:
276:
275:
270:
262:
261:
249:
248:
229:
224:
212:
211:
192:
187:
175:
174:
159:
158:
1584:
1583:
1579:
1578:
1577:
1575:
1574:
1573:
1544:
1543:
1542:
1541:
1419:Drinfeld, V. G.
1413:
1394:
1391:
1374:
1342:
1332:
1319:
1314:
1313:
1271:
1257:
1256:
1180:
1179:
1174:
1168:
1167:
1162:
1152:
1141:
1130:
1129:
1077:
1076:
1063:
1061:
1055:
1054:
1049:
1036:
1029:
1015:
1007:
1006:
975:
962:
956:
955:
931:
930:
914:
891:
875:
849:
839:
836:
835:
825:
812:
810:
800:
787:
785:
775:
763:
762:
724:
723:
707:
702:
686:
677:
676:
666:
664:
654:
647:
630:
625:
624:
606:
579:
571:
559:Nikita Nekrasov
555:identity matrix
526:
525:
518:
513:
511:Generalizations
407:
394:
389:
388:
378:
371:
322:
309:
293:
287:
286:
253:
240:
203:
166:
150:
145:
144:
118:complex matrix
109:
102:
61:
17:
12:
11:
5:
1582:
1580:
1572:
1571:
1566:
1561:
1556:
1554:Gauge theories
1546:
1545:
1540:
1539:
1532:
1525:
1517:
1514:
1513:
1496:
1485:
1484:
1470:
1437:(3): 185–187,
1423:Hitchin, N. J.
1411:
1390:
1387:
1386:
1385:
1383:Twistor theory
1380:
1373:
1370:
1369:
1368:
1357:
1354:
1349:
1345:
1339:
1335:
1331:
1326:
1322:
1299:
1298:
1286:
1283:
1278:
1274:
1270:
1267:
1264:
1226: + 2
1204:
1203:
1192:
1189:
1184:
1178:
1175:
1173:
1170:
1169:
1166:
1163:
1161:
1158:
1157:
1155:
1148:
1144:
1140:
1137:
1115:
1114:
1081:
1073:
1070:
1066:
1062:
1060:
1057:
1056:
1053:
1050:
1046:
1043:
1039:
1035:
1034:
1032:
1027:
1022:
1018:
1014:
990:
987:
982:
978:
974:
969:
965:
952:
951:
940:
935:
926:
921:
917:
910:
905:
900:
896:
892:
887:
882:
878:
871:
866:
861:
857:
853:
850:
846:
842:
838:
837:
832:
828:
824:
819:
815:
811:
807:
803:
799:
794:
790:
786:
784:
781:
780:
778:
773:
770:
754: + 2
746:Consider the 2
733:
728:
719:
714:
710:
703:
698:
693:
689:
682:
679:
678:
673:
669:
665:
661:
657:
653:
652:
650:
645:
640:
637:
633:
605:
602:
586:supersymmetric
577:
570:
567:
563:Albert Schwarz
539:
536:
522:noncommutative
517:
514:
512:
509:
508:
507:
484:
453:
422:
419:
414:
410:
406:
401:
397:
376:
369:
359:
358:
346:
343:
340:
337:
334:
329:
325:
321:
316:
312:
308:
305:
300:
296:
279:
268:
265:
260:
256:
252:
247:
243:
239:
236:
233:
228:
223:
219:
215:
210:
206:
202:
199:
196:
191:
186:
182:
178:
173:
169:
165:
162:
157:
153:
135:
107:
100:
87:
60:
57:
41:Michael Atiyah
15:
13:
10:
9:
6:
4:
3:
2:
1581:
1570:
1567:
1565:
1562:
1560:
1557:
1555:
1552:
1551:
1549:
1538:
1533:
1531:
1526:
1524:
1519:
1518:
1512:
1510:
1506:
1502:
1497:
1494:
1490:
1482:
1478:
1474:
1471:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1409:
1405:
1401:
1397:
1393:
1392:
1388:
1384:
1381:
1379:
1376:
1375:
1371:
1355:
1352:
1347:
1337:
1333:
1329:
1324:
1320:
1312:
1311:
1310:
1308:
1304:
1284:
1281:
1276:
1272:
1268:
1265:
1262:
1255:
1254:
1253:
1250:
1248:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1190:
1182:
1176:
1171:
1164:
1159:
1153:
1146:
1138:
1135:
1128:
1127:
1126:
1124:
1120:
1112:
1109:
1105:
1101:
1097:
1079:
1071:
1068:
1064:
1058:
1051:
1044:
1041:
1037:
1030:
1025:
1020:
1005:
1004:
1003:
988:
985:
980:
976:
972:
967:
963:
938:
933:
919:
915:
908:
903:
898:
894:
880:
876:
869:
864:
859:
855:
851:
844:
840:
830:
826:
822:
817:
813:
805:
801:
797:
792:
788:
782:
776:
771:
761:
760:
759:
757:
753:
749:
744:
731:
726:
712:
708:
691:
687:
680:
671:
667:
659:
655:
648:
643:
638:
635:
631:
622:
618:
615:
611:
603:
601:
599:
596:
592:
587:
583:
576:
568:
566:
564:
560:
556:
534:
523:
515:
510:
505:
501:
497:
493:
489:
485:
482:
478:
474:
470:
466:
462:
458:
454:
451:
447:
444:
442:
436:
420:
417:
412:
408:
404:
399:
395:
386:
382:
375:
368:
364:
363:
362:
344:
341:
338:
335:
327:
323:
319:
314:
310:
303:
298:
294:
284:
280:
266:
263:
258:
254:
250:
245:
241:
237:
234:
226:
221:
217:
213:
208:
204:
197:
189:
184:
180:
176:
171:
167:
160:
155:
151:
143:
140:
136:
133:
129:
125:
121:
117:
113:
106:
99:
95:
91:
88:
85:
81:
78:of dimension
77:
73:
70:
69:vector spaces
66:
65:
64:
58:
56:
54:
53:Yuri I. Manin
50:
49:Nigel Hitchin
46:
42:
38:
34:
30:
26:
22:
1509:expanding it
1498:
1483:89, 145–190.
1480:
1434:
1430:
1399:
1306:
1300:
1251:
1246:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1218:for generic
1215:
1211:
1205:
1122:
1116:
1107:
1103:
1099:
1095:
953:
755:
751:
747:
745:
621:quaternionic
609:
607:
581:
574:
572:
519:
503:
499:
495:
492:moduli space
476:
472:
464:
460:
449:
446:gauge theory
440:
384:
380:
373:
366:
360:
131:
127:
123:
119:
115:
111:
104:
97:
93:
89:
83:
79:
75:
71:
62:
32:
28:
25:gauge theory
18:
1473:Hitchin, N.
481:fundamental
285:moment map
1548:Categories
1389:References
1303:connection
1119:projection
598:condensate
457:instantons
387:such that
142:moment map
37:instantons
1459:0375-9601
1344:∂
1338:†
1277:†
1208:nullspace
1188:Δ
1147:†
1143:Δ
1121:operator
1069:−
1042:−
1021:†
1017:Δ
1013:Δ
977:μ
964:μ
925:¯
904:†
886:¯
870:−
865:†
852:−
845:†
769:Δ
758:) matrix
718:¯
697:¯
681:−
623:notation
617:spacetime
614:Euclidean
565:in 1998.
538:→
535:μ
435:instanton
409:μ
396:μ
295:μ
259:†
251:−
246:†
227:†
190:†
152:μ
59:ADHM data
1475:(1983),
1398:(1979),
1372:See also
573:Setting
569:Vortices
479:via the
67:complex
1467:0598562
1439:Bibcode
1408:0554924
1234:matrix
1106:×
1102:) is a
490:on the
471:and on
467:in the
283:complex
1465:
1457:
1406:
1094:where
595:squark
488:metric
365:Given
122:and a
27:, the
1499:This
1210:of Δ(
520:In a
437:in a
1505:stub
1455:ISSN
1206:The
608:Let
580:and
561:and
502:and
486:The
475:and
139:real
110:, a
82:and
74:and
23:and
1447:doi
498:,
439:SU(
31:or
19:In
1550::
1479:,
1463:MR
1461:,
1453:,
1445:,
1435:65
1433:,
1425:;
1421:;
1417:;
1404:MR
1285:1.
383:,
379:,
372:,
281:a
137:a
103:,
51:,
47:,
43:,
1536:e
1529:t
1522:v
1511:.
1449::
1441::
1356:.
1353:U
1348:m
1334:U
1330:=
1325:m
1321:A
1307:U
1282:=
1273:U
1269:U
1266:+
1263:P
1247:U
1244:U
1240:x
1238:(
1236:U
1232:N
1228:k
1224:N
1220:x
1216:N
1212:x
1191:.
1183:)
1177:f
1172:0
1165:0
1160:f
1154:(
1139:=
1136:P
1123:P
1113:.
1108:k
1104:k
1100:x
1098:(
1096:f
1080:)
1072:1
1065:f
1059:0
1052:0
1045:1
1038:f
1031:(
1026:=
989:0
986:=
981:c
973:=
968:r
939:.
934:)
920:2
916:z
909:+
899:2
895:B
881:1
877:z
860:1
856:B
841:J
831:1
827:z
823:+
818:1
814:B
806:2
802:z
798:+
793:2
789:B
783:I
777:(
772:=
756:k
752:N
748:k
732:.
727:)
713:2
709:z
692:1
688:z
672:1
668:z
660:2
656:z
649:(
644:=
639:j
636:i
632:x
610:x
582:J
578:2
575:B
506:.
504:J
500:I
496:B
477:J
473:I
465:B
461:k
450:k
443:)
441:N
421:0
418:=
413:c
405:=
400:r
385:J
381:I
377:2
374:B
370:1
367:B
345:.
342:J
339:I
336:+
333:]
328:2
324:B
320:,
315:1
311:B
307:[
304:=
299:c
267:,
264:J
255:J
242:I
238:I
235:+
232:]
222:2
218:B
214:,
209:2
205:B
201:[
198:+
195:]
185:1
181:B
177:,
172:1
168:B
164:[
161:=
156:r
134:,
132:J
128:k
124:N
120:I
116:N
112:k
108:2
105:B
101:1
98:B
94:k
90:k
86:,
84:N
80:k
76:W
72:V
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