42:
978:, a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property. This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.
872:. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting. The BFSS matrix model is also considered the worldvolume theory of a large number of D0-
946:
is a branch of mathematics that attempts to generalize this situation. Rather than working with ordinary numbers, one considers some similar objects, such as matrices, whose multiplication does not satisfy the commutative law (that is, objects for which
863:
This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional
959:). One imagines that these noncommuting objects are coordinates on some more general notion of "space" and proves theorems about these generalized spaces by exploiting the analogy with ordinary geometry.
922:. In ordinary geometry, the coordinates of a point are numbers, so they can be multiplied, and the product of two coordinates does not depend on the order of multiplication. That is,
1757:
1359:
254:
814:
1181:
975:
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1352:
188:
1724:
1954:
1689:
193:
2316:
560:
1896:
1402:
1345:
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2341:
1397:
585:
1921:
1117:
Banks, Tom; Fischler, Willy; Schenker, Stephen; Susskind, Leonard (1997). "M theory as a matrix model: A conjecture".
919:
807:
1547:
1842:
1494:
2181:
2136:
788:
420:
380:
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340:
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555:
172:
104:
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2007:
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229:
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660:
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2002:
1969:
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1435:
836:
365:
305:
272:
249:
100:
87:
2291:
535:
1191:
Connes, Alain; Douglas, Michael; Schwarz, Albert (1998). "Noncommutative geometry and matrix theory".
2506:
1504:
1489:
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1263:
1210:
1136:
1003:
755:
224:
162:
96:
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1847:
1742:
1694:
1588:
1177:
899:
832:
675:
580:
415:
325:
295:
868:. These calculations led them to propose that the BFSS matrix model is exactly equivalent to
2441:
2376:
2346:
2226:
2166:
2131:
2076:
2066:
2046:
1979:
1931:
1889:
1794:
1787:
1780:
1773:
1766:
1684:
1474:
1382:
1318:
1271:
1218:
1144:
1086:
N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, "A Large-N Reduced Model as
Superstring",
848:
685:
620:
590:
470:
410:
375:
320:
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167:
56:
2521:
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903:
844:
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330:
234:
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610:
1322:
1314:
1267:
1222:
1214:
1140:
2496:
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2201:
2116:
2111:
2106:
2071:
2056:
1984:
1661:
1524:
1173:
1166:
971:
840:
745:
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635:
625:
545:
465:
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360:
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315:
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203:
1030:
2536:
2486:
2471:
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2351:
2326:
2271:
2256:
2251:
2216:
2191:
2151:
1941:
1593:
1509:
1387:
1368:
1291:
Seiberg, Nathan; Witten, Edward (1999). "String Theory and
Noncommutative Geometry".
735:
720:
695:
680:
650:
595:
570:
515:
500:
495:
460:
435:
425:
395:
239:
61:
33:
1283:
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41:
2516:
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2186:
2156:
2141:
1949:
1916:
1808:
1734:
1714:
1651:
1514:
1230:
963:
865:
765:
600:
480:
430:
400:
385:
244:
1330:
1102:
994:, was constructed in 1996â97 by N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya.
2511:
2481:
2461:
2321:
2276:
2231:
2196:
2146:
1911:
1880:
1641:
1598:
760:
730:
710:
565:
520:
475:
440:
390:
2456:
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1994:
1974:
1873:
1832:
1646:
705:
640:
575:
267:
1148:
2371:
2161:
2101:
1747:
1621:
1542:
1537:
1532:
615:
405:
345:
131:
126:
121:
17:
1275:
974:
showed that some aspects of matrix models and M-theory are described by a
2017:
1906:
1901:
1631:
1626:
1563:
1479:
987:
877:
869:
141:
136:
1238:
Nekrasov, Nikita; Schwarz, Albert (1998). "Instantons on noncommutative
1636:
1616:
1573:
1568:
1305:
1258:
1205:
1131:
71:
1608:
940:
of coordinates is the starting point for much of modern geometry.
873:
66:
1337:
1341:
918:
as the distances between any point in the plane and a pair of
27:
Quantum mechanical model based on mathematical matrices
902:. For example, in order to study the geometry of the
2039:
2016:
1993:
1940:
1825:
1733:
1675:
1607:
1556:
1523:
1418:
1375:
1244:and (2,0) superconformal six dimensional theory".
1165:
986:Another notable matrix model capturing aspects of
932:. This property of multiplication is known as the
936:, and this relationship between geometry and the
1353:
898:In geometry, it is often useful to introduce
808:
8:
1360:
1346:
1338:
815:
801:
40:
29:
1304:
1257:
1204:
1130:
1093:(1997), 467-491 (arXiv:hep-th/9612115).
1014:
180:
154:
113:
79:
48:
32:
1246:Communications in Mathematical Physics
7:
1758:Bogomol'nyiâPrasadâSommerfield bound
258:= 4 supersymmetric YangâMills theory
976:noncommutative quantum field theory
894:Noncommutative quantum field theory
189:Geometric Langlands correspondence
25:
1059:Connes, Douglas, and Schwarz 1998
1955:Eleven-dimensional supergravity
855:, after the authors' initials.
1293:Journal of High Energy Physics
1193:Journal of High Energy Physics
906:, one defines the coordinates
1:
1403:Second superstring revolution
1323:10.1088/1126-6708/1999/09/032
1223:10.1088/1126-6708/1998/02/003
1897:Generalized complex manifold
1398:First superstring revolution
953:is not necessarily equal to
827:In theoretical physics, the
2559:
1495:Non-critical string theory
887:
835:model proposed in 1997 by
1103:IKKT matrix model in nLab
1068:Nekrasov and Schwarz 1998
1031:BFSS matrix model in nLab
2031:Introduction to M-theory
1725:WessâZuminoâWitten model
1667:HananyâWitten transition
1393:History of string theory
1149:10.1103/physrevd.55.5112
114:Non-perturbative results
1710:Vertex operator algebra
1410:String theory landscape
1168:Noncommutative Geometry
1077:Seiberg and Witten 1999
944:Noncommutative geometry
890:Noncommutative geometry
884:Noncommutative geometry
2008:AdS/CFT correspondence
1763:Exceptional Lie groups
1705:Superconformal algebra
1677:Conformal field theory
1548:MontonenâOlive duality
1500:Non-linear sigma model
1164:Connes, Alain (1994).
962:In a paper from 1998,
851:; it is also known as
230:Conformal field theory
147:AdS/CFT correspondence
2003:Holographic principle
1970:Type IIB supergravity
1965:Type IIA supergravity
1817:-form electrodynamics
1436:Bosonic string theory
1276:10.1007/s002200050490
273:Holographic principle
250:Twistor string theory
1922:HoĆavaâWitten theory
1869:HyperkÀhler manifold
1557:Particles and fields
1505:Tachyon condensation
1490:Matrix string theory
1004:Matrix string theory
225:Theory of everything
1960:Type I supergravity
1864:CalabiâYau manifold
1859:Ricci-flat manifold
1838:KaluzaâKlein theory
1579:RamondâRamond field
1485:String field theory
1315:1999JHEP...09..032S
1268:1998CMaPh.198..689N
1215:1998JHEP...02..003C
1141:1997PhRvD..55.5112B
990:string theory, the
938:commutative algebra
263:KaluzaâKlein theory
199:Monstrous moonshine
80:Perturbative theory
49:Fundamental objects
1927:K-theory (physics)
1804:ADE classification
1441:Superstring theory
968:Michael R. Douglas
833:quantum mechanical
2530:
2529:
2312:van Nieuwenhuizen
1848:Why 10 dimensions
1753:ChernâSimons form
1720:KacâMoody algebra
1700:Conformal algebra
1695:Conformal anomaly
1589:Magnetic monopole
1584:KalbâRamond field
1426:NambuâGoto action
1183:978-0-12-185860-5
1119:Physical Review D
1041:Connes 1994, p. 1
1021:Banks et al. 1997
992:IKKT matrix model
853:BFSS matrix model
825:
824:
556:van Nieuwenhuizen
16:(Redirected from
2550:
2040:String theorists
1980:Lie superalgebra
1932:Twisted K-theory
1890:Spin(7)-manifold
1843:Compactification
1685:Virasoro algebra
1468:Heterotic string
1362:
1355:
1348:
1339:
1334:
1308:
1287:
1261:
1243:
1234:
1208:
1187:
1171:
1160:
1134:
1125:(8): 5112â5128.
1105:
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1084:
1078:
1075:
1069:
1066:
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1048:
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1033:
1028:
1022:
1019:
958:
952:
931:
917:
911:
849:Leonard Susskind
817:
810:
803:
219:Related concepts
44:
30:
21:
2558:
2557:
2553:
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2551:
2549:
2548:
2547:
2533:
2532:
2531:
2526:
2035:
2012:
1989:
1936:
1884:
1854:KĂ€hler manifold
1821:
1798:
1791:
1784:
1777:
1770:
1729:
1690:Mirror symmetry
1671:
1657:Brane cosmology
1603:
1552:
1519:
1475:N=2 superstring
1461:Type IIB string
1456:Type IIA string
1431:Polyakov action
1414:
1371:
1366:
1290:
1239:
1237:
1190:
1184:
1163:
1116:
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1016:
1012:
1000:
984:
954:
948:
934:commutative law
923:
913:
907:
904:Euclidean plane
896:
888:Main articles:
886:
880:string theory.
861:
845:Stephen Shenker
821:
776:
775:
286:
278:
277:
235:Quantum gravity
220:
194:Mirror symmetry
28:
23:
22:
15:
12:
11:
5:
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2374:
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2362:Randjbar-Daemi
2359:
2354:
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2344:
2339:
2334:
2329:
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2319:
2314:
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2304:
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2014:
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2011:
2010:
2005:
1999:
1997:
1991:
1990:
1988:
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1985:Lie supergroup
1982:
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1962:
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1662:Quiver diagram
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1525:String duality
1521:
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1512:
1507:
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1492:
1487:
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1471:
1470:
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1451:Type II string
1448:
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1428:
1422:
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1416:
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1407:
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1400:
1390:
1388:Cosmic strings
1385:
1379:
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1373:
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1367:
1365:
1364:
1357:
1350:
1342:
1336:
1335:
1306:hep-th/9908142
1288:
1259:hep-th/9802068
1252:(3): 689â703.
1235:
1206:hep-th/9711162
1188:
1182:
1174:Academic Press
1161:
1132:hep-th/9610043
1112:
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1008:
1007:
1006:
999:
996:
983:
982:Related models
980:
972:Albert Schwarz
885:
882:
860:
857:
841:Willy Fischler
823:
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820:
819:
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805:
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606:Randjbar-Daemi
603:
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583:
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573:
568:
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204:Vertex algebra
201:
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176:
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90:
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37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2555:
2544:
2543:String theory
2541:
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2538:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2507:Zamolodchikov
2505:
2503:
2502:Zamolodchikov
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
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2258:
2255:
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2250:
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2118:
2115:
2113:
2110:
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2100:
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2058:
2055:
2053:
2050:
2048:
2045:
2044:
2042:
2038:
2032:
2029:
2027:
2026:Matrix theory
2024:
2023:
2021:
2019:
2015:
2009:
2006:
2004:
2001:
2000:
1998:
1996:
1992:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1947:
1945:
1943:
1942:Supersymmetry
1939:
1933:
1930:
1928:
1925:
1923:
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1597:
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1594:Dual graviton
1592:
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1539:
1536:
1534:
1531:
1530:
1528:
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1510:RNS formalism
1508:
1506:
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1446:Type I string
1444:
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1370:
1369:String theory
1363:
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1120:
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1024:
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989:
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905:
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883:
881:
879:
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871:
867:
858:
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842:
838:
834:
830:
829:matrix theory
818:
813:
811:
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799:
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795:
790:
787:
785:
782:
781:
780:
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772:
769:
767:
764:
762:
759:
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756:Zamolodchikov
754:
752:
751:Zamolodchikov
749:
747:
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742:
739:
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734:
732:
729:
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724:
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719:
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619:
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584:
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569:
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389:
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334:
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329:
327:
324:
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319:
317:
314:
312:
309:
307:
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302:
299:
297:
294:
292:
289:
288:
282:
281:
274:
271:
269:
266:
264:
261:
259:
257:
253:
251:
248:
246:
243:
241:
240:Supersymmetry
238:
236:
233:
231:
228:
226:
223:
222:
216:
215:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
185:
184:
179:
174:
171:
169:
166:
164:
163:Phenomenology
161:
160:
159:
158:
155:Phenomenology
153:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
119:
118:
117:
112:
106:
102:
98:
94:
91:
89:
86:
85:
84:
83:
78:
73:
70:
68:
65:
63:
62:Cosmic string
60:
58:
55:
54:
53:
52:
47:
43:
39:
38:
35:
34:String theory
31:
19:
2052:Arkani-Hamed
2025:
1950:Supergravity
1917:Moduli space
1814:
1809:Dirac string
1735:Gauge theory
1715:Loop algebra
1652:Black string
1515:GS formalism
1296:
1292:
1249:
1245:
1240:
1196:
1192:
1167:
1122:
1118:
1098:
1090:
1087:
1082:
1073:
1064:
1055:
1046:
1037:
1026:
1017:
991:
985:
964:Alain Connes
961:
955:
949:
942:
928:
924:
914:
908:
897:
866:supergravity
862:
852:
828:
826:
296:Arkani-Hamed
255:
245:Supergravity
18:Matrix field
2412:Silverstein
1912:Orientifold
1647:Black holes
1642:Black brane
1599:Dual photon
1050:Connes 1994
900:coordinates
656:Silverstein
181:Mathematics
93:Superstring
2432:Strominger
2427:Steinhardt
2422:Staudacher
2337:Polchinski
2287:Nanopoulos
2247:Mandelstam
2227:Kontsevich
2067:Berenstein
1995:Holography
1975:Superspace
1874:K3 surface
1833:Worldsheet
1748:Instantons
1376:Background
1299:(9): 032.
1199:(2): 003.
1111:References
1088:Nucl.Phys.
676:Strominger
671:Steinhardt
666:Staudacher
581:Polchinski
531:Nanopoulos
491:Mandelstam
471:Kontsevich
311:Berenstein
268:Multiverse
2467:Veneziano
2347:Rajaraman
2242:Maldacena
2132:Gopakumar
2082:Dijkgraaf
2077:Curtright
1743:Anomalies
1622:NS5-brane
1543:U-duality
1538:S-duality
1533:T-duality
837:Tom Banks
716:Veneziano
591:Rajaraman
486:Maldacena
376:Gopakumar
326:Dijkgraaf
321:Curtright
285:Theorists
173:Landscape
168:Cosmology
132:U-duality
127:T-duality
122:S-duality
105:Heterotic
2537:Category
2522:Zwiebach
2477:Verlinde
2472:Verlinde
2447:Townsend
2442:Susskind
2377:Sagnotti
2342:Polyakov
2297:Nekrasov
2262:Minwalla
2257:Martinec
2222:Knizhnik
2217:Klebanov
2212:Kapustin
2177:'t Hooft
2112:Fischler
2047:AganagiÄ
2018:M-theory
1907:Conifold
1902:Orbifold
1885:manifold
1826:Geometry
1632:M5-brane
1627:M2-brane
1564:Graviton
1480:F-theory
1284:14125789
1157:13073785
998:See also
988:Type IIB
878:Type IIA
870:M-theory
859:Overview
789:Glossary
771:Zwiebach
726:Verlinde
721:Verlinde
696:Townsend
691:'t Hooft
686:Susskind
621:Sagnotti
586:Polyakov
541:Nekrasov
506:Minwalla
501:Martinec
466:Knizhnik
461:Klebanov
456:Kapustin
426:Horowitz
356:Fischler
291:AganagiÄ
209:K-theory
142:F-theory
137:M-theory
2452:Trivedi
2437:Sundrum
2402:Shenker
2392:Seiberg
2387:Schwarz
2357:Randall
2317:Novikov
2307:Nielsen
2292:NÄstase
2202:Kallosh
2187:Gibbons
2127:Gliozzi
2117:Friedan
2107:Ferrara
2092:Douglas
2087:Distler
1637:S-brane
1617:D-brane
1574:Tachyon
1569:Dilaton
1383:Strings
1311:Bibcode
1264:Bibcode
1231:7562354
1211:Bibcode
1137:Bibcode
784:History
701:Trivedi
681:Sundrum
646:Shenker
636:Seiberg
631:Schwarz
601:Randall
561:Novikov
551:Nielsen
536:NÄstase
446:Kallosh
431:Gibbons
371:Gliozzi
361:Friedan
351:Ferrara
336:Douglas
331:Distler
101:Type II
88:Bosonic
72:D-brane
2517:Zumino
2512:Zaslow
2497:Yoneya
2487:Witten
2407:Siegel
2382:Scherk
2352:Ramond
2327:Ooguri
2252:Marolf
2207:Kaluza
2192:Kachru
2182:HoĆava
2172:Harvey
2167:Hanson
2152:Gubser
2142:Greene
2072:Bousso
2057:Atiyah
1609:Branes
1419:Theory
1331:668885
1329:
1282:
1229:
1180:
1155:
970:, and
874:branes
847:, and
766:Zumino
761:Zaslow
746:Yoneya
736:Witten
651:Siegel
626:Scherk
596:Ramond
571:Ooguri
496:Marolf
451:Kaluza
436:Kachru
421:HoĆava
416:Harvey
411:Hanson
396:Gubser
386:Greene
316:Bousso
301:Atiyah
97:Type I
57:String
2457:Turok
2367:RoÄek
2332:Ovrut
2322:Olive
2302:Neveu
2282:Myers
2277:Mukhi
2267:Moore
2237:Linde
2232:Klein
2157:Gukov
2147:Gross
2137:Green
2122:Gates
2102:Dvali
2062:Banks
1327:S2CID
1301:arXiv
1280:S2CID
1254:arXiv
1227:S2CID
1201:arXiv
1197:19981
1153:S2CID
1127:arXiv
1010:Notes
831:is a
706:Turok
611:RoÄek
576:Ovrut
566:Olive
546:Neveu
526:Myers
521:Mukhi
511:Moore
481:Linde
476:Klein
401:Gukov
391:Gross
381:Green
366:Gates
346:Dvali
306:Banks
67:Brane
2482:Wess
2462:Vafa
2372:Rohm
2272:Motl
2197:Kaku
2162:Guth
2097:Duff
1297:1999
1178:ISBN
1091:B498
920:axes
912:and
892:and
731:Wess
711:Vafa
616:Rohm
516:Motl
441:Kaku
406:Guth
341:Duff
2492:Yau
2417:SÆĄn
2397:Sen
1319:doi
1272:doi
1250:198
1219:doi
1145:doi
876:in
741:Yau
661:SÆĄn
641:Sen
2539::
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1779:,
1772:,
1325:.
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1278:.
1270:.
1262:.
1248:.
1225:.
1217:.
1209:.
1195:.
1176:.
1172:.
1151:.
1143:.
1135:.
1123:55
1121:.
966:,
956:yx
950:xy
929:yx
927:=
925:xy
843:,
839:,
103:,
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1883:2
1881:G
1850:?
1815:p
1800:)
1797:8
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1790:7
1788:E
1783:6
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1776:4
1774:F
1769:2
1767:G
1765:(
1361:e
1354:t
1347:v
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1313::
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1286:.
1274::
1266::
1256::
1241:R
1233:.
1221::
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1186:.
1159:.
1147::
1139::
1129::
915:y
909:x
816:e
809:t
802:v
256:N
107:)
95:(
20:)
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