1836:
one can deduce that the surfaces are parametrized by pseudoholomorphic curves, and so the path integrals reduce to integrals over moduli spaces of pseudoholomorphic curves (or rather stable maps), which are finite-dimensional. In closed type IIA string theory, for example, these integrals are
1745:, and described the way in which pseudoholomorphic curves can degenerate when only finite energy is assumed. (The finite energy condition holds most notably for curves with a fixed homology class in a symplectic manifold where J is
1115:
325:
1637:
1832:, one wishes to compute certain integrals over the space of all such surfaces. Because such a space is infinite-dimensional, these path integrals are not mathematically well-defined in general. However, under the
1370:
988:
671:
A pseudoholomorphic curve is, by its definition, always parametrized. In applications one is often truly interested in unparametrized curves, meaning embedded (or immersed) two-submanifolds of
598:
518:
1201:
414:
1248:
1511:
640:
364:
225:
1783:
1763:
1704:
1684:
1462:
1421:
818:
691:, so one mods out by reparametrizations of the domain that preserve the relevant structure. In the case of Gromov–Witten invariants, for example, we consider only
662:
541:
440:
1724:
1664:
1534:
1441:
1398:
910:
890:
858:
838:
780:
753:
733:
713:
689:
460:
193:
169:
141:
121:
101:
999:
233:
1542:
1903:
1259:
922:
1786:
861:
1833:
1793:, makes possible the definition of Gromov–Witten invariants, which count pseudoholomorphic curves in symplectic manifolds.
1380:
Although they can be defined for any almost complex manifold, pseudoholomorphic curves are especially interesting when
1878:
549:
468:
56:
1123:
1937:
1887:, Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae vol. 82, 1985, pgs. 307-347.
52:
372:
1942:
1838:
1209:
64:
1472:
1932:
1884:
1820:
In type II string theory, one considers surfaces traced out by strings as they travel along paths in a
523:
to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term
1825:
1268:
1203:. After multiplying these matrices in two different orders, one sees immediately that the equation
665:
48:
607:
1947:
333:
198:
1768:
1748:
1731:
1689:
1669:
1447:
1406:
788:
783:
647:
526:
8:
1805:
60:
422:
1709:
1649:
1519:
1426:
1383:
895:
875:
843:
823:
765:
738:
718:
698:
674:
603:
A pseudoholomorphic curve satisfying this equation can be called, more specifically, a
445:
178:
154:
126:
106:
86:
1874:
1850:
1829:
1643:
1809:
1804:(and later authors, in greater generality) used to prove the famous conjecture of
1110:{\displaystyle df={\begin{bmatrix}du/dx&du/dy\\dv/dx&dv/dy\end{bmatrix}},}
1952:
1895:
1891:
1866:
1401:
692:
320:{\displaystyle {\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=0.}
148:
144:
44:
1821:
1797:
913:
68:
1926:
1801:
1796:
Compact moduli spaces of pseudoholomorphic curves are also used to construct
1742:
1741:
of pseudoholomorphic curves (satisfying additional specified conditions) are
72:
1632:{\displaystyle (v,w)={\frac {1}{2}}\left(\omega (v,Jw)+\omega (w,Jv)\right)}
820:
is negative, there are only finitely many holomorphic reparametrizations of
1738:
1727:
1862:
17:
1790:
40:
1253:
written above is equivalent to the classical Cauchy–Riemann equations
867:
1365:{\displaystyle {\begin{cases}du/dx=dv/dy\\dv/dx=-du/dy.\end{cases}}}
983:{\displaystyle j=J={\begin{bmatrix}0&-1\\1&0\end{bmatrix}},}
543:
and to study maps satisfying the perturbed Cauchy–Riemann equation
25:
21:
59:, pseudoholomorphic curves have since revolutionized the study of
1873:, American Mathematical Society colloquium publications, 2004.
668:(particularly in Floer theory), but in general it need not be.
1734:
concerning symplectic embeddings of spheres into cylinders.
103:
be an almost complex manifold with almost complex structure
1358:
1375:
1017:
943:
1771:
1751:
1712:
1692:
1672:
1652:
1545:
1522:
1475:
1450:
1429:
1409:
1386:
1262:
1212:
1126:
1002:
925:
898:
878:
846:
826:
791:
768:
741:
721:
701:
677:
650:
610:
552:
529:
471:
448:
425:
375:
336:
236:
201:
181:
157:
129:
109:
89:
868:
Analogy with the classical Cauchy–Riemann equations
1777:
1757:
1718:
1698:
1678:
1658:
1631:
1528:
1505:
1456:
1435:
1415:
1392:
1364:
1242:
1195:
1109:
982:
904:
884:
852:
840:that preserve the marked points. The domain curve
832:
812:
774:
747:
727:
707:
683:
656:
634:
592:
535:
512:
454:
434:
408:
358:
319:
219:
187:
163:
135:
115:
95:
1924:
593:{\displaystyle {\bar {\partial }}_{j,J}f=\nu .}
1904:Notices of the American Mathematical Society
1871:J-Holomorphic Curves and Symplectic Topology
227:that satisfies the Cauchy–Riemann equation
1815:
664:is sometimes assumed to be generated by a
1890:
1808:concerning the number of fixed points of
513:{\displaystyle T_{x}f(C)\subseteq T_{x}X}
419:which simply means that the differential
1925:
1896:"What Is...a Pseudoholomorphic Curve?"
1196:{\displaystyle f(x,y)=(u(x,y),v(x,y))}
862:Deligne–Mumford moduli space of curves
1536:. Tameness implies that the formula
409:{\displaystyle J\circ df=df\circ j,}
1376:Applications in symplectic topology
1243:{\displaystyle J\circ df=df\circ j}
13:
1666:. Gromov showed that, for a given
1506:{\displaystyle \omega (v,Jv)>0}
557:
366:, this condition is equivalent to
241:
63:. In particular, they lead to the
14:
1964:
1730:. He used this theory to prove a
1789:, now greatly generalized using
1516:for all nonzero tangent vectors
872:The classical case occurs when
71:, and play a prominent role in
1621:
1606:
1597:
1582:
1558:
1546:
1494:
1479:
1423:. An almost complex structure
1190:
1187:
1175:
1166:
1154:
1148:
1142:
1130:
629:
611:
560:
491:
485:
308:
278:
244:
211:
1:
1856:
78:
442:is complex-linear, that is,
7:
1844:
1737:Gromov showed that certain
916:plane. In real coordinates
782:. As soon as the punctured
10:
1969:
1885:Mikhail Leonidovich Gromov
1787:Gromov compactness theorem
635:{\displaystyle (j,J,\nu )}
1826:path integral formulation
151:) with complex structure
1839:Gromov–Witten invariants
462:maps each tangent space
359:{\displaystyle J^{2}=-1}
220:{\displaystyle f:C\to X}
65:Gromov–Witten invariants
55:. Introduced in 1985 by
1816:Applications in physics
1778:{\displaystyle \omega }
1758:{\displaystyle \omega }
1699:{\displaystyle \omega }
1679:{\displaystyle \omega }
1457:{\displaystyle \omega }
1416:{\displaystyle \omega }
173:pseudoholomorphic curve
53:Cauchy–Riemann equation
49:almost complex manifold
30:pseudoholomorphic curve
1824:3-fold. Following the
1779:
1759:
1720:
1700:
1680:
1660:
1633:
1530:
1507:
1458:
1437:
1417:
1394:
1366:
1244:
1197:
1111:
984:
906:
886:
854:
834:
814:
813:{\displaystyle 2-2g-n}
776:
749:
729:
709:
685:
658:
636:
594:
537:
514:
456:
436:
410:
360:
321:
221:
189:
165:
137:
117:
97:
1780:
1760:
1732:non-squeezing theorem
1721:
1701:
1681:
1661:
1634:
1531:
1508:
1459:
1438:
1418:
1395:
1367:
1245:
1198:
1112:
985:
907:
887:
860:is an element of the
855:
835:
815:
777:
750:
730:
710:
686:
659:
637:
595:
538:
515:
457:
437:
411:
361:
322:
222:
190:
166:
138:
118:
98:
1785:-compatible). This
1769:
1749:
1710:
1690:
1670:
1650:
1543:
1520:
1473:
1448:
1427:
1407:
1384:
1260:
1210:
1124:
1000:
923:
912:are both simply the
896:
876:
844:
824:
789:
784:Euler characteristic
766:
739:
719:
699:
675:
657:{\displaystyle \nu }
648:
608:
550:
536:{\displaystyle \nu }
527:
469:
446:
423:
373:
334:
234:
199:
179:
155:
127:
107:
87:
61:symplectic manifolds
1938:Symplectic topology
1892:Donaldson, Simon K.
644:. The perturbation
51:that satisfies the
1943:Algebraic geometry
1775:
1755:
1716:
1696:
1676:
1656:
1629:
1526:
1503:
1454:
1433:
1413:
1390:
1362:
1357:
1240:
1193:
1107:
1098:
980:
971:
902:
882:
850:
830:
810:
772:
745:
725:
705:
681:
654:
642:-holomorphic curve
632:
590:
533:
510:
452:
435:{\displaystyle df}
432:
406:
356:
317:
217:
185:
161:
133:
113:
93:
37:-holomorphic curve
20:, specifically in
1933:Complex manifolds
1851:Holomorphic curve
1830:quantum mechanics
1810:Hamiltonian flows
1719:{\displaystyle J}
1659:{\displaystyle X}
1644:Riemannian metric
1572:
1529:{\displaystyle v}
1436:{\displaystyle J}
1400:interacts with a
1393:{\displaystyle J}
905:{\displaystyle C}
885:{\displaystyle X}
853:{\displaystyle C}
833:{\displaystyle C}
775:{\displaystyle C}
748:{\displaystyle n}
735:and we introduce
728:{\displaystyle g}
708:{\displaystyle C}
684:{\displaystyle X}
563:
455:{\displaystyle J}
276:
247:
188:{\displaystyle X}
164:{\displaystyle j}
136:{\displaystyle C}
116:{\displaystyle J}
96:{\displaystyle X}
1960:
1919:
1917:
1916:
1900:
1894:(October 2005).
1806:Vladimir Arnol'd
1784:
1782:
1781:
1776:
1764:
1762:
1761:
1756:
1726:is nonempty and
1725:
1723:
1722:
1717:
1705:
1703:
1702:
1697:
1685:
1683:
1682:
1677:
1665:
1663:
1662:
1657:
1638:
1636:
1635:
1630:
1628:
1624:
1573:
1565:
1535:
1533:
1532:
1527:
1512:
1510:
1509:
1504:
1463:
1461:
1460:
1455:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1399:
1397:
1396:
1391:
1371:
1369:
1368:
1363:
1361:
1360:
1345:
1322:
1301:
1281:
1249:
1247:
1246:
1241:
1202:
1200:
1199:
1194:
1116:
1114:
1113:
1108:
1103:
1102:
1089:
1070:
1049:
1030:
989:
987:
986:
981:
976:
975:
911:
909:
908:
903:
891:
889:
888:
883:
859:
857:
856:
851:
839:
837:
836:
831:
819:
817:
816:
811:
781:
779:
778:
773:
754:
752:
751:
746:
734:
732:
731:
726:
714:
712:
711:
706:
690:
688:
687:
682:
663:
661:
660:
655:
641:
639:
638:
633:
599:
597:
596:
591:
577:
576:
565:
564:
556:
542:
540:
539:
534:
519:
517:
516:
511:
506:
505:
481:
480:
461:
459:
458:
453:
441:
439:
438:
433:
415:
413:
412:
407:
365:
363:
362:
357:
346:
345:
326:
324:
323:
318:
277:
269:
261:
260:
249:
248:
240:
226:
224:
223:
218:
194:
192:
191:
186:
170:
168:
167:
162:
142:
140:
139:
134:
122:
120:
119:
114:
102:
100:
99:
94:
1968:
1967:
1963:
1962:
1961:
1959:
1958:
1957:
1923:
1922:
1914:
1912:
1898:
1867:Dietmar Salamon
1859:
1847:
1818:
1770:
1767:
1766:
1750:
1747:
1746:
1711:
1708:
1707:
1691:
1688:
1687:
1686:, the space of
1671:
1668:
1667:
1651:
1648:
1647:
1578:
1574:
1564:
1544:
1541:
1540:
1521:
1518:
1517:
1474:
1471:
1470:
1466:if and only if
1449:
1446:
1445:
1428:
1425:
1424:
1408:
1405:
1404:
1402:symplectic form
1385:
1382:
1381:
1378:
1356:
1355:
1341:
1318:
1309:
1308:
1297:
1277:
1264:
1263:
1261:
1258:
1257:
1211:
1208:
1207:
1125:
1122:
1121:
1097:
1096:
1085:
1077:
1066:
1057:
1056:
1045:
1037:
1026:
1013:
1012:
1001:
998:
997:
970:
969:
964:
958:
957:
949:
939:
938:
924:
921:
920:
897:
894:
893:
877:
874:
873:
870:
845:
842:
841:
825:
822:
821:
790:
787:
786:
767:
764:
763:
740:
737:
736:
720:
717:
716:
715:of fixed genus
700:
697:
696:
676:
673:
672:
649:
646:
645:
609:
606:
605:
566:
555:
554:
553:
551:
548:
547:
528:
525:
524:
501:
497:
476:
472:
470:
467:
466:
447:
444:
443:
424:
421:
420:
374:
371:
370:
341:
337:
335:
332:
331:
268:
250:
239:
238:
237:
235:
232:
231:
200:
197:
196:
180:
177:
176:
156:
153:
152:
147:(also called a
145:Riemann surface
128:
125:
124:
108:
105:
104:
88:
85:
84:
81:
45:Riemann surface
12:
11:
5:
1966:
1956:
1955:
1950:
1945:
1940:
1935:
1921:
1920:
1911:(9): 1026–1027
1888:
1882:
1858:
1855:
1854:
1853:
1846:
1843:
1837:precisely the
1817:
1814:
1798:Floer homology
1774:
1754:
1715:
1695:
1675:
1655:
1640:
1639:
1627:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1577:
1571:
1568:
1563:
1560:
1557:
1554:
1551:
1548:
1525:
1514:
1513:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1453:
1443:is said to be
1432:
1412:
1389:
1377:
1374:
1373:
1372:
1359:
1354:
1351:
1348:
1344:
1340:
1337:
1334:
1331:
1328:
1325:
1321:
1317:
1314:
1311:
1310:
1307:
1304:
1300:
1296:
1293:
1290:
1287:
1284:
1280:
1276:
1273:
1270:
1269:
1267:
1251:
1250:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1118:
1117:
1106:
1101:
1095:
1092:
1088:
1084:
1081:
1078:
1076:
1073:
1069:
1065:
1062:
1059:
1058:
1055:
1052:
1048:
1044:
1041:
1038:
1036:
1033:
1029:
1025:
1022:
1019:
1018:
1016:
1011:
1008:
1005:
991:
990:
979:
974:
968:
965:
963:
960:
959:
956:
953:
950:
948:
945:
944:
942:
937:
934:
931:
928:
914:complex number
901:
881:
869:
866:
849:
829:
809:
806:
803:
800:
797:
794:
771:
744:
724:
704:
680:
653:
631:
628:
625:
622:
619:
616:
613:
601:
600:
589:
586:
583:
580:
575:
572:
569:
562:
559:
532:
521:
520:
509:
504:
500:
496:
493:
490:
487:
484:
479:
475:
451:
431:
428:
417:
416:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
355:
352:
349:
344:
340:
328:
327:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
275:
272:
267:
264:
259:
256:
253:
246:
243:
216:
213:
210:
207:
204:
184:
160:
132:
112:
92:
80:
77:
69:Floer homology
57:Mikhail Gromov
9:
6:
4:
3:
2:
1965:
1954:
1951:
1949:
1948:String theory
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1930:
1928:
1910:
1906:
1905:
1897:
1893:
1889:
1886:
1883:
1880:
1879:0-8218-3485-1
1876:
1872:
1868:
1864:
1861:
1860:
1852:
1849:
1848:
1842:
1840:
1835:
1831:
1827:
1823:
1813:
1811:
1807:
1803:
1802:Andreas Floer
1799:
1794:
1792:
1788:
1772:
1752:
1744:
1740:
1739:moduli spaces
1735:
1733:
1729:
1713:
1693:
1673:
1653:
1645:
1625:
1618:
1615:
1612:
1609:
1603:
1600:
1594:
1591:
1588:
1585:
1579:
1575:
1569:
1566:
1561:
1555:
1552:
1549:
1539:
1538:
1537:
1523:
1500:
1497:
1491:
1488:
1485:
1482:
1476:
1469:
1468:
1467:
1465:
1451:
1430:
1410:
1403:
1387:
1352:
1349:
1346:
1342:
1338:
1335:
1332:
1329:
1326:
1323:
1319:
1315:
1312:
1305:
1302:
1298:
1294:
1291:
1288:
1285:
1282:
1278:
1274:
1271:
1265:
1256:
1255:
1254:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1206:
1205:
1204:
1184:
1181:
1178:
1172:
1169:
1163:
1160:
1157:
1151:
1145:
1139:
1136:
1133:
1127:
1104:
1099:
1093:
1090:
1086:
1082:
1079:
1074:
1071:
1067:
1063:
1060:
1053:
1050:
1046:
1042:
1039:
1034:
1031:
1027:
1023:
1020:
1014:
1009:
1006:
1003:
996:
995:
994:
977:
972:
966:
961:
954:
951:
946:
940:
935:
932:
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1927:Categories
1915:2008-01-17
1857:References
1822:Calabi–Yau
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