332:
Choquet-Bruhat and Geroch's work. Moreover, the analyticity and corresponding unique continuation of a Ricci-flat
Riemannian metric has a fundamentally different character than Ricci-flat Lorentzian metrics, which have finite speeds of propagation and fully localizable phenomena. This can be viewed as a nonlinear geometric analogue of the difference between the
464:, every such metric is Ricci-flat. The Calabi–Yau theorem specializes to this context, giving a general existence and uniqueness theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact spaces had earlier been obtained by
547:
commented that all known examples of irreducible Ricci-flat
Riemannian metrics on simply-connected closed manifolds have special holonomy groups, according to the above possibilities. It is not known whether this suggests an unknown general theorem or simply a limitation of known techniques. For this
331:
The study of Ricci-flatness in the
Riemannian and Lorentzian cases are quite distinct. This is already indicated by the fundamental distinction between the geodesically complete metrics which are typical of Riemannian geometry and the maximal globally hyperbolic developments which arise from
164:. Conversely, it is automatic from the definitions that any flat metric is Ricci-flat. The study of flat metrics is usually considered as a topic unto itself. As such, the study of Ricci-flat metrics is only a distinct topic in dimension four and above.
222:. However, these constructions are not directly helpful for Ricci-flat Riemannian metrics, in the sense that any homogeneous Riemannian manifold which is Ricci-flat must be flat. However, there are homogeneous (and even
371:
on its topological data. As particular cases of well-known theorems on
Riemannian manifolds of nonnegative Ricci curvature, any manifold with a complete Ricci-flat Riemannian metric must:
153:, it is straightforward to see that the converse also holds. This may also be phrased as saying that Ricci-flatness is characterized by the vanishing of the two non-Weyl parts of the
195:, a two-parameter family containing the Schwarzschild metrics as a special case. These metrics are fully explicit and are of fundamental interest in the mathematics and physics of
367:
Beyond Kähler geometry, the situation is not as well understood. A four-dimensional closed and oriented manifold supporting any
Einstein Riemannian metric must satisfy the
1767:
1369:
1310:
325:
963:
348:
Yau's existence theorem for Ricci-flat Kähler metrics established the precise topological condition under which such a metric exists on a given closed
297:
274:
409:
with an arbitrary closed manifold. Every Ricci-flat
Riemannian manifold in this class is flat, which is a corollary of Cheeger and Gromoll's
172:
As noted above, any flat metric is Ricci-flat. However it is nontrivial to identify Ricci-flat manifolds whose full curvature is nonzero.
257:. Due to his analytical techniques, the metrics are non-explicit even in the simplest cases. Such Riemannian manifolds are often called
1362:
1267:
1734:
1216:
1167:
1115:
1027:
985:
901:
851:
1964:
1699:
2326:
390:
368:
2552:
1906:
1412:
1355:
476:
246:
208:
1676:
452:. This condition on a Riemannian manifold may also be characterized (roughly speaking) by the existence of a 2-sphere of
2351:
1407:
319:
64:
1931:
296:
Analogously, relative to harmonic coordinates, Ricci-flatness of a
Lorentzian metric can be interpreted as a system of
226:) Lorentzian manifolds which are Ricci-flat but not flat, as follows from an explicit construction and computation of
1557:
1248:
Sitzungsberichte der Königlich
Preussischen Akademie der Wissenschaften zu Berlin, Physikalisch-Mathematische Klasse
1852:
1504:
1107:
969:
357:
115:
2191:
2146:
1159:
537:
implies that any such manifold is Ricci-flat. The existence of closed manifolds of this type was established by
2371:
2291:
2106:
2040:
1402:
146:
2251:
1873:
1847:
1588:
469:
258:
490:
showed that any such metric must be
Einstein. Furthermore, any Ricci-flat quaternion-Kähler manifold must be
160:
Since the Weyl curvature vanishes in two or three dimensions, every Ricci-flat metric in these dimensions is
2511:
2321:
2035:
1878:
1719:
1477:
1419:
1099:
1058:
1019:
445:
316:
Ricci-flat
Lorentzian metrics are prescribed and constructed by certain Riemannian data. These are known as
2276:
2017:
1823:
1714:
1686:
1509:
1149:
150:
2426:
1762:
1729:
1593:
1435:
361:
126:
is zero. It is direct to verify that, except in dimension two, a metric is Ricci-flat if and only if its
2131:
2071:
2012:
1979:
1974:
1772:
1470:
1465:
1460:
1445:
1308:(1978). "On the Ricci curvature of a compact Kähler manifold and the complex Monge−Ampère equation. I".
430:
301:
212:
142:
87:
72:
32:
2301:
2516:
1514:
1499:
1455:
402:
180:
441:
holonomy group of a Ricci-flat Kähler metric is necessarily contained in the special unitary group.
2431:
2316:
1969:
1494:
282:
270:
184:
154:
91:
60:
56:
44:
281:
results that any Ricci-flat Riemannian metric on a smooth manifold is analytic, in the sense that
2476:
2396:
2296:
2256:
2136:
2101:
1936:
1813:
1709:
1450:
1262:
1243:
1153:
1056:(1963). "Gravitational field of a spinning mass as an example of algebraically special metrics".
959:
884:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 10. Reprinted in 2008. Berlin:
457:
305:
200:
176:
79:
2186:
1863:
1193:
103:
2441:
2346:
2181:
2091:
2061:
1857:
1752:
1704:
1598:
1212:
1163:
1111:
1023:
981:
897:
847:
410:
383:
353:
290:
286:
238:
219:
131:
99:
52:
249:, established a comprehensive existence theory for Ricci-flat metrics in the special case of
2451:
2386:
2356:
2236:
2176:
2141:
2086:
2076:
2056:
1989:
1941:
1899:
1804:
1797:
1790:
1783:
1776:
1694:
1484:
1392:
1335:
1319:
1292:
1276:
1251:
1230:
1204:
1181:
1129:
1095:
1083:
1067:
1041:
999:
973:
943:
931:
915:
889:
865:
839:
525:
512:
453:
449:
422:
349:
333:
273:, the condition of Ricci-flatness for a Riemannian metric can be interpreted as a system of
254:
250:
135:
1331:
1288:
1226:
1177:
1125:
1079:
1037:
995:
911:
861:
2531:
2486:
2436:
2421:
2411:
2306:
2271:
2096:
1666:
1440:
1339:
1327:
1296:
1284:
1255:
1234:
1222:
1185:
1173:
1133:
1121:
1087:
1075:
1045:
1033:
1003:
991:
955:
947:
927:
919:
907:
885:
869:
857:
835:
242:
223:
204:
127:
123:
2376:
1246:(1916). "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie".
460:. This says in particular that every hyperkähler metric is Kähler; furthermore, via the
2506:
2501:
2461:
2401:
2391:
2311:
2231:
2221:
2216:
2211:
2126:
2121:
2116:
2081:
2066:
1994:
1671:
1534:
1305:
1201:
1141:
534:
465:
461:
426:
308:
of the Ricci-flatness condition. She reached a definitive result in collaboration with
234:
95:
1208:
2546:
2496:
2481:
2456:
2446:
2416:
2361:
2336:
2281:
2266:
2261:
2226:
2201:
2161:
1951:
1603:
1519:
1397:
1378:
1265:(2003). "On the gravitational field of a mass point according to Einstein's theory".
1011:
827:
544:
538:
487:
406:
394:
337:
309:
161:
2526:
2366:
2246:
2196:
2166:
2151:
1959:
1926:
1818:
1744:
1724:
1661:
1524:
877:
376:
2521:
2491:
2471:
2331:
2286:
2241:
2206:
2156:
1921:
1890:
1651:
1608:
502:
227:
192:
28:
1071:
241:
in the 1970s, it was not known whether every Ricci-flat Riemannian metric on a
78:
In Lorentzian geometry, a number of Ricci-flat metrics are known from works of
2466:
2406:
2341:
2004:
1984:
1883:
1842:
1656:
1280:
1145:
968:. Cambridge Monographs on Mathematical Physics. Vol. 1. London−New York:
893:
843:
519:
196:
977:
548:
reason, Berger considered Ricci-flat manifolds to be "extremely mysterious."
518:
is a Riemannian manifold whose holonomy group is contained in the Lie groups
145:, it is direct to see that any Ricci-flat metric has Weyl curvature equal to
2381:
2171:
2111:
1757:
1631:
1552:
1547:
1542:
480:
40:
1323:
17:
2027:
1916:
1911:
1641:
1636:
1573:
1489:
1053:
401:
of a closed manifold. The class of enlargeable manifolds is closed under
293:. This also holds in the broader setting of Einstein Riemannian metrics.
188:
83:
1646:
1626:
1583:
1578:
472:, discovered at the same time, is a special case of his construction.
360:
must be zero. The necessity of this condition was previously known by
261:, although various authors use this name in slightly different ways.
68:
379:
less than or equal to the dimension, whenever the manifold is closed
1618:
479:
is a Riemannian manifold whose holonomy group is contained in the
448:
is a Riemannian manifold whose holonomy group is contained in the
425:
Kähler manifold, a Kähler metric is Ricci-flat if and only if the
1347:
130:
is zero. Ricci-flat manifolds are one of three special types of
1351:
1106:. Princeton Mathematical Series. Vol. 38. Princeton, NJ:
934:[The foundation of the general theory of relativity].
324:. In general relativity, this is typically interpreted as an
63:
are of fundamental interest, as they are the solutions of
1200:. Pure and Applied Mathematics. Vol. 103. New York:
1198:
Semi-Riemannian geometry. With applications to relativity
942:(7). Translated by Perrett, W.; Jeffery, G. B.: 769–822.
1275:(5). Translated by Antoci, S.; Loinger, A.: 951–959.
498:
holonomy group is contained in the symplectic group.
218:
Many pseudo-Riemannian manifolds are constructed as
2049:
2026:
2003:
1950:
1835:
1743:
1685:
1617:
1566:
1533:
1428:
1385:
932:"Die Grundlage der allgemeinen Relativitätstheorie"
621:
597:
312:in the 1960s, establishing how a certain class of
277:. It is a straightforward consequence of standard
733:
289:, and the local representation of the metric is
203:, Ricci-flat Lorentzian manifolds represent the
328:of Einstein's field equations for gravitation.
1311:Communications on Pure and Applied Mathematics
1363:
605:
8:
709:
601:
581:
569:
344:Topology of Ricci-flat Riemannian manifolds
102:produced a number of Ricci-flat metrics on
1370:
1356:
1348:
1018:. Oxford Mathematical Monographs. Oxford:
298:hyperbolic partial differential equations
405:, the taking of products, and under the
1016:Compact manifolds with special holonomy
965:The large scale structure of space-time
832:A panoramic view of Riemannian geometry
562:
275:elliptic partial differential equations
245:is flat. His work, using techniques of
809:
793:
797:
781:
769:
757:
745:
721:
697:
681:
669:
657:
645:
633:
593:
7:
1768:Bogomol'nyi–Prasad–Sommerfield bound
617:
437:direction still holds, but only the
433:. On a general Kähler manifold, the
685:
1268:General Relativity and Gravitation
25:
622:Misner, Thorne & Wheeler 1973
598:Misner, Thorne & Wheeler 1973
134:, arising as the special case of
1965:Eleven-dimensional supergravity
494:hyperkähler, meaning that the
247:partial differential equations
1:
1413:Second superstring revolution
1209:10.1016/s0079-8169(08)x6002-7
300:. Based on this perspective,
1907:Generalized complex manifold
1408:First superstring revolution
734:Lawson & Michelsohn 1989
417:Ricci-flatness and holonomy
141:From the definition of the
2569:
1505:Non-critical string theory
1108:Princeton University Press
1072:10.1103/PhysRevLett.11.237
970:Cambridge University Press
477:quaternion-Kähler manifold
358:holomorphic tangent bundle
209:Einstein's field equations
116:pseudo-Riemannian manifold
65:Einstein's field equations
1160:W. H. Freeman and Company
894:10.1007/978-3-540-74311-8
844:10.1007/978-3-642-18245-7
397:introduced the notion of
369:Hitchin–Thorpe inequality
326:initial value formulation
2041:Introduction to M-theory
1735:Wess–Zumino–Witten model
1677:Hanany–Witten transition
1403:History of string theory
1100:Michelsohn, Marie-Louise
978:10.1017/CBO9780511524646
710:Hawking & Ellis 1973
147:Riemann curvature tensor
1720:Vertex operator algebra
1420:String theory landscape
1281:10.1023/A:1022971926521
1150:Wheeler, John Archibald
1059:Physical Review Letters
1020:Oxford University Press
183:, which are Ricci-flat
2018:AdS/CFT correspondence
1773:Exceptional Lie groups
1715:Superconformal algebra
1687:Conformal field theory
1558:Montonen–Olive duality
1510:Non-linear sigma model
1324:10.1002/cpa.3160310304
535:Ambrose–Singer theorem
462:Ambrose–Singer theorem
187:of nonzero curvature.
51:are a special kind of
39:is a condition on the
2013:Holographic principle
1980:Type IIB supergravity
1975:Type IIA supergravity
1827:-form electrodynamics
1446:Bosonic string theory
1158:. San Francisco, CA:
1096:Lawson, H. Blaine Jr.
431:special unitary group
386:of polynomial growth.
302:Yvonne Choquet-Bruhat
237:'s resolution of the
213:cosmological constant
199:. More generally, in
181:Schwarzschild metrics
143:Weyl curvature tensor
98:'s resolution of the
88:Yvonne Choquet-Bruhat
73:cosmological constant
33:differential geometry
2553:Riemannian manifolds
1932:Hořava–Witten theory
1879:Hyperkähler manifold
1567:Particles and fields
1515:Tachyon condensation
1500:Matrix string theory
1202:Academic Press, Inc.
748:, Proposition 10.29.
446:hyperkähler manifold
429:is contained in the
403:homotopy equivalence
285:define a compatible
283:harmonic coordinates
271:harmonic coordinates
265:Analytical character
259:Calabi–Yau manifolds
185:Lorentzian manifolds
61:Lorentzian manifolds
49:Ricci-flat manifolds
1970:Type I supergravity
1874:Calabi–Yau manifold
1869:Ricci-flat manifold
1848:Kaluza–Klein theory
1589:Ramond–Ramond field
1495:String field theory
712:, Sections 7.5–7.6.
470:Eguchi–Hanson space
320:globally hyperbolic
279:elliptic regularity
155:Ricci decomposition
92:Riemannian geometry
57:theoretical physics
45:Riemannian manifold
1937:K-theory (physics)
1814:ADE classification
1451:Superstring theory
1142:Misner, Charles W.
936:Annalen der Physik
882:Einstein manifolds
796:, Section 13.5.1;
684:, Sections 11B–C;
606:Schwarzschild 1916
454:complex structures
314:maximally extended
287:analytic structure
220:homogeneous spaces
201:general relativity
177:Karl Schwarzschild
80:Karl Schwarzschild
2540:
2539:
2322:van Nieuwenhuizen
1858:Why 10 dimensions
1763:Chern–Simons form
1730:Kac–Moody algebra
1710:Conformal algebra
1705:Conformal anomaly
1599:Magnetic monopole
1594:Kalb–Ramond field
1436:Nambu–Goto action
1263:Schwarzschild, K.
1244:Schwarzschild, K.
1012:Joyce, Dominic D.
812:, Section 11.4.6.
760:, Sections 14A–C.
672:, Paragraph 0.30.
411:splitting theorem
384:fundamental group
362:Chern–Weil theory
354:first Chern class
255:complex manifolds
239:Calabi conjecture
132:Einstein manifold
100:Calabi conjecture
53:Einstein manifold
16:(Redirected from
2560:
2050:String theorists
1990:Lie superalgebra
1942:Twisted K-theory
1900:Spin(7)-manifold
1853:Compactification
1695:Virasoro algebra
1478:Heterotic string
1372:
1365:
1358:
1349:
1343:
1301:
1300:
1259:
1238:
1194:O'Neill, Barrett
1189:
1137:
1091:
1049:
1007:
951:
923:
878:Besse, Arthur L.
873:
813:
807:
801:
791:
785:
779:
773:
767:
761:
755:
749:
743:
737:
731:
725:
724:, Sections 6D–E.
719:
713:
707:
701:
695:
689:
679:
673:
667:
661:
660:, Theorem 7.118.
655:
649:
643:
637:
631:
625:
615:
609:
591:
585:
579:
573:
567:
531:
522:
515:
508:
485:
450:symplectic group
423:simply-connected
350:complex manifold
334:Laplace equation
205:vacuum solutions
191:later found the
136:scalar curvature
104:Kähler manifolds
21:
2568:
2567:
2563:
2562:
2561:
2559:
2558:
2557:
2543:
2542:
2541:
2536:
2045:
2022:
1999:
1946:
1894:
1864:Kähler manifold
1831:
1808:
1801:
1794:
1787:
1780:
1739:
1700:Mirror symmetry
1681:
1667:Brane cosmology
1613:
1562:
1529:
1485:N=2 superstring
1471:Type IIB string
1466:Type IIA string
1441:Polyakov action
1424:
1381:
1376:
1346:
1306:Yau, Shing Tung
1304:
1261:
1260:
1242:
1241:
1219:
1192:
1170:
1140:
1118:
1094:
1052:
1030:
1010:
988:
960:Ellis, G. F. R.
954:
926:
904:
886:Springer-Verlag
876:
854:
836:Springer-Verlag
826:
817:
816:
808:
804:
792:
788:
780:
776:
768:
764:
756:
752:
744:
740:
736:, Section IV.5.
732:
728:
720:
716:
708:
704:
696:
692:
680:
676:
668:
664:
656:
652:
648:, Theorem 7.61.
644:
640:
632:
628:
616:
612:
592:
588:
580:
576:
568:
564:
554:
530:
526:
520:
513:
507:
503:
483:
419:
346:
267:
243:closed manifold
211:with vanishing
170:
138:equaling zero.
128:Einstein tensor
124:Ricci curvature
112:
71:with vanishing
23:
22:
15:
12:
11:
5:
2566:
2564:
2556:
2555:
2545:
2544:
2538:
2537:
2535:
2534:
2529:
2524:
2519:
2514:
2509:
2504:
2499:
2494:
2489:
2484:
2479:
2474:
2469:
2464:
2459:
2454:
2449:
2444:
2439:
2434:
2429:
2424:
2419:
2414:
2409:
2404:
2399:
2394:
2389:
2384:
2379:
2374:
2372:Randjbar-Daemi
2369:
2364:
2359:
2354:
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2244:
2239:
2234:
2229:
2224:
2219:
2214:
2209:
2204:
2199:
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2059:
2053:
2051:
2047:
2046:
2044:
2043:
2038:
2032:
2030:
2024:
2023:
2021:
2020:
2015:
2009:
2007:
2001:
2000:
1998:
1997:
1995:Lie supergroup
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1956:
1954:
1948:
1947:
1945:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1903:
1902:
1897:
1892:
1888:
1887:
1886:
1876:
1866:
1861:
1855:
1850:
1845:
1839:
1837:
1833:
1832:
1830:
1829:
1821:
1816:
1811:
1806:
1799:
1792:
1785:
1778:
1770:
1765:
1760:
1755:
1749:
1747:
1741:
1740:
1738:
1737:
1732:
1727:
1722:
1717:
1712:
1707:
1702:
1697:
1691:
1689:
1683:
1682:
1680:
1679:
1674:
1672:Quiver diagram
1669:
1664:
1659:
1654:
1649:
1644:
1639:
1634:
1629:
1623:
1621:
1615:
1614:
1612:
1611:
1606:
1601:
1596:
1591:
1586:
1581:
1576:
1570:
1568:
1564:
1563:
1561:
1560:
1555:
1550:
1545:
1539:
1537:
1535:String duality
1531:
1530:
1528:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1481:
1480:
1475:
1474:
1473:
1468:
1461:Type II string
1458:
1448:
1443:
1438:
1432:
1430:
1426:
1425:
1423:
1422:
1417:
1416:
1415:
1410:
1400:
1398:Cosmic strings
1395:
1389:
1387:
1383:
1382:
1377:
1375:
1374:
1367:
1360:
1352:
1345:
1344:
1318:(3): 339–411.
1302:
1239:
1217:
1190:
1168:
1146:Thorne, Kip S.
1138:
1116:
1092:
1066:(5): 237–238.
1050:
1028:
1008:
986:
956:Hawking, S. W.
952:
924:
902:
874:
852:
828:Berger, Marcel
823:
815:
814:
802:
786:
784:, Section 10F.
774:
772:, Section 14D.
762:
750:
738:
726:
714:
702:
690:
674:
662:
650:
638:
626:
610:
604:, Chapter 13;
600:, Chapter 31;
596:, Section 3F;
586:
584:, p. 336.
574:
561:
560:
553:
550:
541:in the 1990s.
528:
505:
466:Eugenio Calabi
456:which are all
427:holonomy group
418:
415:
399:enlargeability
391:Mikhael Gromov
388:
387:
380:
345:
342:
306:well-posedness
304:developed the
266:
263:
251:Kähler metrics
235:Shing-Tung Yau
169:
166:
118:is said to be
111:
108:
96:Shing-Tung Yau
37:Ricci-flatness
24:
14:
13:
10:
9:
6:
4:
3:
2:
2565:
2554:
2551:
2550:
2548:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2517:Zamolodchikov
2515:
2513:
2512:Zamolodchikov
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2054:
2052:
2048:
2042:
2039:
2037:
2036:Matrix theory
2034:
2033:
2031:
2029:
2025:
2019:
2016:
2014:
2011:
2010:
2008:
2006:
2002:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1957:
1955:
1953:
1952:Supersymmetry
1949:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1901:
1898:
1896:
1889:
1885:
1882:
1881:
1880:
1877:
1875:
1872:
1871:
1870:
1867:
1865:
1862:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1840:
1838:
1834:
1828:
1826:
1822:
1820:
1817:
1815:
1812:
1809:
1802:
1795:
1788:
1781:
1774:
1771:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1750:
1748:
1746:
1742:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1692:
1690:
1688:
1684:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1624:
1622:
1620:
1616:
1610:
1607:
1605:
1604:Dual graviton
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
1575:
1572:
1571:
1569:
1565:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1540:
1538:
1536:
1532:
1526:
1523:
1521:
1520:RNS formalism
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1479:
1476:
1472:
1469:
1467:
1464:
1463:
1462:
1459:
1457:
1456:Type I string
1454:
1453:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1433:
1431:
1427:
1421:
1418:
1414:
1411:
1409:
1406:
1405:
1404:
1401:
1399:
1396:
1394:
1391:
1390:
1388:
1384:
1380:
1379:String theory
1373:
1368:
1366:
1361:
1359:
1354:
1353:
1350:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1312:
1307:
1303:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1269:
1264:
1257:
1253:
1249:
1245:
1240:
1236:
1232:
1228:
1224:
1220:
1218:0-12-526740-1
1214:
1210:
1206:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1169:0-7503-0948-2
1165:
1161:
1157:
1156:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1117:0-691-08542-0
1113:
1109:
1105:
1104:Spin geometry
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1060:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1029:0-19-850601-5
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
987:0-521-20016-4
983:
979:
975:
971:
967:
966:
961:
957:
953:
949:
945:
941:
937:
933:
929:
925:
921:
917:
913:
909:
905:
903:3-540-15279-2
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
853:3-540-65317-1
849:
845:
841:
837:
833:
829:
825:
824:
822:
821:
811:
806:
803:
799:
795:
790:
787:
783:
778:
775:
771:
766:
763:
759:
754:
751:
747:
742:
739:
735:
730:
727:
723:
718:
715:
711:
706:
703:
700:, Section 5F.
699:
694:
691:
687:
683:
678:
675:
671:
666:
663:
659:
654:
651:
647:
642:
639:
636:, Section 3C.
635:
630:
627:
624:, Chapter 33.
623:
619:
614:
611:
607:
603:
599:
595:
590:
587:
583:
578:
575:
572:, p. 87.
571:
566:
563:
559:
558:
551:
549:
546:
545:Marcel Berger
542:
540:
539:Dominic Joyce
536:
532:
523:
517:
510:
499:
497:
493:
489:
488:Marcel Berger
482:
478:
473:
471:
467:
463:
459:
455:
451:
447:
442:
440:
436:
432:
428:
424:
416:
414:
412:
408:
407:connected sum
404:
400:
396:
395:Blaine Lawson
392:
385:
381:
378:
374:
373:
372:
370:
365:
363:
359:
355:
351:
343:
341:
339:
338:wave equation
335:
329:
327:
323:
321:
315:
311:
310:Robert Geroch
307:
303:
299:
294:
292:
291:real-analytic
288:
284:
280:
276:
272:
264:
262:
260:
256:
252:
248:
244:
240:
236:
231:
229:
225:
221:
216:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
173:
167:
165:
163:
158:
156:
152:
148:
144:
139:
137:
133:
129:
125:
121:
117:
109:
107:
105:
101:
97:
93:
89:
85:
81:
76:
74:
70:
66:
62:
59:, Ricci-flat
58:
54:
50:
46:
42:
38:
34:
30:
19:
2062:Arkani-Hamed
1960:Supergravity
1927:Moduli space
1868:
1824:
1819:Dirac string
1745:Gauge theory
1725:Loop algebra
1662:Black string
1525:GS formalism
1315:
1309:
1272:
1266:
1247:
1197:
1154:
1103:
1063:
1057:
1054:Kerr, Roy P.
1015:
964:
939:
935:
928:Einstein, A.
881:
831:
819:
818:
805:
789:
777:
765:
753:
741:
729:
717:
705:
693:
677:
665:
653:
641:
629:
613:
602:O'Neill 1983
589:
582:O'Neill 1983
577:
570:O'Neill 1983
565:
556:
555:
543:
500:
495:
491:
474:
443:
438:
434:
420:
398:
389:
377:Betti number
366:
347:
330:
322:developments
317:
313:
295:
278:
269:Relative to
268:
232:
228:Lie algebras
217:
193:Kerr metrics
174:
171:
159:
149:. By taking
140:
119:
113:
77:
48:
36:
29:mathematical
26:
2422:Silverstein
1922:Orientifold
1657:Black holes
1652:Black brane
1609:Dual photon
1250:: 189–196.
1155:Gravitation
810:Berger 2003
794:Berger 2003
484:Sp(n)·Sp(1)
375:have first
197:black holes
2442:Strominger
2437:Steinhardt
2432:Staudacher
2347:Polchinski
2297:Nanopoulos
2257:Mandelstam
2237:Kontsevich
2077:Berenstein
2005:Holography
1985:Superspace
1884:K3 surface
1843:Worldsheet
1758:Instantons
1386:Background
1340:0369.53059
1297:1020.83005
1256:46.1296.02
1235:0531.53051
1186:1375.83002
1134:0688.57001
1088:0112.21904
1046:1027.53052
1004:0265.53054
948:46.1293.01
920:0613.53001
870:1038.53002
834:. Berlin:
798:Joyce 2000
782:Besse 1987
770:Besse 1987
758:Besse 1987
746:Besse 1987
722:Besse 1987
698:Besse 1987
682:Besse 1987
670:Besse 1987
658:Besse 1987
646:Besse 1987
634:Besse 1987
594:Besse 1987
552:References
496:restricted
439:restricted
253:on closed
179:found the
120:Ricci-flat
110:Definition
18:Ricci-flat
2477:Veneziano
2357:Rajaraman
2252:Maldacena
2142:Gopakumar
2092:Dijkgraaf
2087:Curtright
1753:Anomalies
1632:NS5-brane
1553:U-duality
1548:S-duality
1543:T-duality
618:Kerr 1963
481:Lie group
224:symmetric
175:In 1916,
41:curvature
31:field of
2547:Category
2532:Zwiebach
2487:Verlinde
2482:Verlinde
2457:Townsend
2452:Susskind
2387:Sagnotti
2352:Polyakov
2307:Nekrasov
2272:Minwalla
2267:Martinec
2232:Knizhnik
2227:Klebanov
2222:Kapustin
2187:'t Hooft
2122:Fischler
2057:Aganagić
2028:M-theory
1917:Conifold
1912:Orbifold
1895:manifold
1836:Geometry
1642:M5-brane
1637:M2-brane
1574:Graviton
1490:F-theory
1196:(1983).
1152:(1973).
1102:(1989).
1014:(2000).
962:(1973).
930:(1916).
880:(1987).
830:(2003).
820:Sources.
686:Yau 1978
516:manifold
509:manifold
458:parallel
336:and the
318:maximal
189:Roy Kerr
168:Examples
84:Roy Kerr
2462:Trivedi
2447:Sundrum
2412:Shenker
2402:Seiberg
2397:Schwarz
2367:Randall
2327:Novikov
2317:Nielsen
2302:Năstase
2212:Kallosh
2197:Gibbons
2137:Gliozzi
2127:Friedan
2117:Ferrara
2102:Douglas
2097:Distler
1647:S-brane
1627:D-brane
1584:Tachyon
1579:Dilaton
1393:Strings
1332:0480350
1289:1982197
1227:0719023
1178:0418833
1126:1031992
1080:0156674
1038:1787733
996:0424186
912:0867684
862:2002701
521:Spin(7)
514:Spin(7)
492:locally
356:of the
122:if its
27:In the
2527:Zumino
2522:Zaslow
2507:Yoneya
2497:Witten
2417:Siegel
2392:Scherk
2362:Ramond
2337:Ooguri
2262:Marolf
2217:Kaluza
2202:Kachru
2192:Hořava
2182:Harvey
2177:Hanson
2162:Gubser
2152:Greene
2082:Bousso
2067:Atiyah
1619:Branes
1429:Theory
1338:
1330:
1295:
1287:
1254:
1233:
1225:
1215:
1184:
1176:
1166:
1132:
1124:
1114:
1086:
1078:
1044:
1036:
1026:
1002:
994:
984:
946:
918:
910:
900:
868:
860:
850:
557:Notes.
533:. The
468:. The
352:: the
233:Until
151:traces
86:, and
69:vacuum
2467:Turok
2377:Roček
2342:Ovrut
2332:Olive
2312:Neveu
2292:Myers
2287:Mukhi
2277:Moore
2247:Linde
2242:Klein
2167:Gukov
2157:Gross
2147:Green
2132:Gates
2112:Dvali
2072:Banks
421:On a
382:have
90:. In
67:in a
55:. In
43:of a
2492:Wess
2472:Vafa
2382:Rohm
2282:Motl
2207:Kaku
2172:Guth
2107:Duff
1213:ISBN
1164:ISBN
1112:ISBN
1024:ISBN
982:ISBN
898:ISBN
848:ISBN
393:and
162:flat
2502:Yau
2427:Sơn
2407:Sen
1336:Zbl
1320:doi
1293:Zbl
1277:doi
1252:JFM
1231:Zbl
1205:doi
1182:Zbl
1130:Zbl
1084:Zbl
1068:doi
1042:Zbl
1000:Zbl
974:doi
944:JFM
940:354
916:Zbl
890:doi
866:Zbl
840:doi
524:or
511:or
207:of
2549::
1803:,
1796:,
1789:,
1782:,
1334:.
1328:MR
1326:.
1316:31
1314:.
1291:.
1285:MR
1283:.
1273:35
1271:.
1229:.
1223:MR
1221:.
1211:.
1180:.
1174:MR
1172:.
1162:.
1148:;
1144:;
1128:.
1122:MR
1120:.
1110:.
1098:;
1082:.
1076:MR
1074:.
1064:11
1062:.
1040:.
1034:MR
1032:.
1022:.
998:.
992:MR
990:.
980:.
972:.
958:;
938:.
914:.
908:MR
906:.
896:.
888:.
864:.
858:MR
856:.
846:.
838:.
620:;
501:A
486:.
475:A
444:A
435:if
413:.
364:.
340:.
230:.
215:.
157:.
114:A
106:.
94:,
82:,
75:.
47:.
35:,
1893:2
1891:G
1860:?
1825:p
1810:)
1807:8
1805:E
1800:7
1798:E
1793:6
1791:E
1786:4
1784:F
1779:2
1777:G
1775:(
1371:e
1364:t
1357:v
1342:.
1322::
1299:.
1279::
1258:.
1237:.
1207::
1188:.
1136:.
1090:.
1070::
1048:.
1006:.
976::
950:.
922:.
892::
872:.
842::
800:.
688:.
608:.
529:2
527:G
506:2
504:G
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.