3570:
983:. This form of entropy provides a measure of how far from a pure state a given quantum state is, or, equivalently, how entangled it is. The entanglement entropy is a useful concept in many areas, such as in condensed matter physics and quantum many-body systems. Given its use, and its suggestive similarity to the Bekenstein–Hawking entropy, it is desirable to have a holographic description of entanglement entropy in terms of gravity.
28:
1080:
system to a quantum system which does admit such a microscopic description. In this case, the CFT has discrete eigenstates and the thermal state is the canonical ensemble of these states. The entropy of this ensemble can be calculated through normal means, and yields the same result as predicted by the area law. This turns out to be a special case of the Ryu–Takayanagi conjecture.
1079:
The
Bekenstein–Hawking area law, while claiming that the area of the black hole horizon is proportional to the black hole's entropy, fails to provide a sufficient microscopic description of how this entropy arises. The holographic principle provides such a description by relating the black hole
1005:
is one example of such duality. Here, the field theory is defined on a fixed background and is equivalent to a quantum gravitational theory whose different states each correspond to a possible spacetime geometry. The conformal field theory is often viewed as living on the boundary of the higher
2740:
976:(in this case the exterior of the black hole) that is not affected by another region (in this case the interior). The Bekenstein–Hawking area law states that the area of this surface is proportional to the entropy of the information lost behind it.
1894:
2495:
833:
spacetime. The formula characterizes "holographic screens" in the bulk; that is, it specifies which regions of the bulk geometry are "responsible to particular information in the dual CFT". The conjecture is named after
940:
2573:
1170:
1556:
1030:
the vacuum state corresponds to pure AdS space, whereas the thermal state corresponds to a planar black hole. Important for the present discussion is that the thermal state of a CFT defined on the
979:
The
Bekenstein–Hawking entropy is a statement about the gravitational entropy of a system; however, there is another type of entropy that is important in quantum information theory, namely the
2193:
1610:
1242:
2551:
2127:
2017:
1711:
972:
is a measure of the information lost to external observers due to the presence of the horizon. The horizon of the black hole acts as a "screen" distinguishing one region of the
1934:
1006:
dimensional space whose gravitational theory it defines. The result of such a duality is a dictionary between the two equivalent descriptions. For example, in a CFT defined on
2371:
1964:
1756:
1639:
970:
2265:
1476:
1428:
1401:
1368:
1319:
1289:
2331:
870:
of black holes and their geometry. Specifically, the
Bekenstein–Hawking area formula conjectures that the entropy of a black hole is proportional to its surface area:
1106:
1766:
2037:
1984:
1213:
2381:
1074:
2305:
2285:
2213:
2077:
2057:
1659:
1339:
1262:
1048:
1024:
1558:. The conjecture provides an explicit geometric interpretation of the entanglement entropy of the boundary CFT, namely as the area of a surface in the bulk.
240:
3158:
843:
3299:
2735:{\displaystyle S_{A}={\frac {R}{4G}}\log {(e^{2\rho _{0}}\sin ^{2}{\frac {\pi l}{L}})}={\frac {c}{3}}\log {(e^{\rho _{0}}\sin {\frac {\pi l}{L}})}}
876:
800:
3264:
3067:
2553:, then using the Ryu–Takayanagi formula to compute the entanglement entropy. Plugging in the length of the minimal surface calculated in (
1438:
when the context is clear. Furthermore, property (1) ensures that the formula preserves certain features of entanglement entropy, such as
1116:
174:
2982:
Hubeny, Veronika E.; Rangamani, Mukund; Takayanagi, Tadashi (23 July 2007). "A Covariant
Holographic Entanglement Entropy Proposal".
3573:
3186:
3289:
179:
3479:
3347:
1481:
546:
3274:
2059:
is the length of the CFT system, in this case the circumference of the cylinder calculated with the appropriate metric, and
3362:
3151:
3367:
850:
for "their insights on quantum entropy in quantum gravity and quantum field theories". The formula was generalized to a
571:
3445:
3246:
2135:
1569:
1759:
793:
3342:
2864:
2307:. The minimal surface is easy to identify in this case, as it is just the geodesic through the bulk that connects
3455:
3432:
3382:
3352:
863:
774:
406:
366:
3463:
3509:
3144:
1218:
769:
591:
511:
326:
3309:
3201:
471:
248:
2883:
Ryu, Shinsei; Takayanagi, Tadashi (May 2006). "Holographic
Derivation of Entanglement Entropy from AdS/CFT".
3499:
3241:
2962:"ICTP Announces 2024 Dirac Medallists Four physicists recognized for their contributions to quantum entropy"
2516:
980:
822:
736:
541:
158:
90:
2084:
3600:
3181:
1937:
1371:
1002:
826:
786:
496:
215:
132:
1989:
646:
3407:
3319:
3216:
3206:
1673:
992:
846:
for "fundamental ideas about entropy in quantum field theory and quantum gravity", and awarded the 2024
818:
351:
291:
258:
235:
86:
73:
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1901:
521:
2336:
1943:
1732:
1615:
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3106:
3001:
2799:
830:
741:
210:
148:
82:
948:
651:
536:
184:
1108:
of an AdS space time on whose boundary we define the dual CFT. The Ryu–Takayanagi formula states:
3494:
3422:
3377:
3122:
3096:
3073:
3045:
3017:
2991:
2918:
2892:
2823:
2789:
2218:
1441:
1406:
1379:
1346:
1297:
1267:
851:
847:
839:
701:
616:
516:
476:
356:
321:
194:
78:
3091:
Calabrese, Pasquale; Cardy, John (2004-06-11). "Entanglement entropy and quantum field theory".
997:
The holographic principle states that gravitational theories in a given dimension are dual to a
676:
2937:"Recipients of the 2015 Breakthrough Prizes in Fundamental Physics and Life Sciences Announced"
2780:
Ryu, Shinsei; Takayanagi, Tadashi (2006-08-21). "Aspects of
Holographic Entanglement Entropy".
2310:
1889:{\displaystyle ds^{2}=R^{2}(-\cosh {\rho ^{2}dt^{2}}+d\rho ^{2}+\sinh {\rho ^{2}d\theta ^{2}})}
3539:
3534:
3314:
3279:
3063:
2910:
2815:
661:
566:
401:
311:
281:
2490:{\displaystyle \cosh {(L_{\gamma _{A}}/R)}=1+2\sinh ^{2}\rho _{0}\sin ^{2}{\frac {\pi l}{L}}}
1091:
3357:
3114:
3055:
3009:
2902:
2807:
671:
606:
576:
456:
396:
361:
306:
296:
276:
153:
42:
3118:
2022:
1969:
1191:
842:, who jointly published the result in 2006. As a result, the authors were awarded the 2015
3544:
3167:
1566:
In their original paper, Ryu and
Takayanagi show this result explicitly for an example in
1027:
756:
711:
656:
641:
631:
526:
491:
316:
220:
1053:
596:
3110:
3013:
3005:
2811:
2803:
3514:
2290:
2270:
2198:
2062:
2042:
1662:
1644:
1324:
1291:
is the Ryu–Takayanagi surface in the bulk. This surface must satisfy three properties:
1247:
1033:
1009:
731:
726:
686:
621:
611:
531:
451:
441:
436:
431:
346:
341:
336:
301:
286:
189:
3589:
3489:
3399:
3077:
2922:
721:
706:
681:
666:
636:
581:
556:
501:
486:
481:
446:
421:
411:
381:
225:
47:
19:
3126:
3021:
2827:
27:
3417:
3324:
3231:
3226:
3221:
3196:
998:
751:
586:
466:
416:
386:
371:
230:
2906:
2373:. Remembering the lattice cutoff, the length of the geodesic can be calculated as
3484:
835:
746:
716:
696:
551:
506:
461:
426:
376:
3059:
3549:
3334:
3269:
3191:
3040:
Van
Raamsdonk, Mark (31 August 2016). "Lectures on Gravity and Entanglement".
2842:
691:
626:
561:
253:
2819:
3504:
3372:
3284:
3256:
973:
935:{\displaystyle S_{\text{BH}}={\frac {k_{\text{B}}A}{4\ell _{\text{P}}^{2}}}}
601:
391:
331:
117:
112:
107:
2914:
1612:
where an expression for the entanglement entropy is already known. For an
3412:
3211:
127:
122:
2936:
3101:
2897:
2794:
867:
57:
3050:
3136:
2996:
1215:
is the entanglement entropy of the CFT in some spatial sub-region
52:
1165:{\displaystyle S_{A}={\frac {{\text{Area of }}\gamma _{A}}{4G}}}
3140:
1403:
extremizes the area. If there are multiple extremal surfaces,
1434:
Because of property (3), this surface is typically called the
2039:
is analogous to the corresponding CFT having a UV cutoff. If
2841:
Stanford
Institute for Theoretical Physics (2015-10-15),
3093:
Journal of
Statistical Mechanics: Theory and Experiment
2758:
This agrees with the result calculated by usual means.
1551:{\displaystyle S_{A_{1}+A_{2}}\geq S_{A_{1}\cup A_{2}}}
2576:
2519:
2384:
2339:
2313:
2293:
2273:
2221:
2215:
slice and take the subregion A of the boundary to be
2201:
2138:
2087:
2065:
2045:
2025:
1992:
1972:
1946:
1904:
1769:
1735:
1676:
1647:
1618:
1572:
1484:
1444:
1409:
1382:
1349:
1327:
1300:
1270:
1250:
1221:
1194:
1119:
1094:
1056:
1036:
1012:
951:
879:
2132:
In this case, the boundary CFT lives at coordinates
821:
that posits a quantitative relationship between the
3527:
3472:
3454:
3431:
3398:
3391:
3333:
3298:
3255:
3174:
1076:dimensional Schwarzschild black hole in AdS space.
2734:
2545:
2489:
2365:
2325:
2299:
2279:
2259:
2207:
2187:
2121:
2071:
2051:
2031:
2011:
1978:
1958:
1928:
1888:
1750:
1705:
1653:
1633:
1604:
1550:
1470:
1422:
1395:
1362:
1333:
1313:
1283:
1256:
1236:
1207:
1164:
1100:
1068:
1042:
1018:
964:
934:
2188:{\displaystyle (t,\rho _{0},\theta )=(t,\theta )}
1605:{\displaystyle {\text{AdS}}_{3}/{\text{CFT}}_{2}}
2775:
2773:
2771:
3152:
794:
8:
866:suggests certain relationships between the
3395:
3159:
3145:
3137:
2866:Introduction to the Ryu–Takayanagi Formula
801:
787:
26:
15:
3100:
3049:
2995:
2896:
2793:
2713:
2699:
2694:
2686:
2670:
2648:
2639:
2627:
2619:
2611:
2590:
2581:
2575:
2529:
2524:
2518:
2472:
2463:
2453:
2440:
2412:
2404:
2399:
2391:
2383:
2355:
2338:
2312:
2292:
2272:
2246:
2220:
2200:
2152:
2137:
2111:
2097:
2092:
2086:
2064:
2044:
2024:
2003:
1991:
1971:
1945:
1903:
1876:
1863:
1858:
1843:
1826:
1813:
1808:
1790:
1777:
1768:
1742:
1737:
1734:
1683:
1675:
1646:
1625:
1620:
1617:
1596:
1591:
1585:
1579:
1574:
1571:
1540:
1527:
1522:
1507:
1494:
1489:
1483:
1462:
1449:
1443:
1414:
1408:
1387:
1381:
1354:
1348:
1326:
1305:
1299:
1275:
1269:
1249:
1237:{\displaystyle A\subset \partial \Sigma }
1220:
1199:
1193:
1145:
1136:
1133:
1124:
1118:
1093:
1055:
1035:
1011:
956:
950:
923:
918:
900:
893:
884:
878:
844:Breakthrough Prize in Fundamental Physics
2565:), the entanglement entropy is given by
2767:
166:
140:
99:
65:
34:
18:
2546:{\displaystyle e^{\rho _{0}}>>1}
1050:dimensional sphere corresponds to the
2122:{\displaystyle e^{\rho _{0}}\sim L/a}
981:entanglement (or von Neumann) entropy
7:
3035:
3033:
3031:
2567:
2559:) and recalling the central charge (
2375:
1667:
1110:
244:= 4 supersymmetric Yang–Mills theory
3042:New Frontiers in Fields and Strings
2012:{\displaystyle \rho \leq \rho _{0}}
1953:
1706:{\displaystyle c={\frac {3R}{2G}}}
1231:
1228:
1095:
829:and the geometry of an associated
175:Geometric Langlands correspondence
14:
3574:Template:Quantum mechanics topics
2019:. This act of imposing a maximum
1940:). Since this metric diverges at
1929:{\displaystyle (t,\rho ,\theta )}
3569:
3568:
3290:Topological quantum field theory
3119:10.1088/1742-5468/2004/06/P06002
2366:{\displaystyle \theta =2\pi l/L}
2079:is the lattice spacing, we have
1959:{\displaystyle \rho \to \infty }
1751:{\displaystyle {\text{AdS}}_{3}}
1634:{\displaystyle {\text{AdS}}_{3}}
3275:Jackiw–Teitelboim gravity
1430:is the one with the least area.
945:The Bekenstein–Hawking entropy
3480:Causal dynamical triangulation
3348:Black hole information paradox
2782:Journal of High Energy Physics
2728:
2687:
2663:
2612:
2420:
2392:
2254:
2228:
2182:
2170:
2164:
2139:
1950:
1923:
1905:
1883:
1796:
1:
3446:Wheeler–DeWitt equation
3014:10.1088/1126-6708/2007/07/062
2907:10.1103/PhysRevLett.96.181602
2863:Fukami, Masaya (March 2018),
2812:10.1088/1126-6708/2006/08/045
965:{\displaystyle S_{\text{BH}}}
864:thermodynamics of black holes
3368:Cosmic censorship hypothesis
3187:Batalin–Vilkovisky formalism
1001:in one lower dimension. The
2561:
2555:
2260:{\displaystyle \theta \in }
1471:{\displaystyle S_{A}=S_{B}}
1423:{\displaystyle \gamma _{A}}
1396:{\displaystyle \gamma _{A}}
1363:{\displaystyle \gamma _{A}}
1314:{\displaystyle \gamma _{A}}
1284:{\displaystyle \gamma _{A}}
3617:
3464:Hartle–Hawking state
3363:Bousso's holographic bound
3343:Black hole complementarity
3060:10.1142/9789813149441_0005
990:
3560:
3456:Euclidean quantum gravity
3433:Canonical quantum gravity
3383:Gravitational singularity
3353:Black-hole thermodynamics
3237:Ryu–Takayanagi conjecture
2941:www.breakthroughprize.org
2326:{\displaystyle \theta =0}
1321:has the same boundary as
1088:Consider a spatial slice
987:Holographic preliminaries
815:Ryu–Takayanagi conjecture
3510:Superfluid vacuum theory
3265:2+1D topological gravity
2844:Gravity and Entanglement
1936:(essentially a stack of
100:Non-perturbative results
3500:Noncommutative geometry
3247:Weinberg–Witten theorem
3242:Trans-Planckian problem
1101:{\displaystyle \Sigma }
848:Dirac Medal of the ICTP
817:is a conjecture within
3182:AdS/CFT correspondence
3095:. P06002 (6): P06002.
2736:
2547:
2513:If it is assumed that
2491:
2367:
2327:
2301:
2281:
2261:
2209:
2189:
2123:
2073:
2053:
2033:
2013:
1980:
1960:
1930:
1890:
1752:
1707:
1655:
1635:
1606:
1552:
1472:
1424:
1397:
1364:
1335:
1315:
1285:
1258:
1238:
1209:
1166:
1102:
1070:
1044:
1020:
1003:AdS/CFT correspondence
966:
936:
827:conformal field theory
216:Conformal field theory
133:AdS/CFT correspondence
3408:Bosonic string theory
3320:Semiclassical gravity
3217:Holographic principle
3207:Gravitational anomaly
2737:
2548:
2492:
2368:
2328:
2302:
2282:
2262:
2210:
2190:
2124:
2074:
2054:
2034:
2032:{\displaystyle \rho }
2014:
1981:
1979:{\displaystyle \rho }
1961:
1931:
1891:
1753:
1708:
1661:, the dual CFT has a
1656:
1636:
1607:
1553:
1473:
1425:
1398:
1365:
1336:
1316:
1286:
1259:
1239:
1210:
1208:{\displaystyle S_{A}}
1167:
1103:
1071:
1045:
1021:
993:Holographic principle
967:
937:
259:Holographic principle
236:Twistor string theory
3441:Loop quantum gravity
3300:Quantum field theory
3044:. pp. 297–351.
2574:
2517:
2382:
2337:
2311:
2291:
2271:
2219:
2199:
2136:
2085:
2063:
2043:
2023:
1990:
1970:
1944:
1902:
1767:
1733:
1674:
1645:
1616:
1570:
1482:
1442:
1407:
1380:
1347:
1325:
1298:
1268:
1248:
1244:with its complement
1219:
1192:
1117:
1092:
1054:
1034:
1010:
949:
877:
823:entanglement entropy
211:Theory of everything
3310:Bunch–Davies vacuum
3302:in curved spacetime
3202:Faddeev–Popov ghost
3111:2004JSMTE..06..002C
3006:2007JHEP...07..062H
2804:2006JHEP...08..045R
2195:. Consider a fixed
1069:{\displaystyle d+1}
928:
249:Kaluza–Klein theory
185:Monstrous moonshine
66:Perturbative theory
35:Fundamental objects
3495:Group field theory
3423:Superstring theory
3378:Firewall (physics)
2732:
2543:
2487:
2363:
2323:
2297:
2277:
2257:
2205:
2185:
2119:
2069:
2049:
2029:
2009:
1976:
1956:
1926:
1886:
1748:
1703:
1651:
1631:
1602:
1548:
1468:
1420:
1393:
1360:
1331:
1311:
1281:
1254:
1234:
1205:
1162:
1098:
1066:
1040:
1016:
962:
932:
914:
840:Tadashi Takayanagi
3583:
3582:
3540:Eternal inflation
3535:Quantum cosmology
3523:
3522:
3315:Hawking radiation
3280:Liouville gravity
3069:978-981-314-943-4
2756:
2755:
2726:
2678:
2661:
2603:
2511:
2510:
2485:
2300:{\displaystyle A}
2287:is the length of
2280:{\displaystyle l}
2208:{\displaystyle t}
2072:{\displaystyle a}
2052:{\displaystyle L}
1986:is restricted to
1740:
1727:
1726:
1701:
1654:{\displaystyle R}
1623:
1594:
1577:
1334:{\displaystyle A}
1257:{\displaystyle B}
1186:
1185:
1160:
1139:
1043:{\displaystyle d}
1019:{\displaystyle d}
959:
930:
921:
903:
887:
811:
810:
542:van Nieuwenhuizen
3608:
3572:
3571:
3396:
3358:Bekenstein bound
3175:Central concepts
3161:
3154:
3147:
3138:
3131:
3130:
3104:
3088:
3082:
3081:
3053:
3037:
3026:
3025:
2999:
2979:
2973:
2972:
2970:
2968:
2958:
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2933:
2927:
2926:
2900:
2880:
2874:
2873:
2871:
2860:
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2853:
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2851:
2838:
2832:
2831:
2797:
2777:
2750:
2741:
2739:
2738:
2733:
2731:
2727:
2722:
2714:
2706:
2705:
2704:
2703:
2679:
2671:
2666:
2662:
2657:
2649:
2644:
2643:
2634:
2633:
2632:
2631:
2604:
2602:
2591:
2586:
2585:
2568:
2552:
2550:
2549:
2544:
2536:
2535:
2534:
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2505:
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2473:
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2444:
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2416:
2411:
2410:
2409:
2408:
2376:
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2332:
2330:
2329:
2324:
2306:
2304:
2303:
2298:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2250:
2214:
2212:
2211:
2206:
2194:
2192:
2191:
2186:
2157:
2156:
2128:
2126:
2125:
2120:
2115:
2104:
2103:
2102:
2101:
2078:
2076:
2075:
2070:
2058:
2056:
2055:
2050:
2038:
2036:
2035:
2030:
2018:
2016:
2015:
2010:
2008:
2007:
1985:
1983:
1982:
1977:
1965:
1963:
1962:
1957:
1938:hyperbolic disks
1935:
1933:
1932:
1927:
1895:
1893:
1892:
1887:
1882:
1881:
1880:
1868:
1867:
1848:
1847:
1832:
1831:
1830:
1818:
1817:
1795:
1794:
1782:
1781:
1757:
1755:
1754:
1749:
1747:
1746:
1741:
1738:
1721:
1712:
1710:
1709:
1704:
1702:
1700:
1692:
1684:
1668:
1660:
1658:
1657:
1652:
1641:space of radius
1640:
1638:
1637:
1632:
1630:
1629:
1624:
1621:
1611:
1609:
1608:
1603:
1601:
1600:
1595:
1592:
1589:
1584:
1583:
1578:
1575:
1557:
1555:
1554:
1549:
1547:
1546:
1545:
1544:
1532:
1531:
1514:
1513:
1512:
1511:
1499:
1498:
1477:
1475:
1474:
1469:
1467:
1466:
1454:
1453:
1429:
1427:
1426:
1421:
1419:
1418:
1402:
1400:
1399:
1394:
1392:
1391:
1369:
1367:
1366:
1361:
1359:
1358:
1340:
1338:
1337:
1332:
1320:
1318:
1317:
1312:
1310:
1309:
1290:
1288:
1287:
1282:
1280:
1279:
1263:
1261:
1260:
1255:
1243:
1241:
1240:
1235:
1214:
1212:
1211:
1206:
1204:
1203:
1180:
1171:
1169:
1168:
1163:
1161:
1159:
1151:
1150:
1149:
1140:
1137:
1134:
1129:
1128:
1111:
1107:
1105:
1104:
1099:
1075:
1073:
1072:
1067:
1049:
1047:
1046:
1041:
1025:
1023:
1022:
1017:
971:
969:
968:
963:
961:
960:
957:
941:
939:
938:
933:
931:
929:
927:
922:
919:
909:
905:
904:
901:
894:
889:
888:
885:
803:
796:
789:
205:Related concepts
30:
16:
3616:
3615:
3611:
3610:
3609:
3607:
3606:
3605:
3586:
3585:
3584:
3579:
3578:
3556:
3545:FRW/CFT duality
3519:
3468:
3450:
3427:
3387:
3329:
3301:
3294:
3251:
3170:
3168:Quantum gravity
3165:
3135:
3134:
3090:
3089:
3085:
3070:
3039:
3038:
3029:
2981:
2980:
2976:
2966:
2964:
2960:
2959:
2955:
2945:
2943:
2935:
2934:
2930:
2885:Phys. Rev. Lett
2882:
2881:
2877:
2869:
2862:
2861:
2857:
2849:
2847:
2840:
2839:
2835:
2779:
2778:
2769:
2764:
2748:
2715:
2695:
2690:
2650:
2635:
2623:
2615:
2595:
2577:
2572:
2571:
2525:
2520:
2515:
2514:
2503:
2474:
2459:
2449:
2436:
2400:
2395:
2380:
2379:
2335:
2334:
2309:
2308:
2289:
2288:
2269:
2268:
2217:
2216:
2197:
2196:
2148:
2134:
2133:
2093:
2088:
2083:
2082:
2061:
2060:
2041:
2040:
2021:
2020:
1999:
1988:
1987:
1968:
1967:
1942:
1941:
1900:
1899:
1872:
1859:
1839:
1822:
1809:
1786:
1773:
1765:
1764:
1736:
1731:
1730:
1719:
1693:
1685:
1672:
1671:
1643:
1642:
1619:
1614:
1613:
1590:
1573:
1568:
1567:
1564:
1536:
1523:
1518:
1503:
1490:
1485:
1480:
1479:
1458:
1445:
1440:
1439:
1436:minimal surface
1410:
1405:
1404:
1383:
1378:
1377:
1350:
1345:
1344:
1323:
1322:
1301:
1296:
1295:
1271:
1266:
1265:
1246:
1245:
1217:
1216:
1195:
1190:
1189:
1178:
1152:
1141:
1135:
1120:
1115:
1114:
1090:
1089:
1086:
1052:
1051:
1032:
1031:
1028:Minkowski space
1008:
1007:
995:
989:
952:
947:
946:
910:
896:
895:
880:
875:
874:
860:
807:
762:
761:
272:
264:
263:
221:Quantum gravity
206:
180:Mirror symmetry
12:
11:
5:
3614:
3612:
3604:
3603:
3598:
3588:
3587:
3581:
3580:
3577:
3576:
3562:
3561:
3558:
3557:
3555:
3554:
3553:
3552:
3547:
3542:
3531:
3529:
3525:
3524:
3521:
3520:
3518:
3517:
3515:Twistor theory
3512:
3507:
3502:
3497:
3492:
3487:
3482:
3476:
3474:
3470:
3469:
3467:
3466:
3460:
3458:
3452:
3451:
3449:
3448:
3443:
3437:
3435:
3429:
3428:
3426:
3425:
3420:
3415:
3410:
3404:
3402:
3393:
3389:
3388:
3386:
3385:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3339:
3337:
3331:
3330:
3328:
3327:
3322:
3317:
3312:
3306:
3304:
3296:
3295:
3293:
3292:
3287:
3282:
3277:
3272:
3267:
3261:
3259:
3253:
3252:
3250:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3178:
3176:
3172:
3171:
3166:
3164:
3163:
3156:
3149:
3141:
3133:
3132:
3102:hep-th/0405152
3083:
3068:
3027:
2974:
2953:
2928:
2898:hep-th/0603001
2891:(18): 181602.
2875:
2855:
2833:
2795:hep-th/0605073
2766:
2765:
2763:
2760:
2754:
2753:
2744:
2742:
2730:
2725:
2721:
2718:
2712:
2709:
2702:
2698:
2693:
2689:
2685:
2682:
2677:
2674:
2669:
2665:
2660:
2656:
2653:
2647:
2642:
2638:
2630:
2626:
2622:
2618:
2614:
2610:
2607:
2601:
2598:
2594:
2589:
2584:
2580:
2542:
2539:
2532:
2528:
2523:
2509:
2508:
2499:
2497:
2484:
2480:
2477:
2471:
2466:
2462:
2456:
2452:
2448:
2443:
2439:
2435:
2432:
2429:
2426:
2422:
2419:
2415:
2407:
2403:
2398:
2394:
2390:
2387:
2362:
2358:
2354:
2351:
2348:
2345:
2342:
2322:
2319:
2316:
2296:
2276:
2256:
2253:
2249:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2204:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2151:
2147:
2144:
2141:
2118:
2114:
2110:
2107:
2100:
2096:
2091:
2068:
2048:
2028:
2006:
2002:
1998:
1995:
1975:
1955:
1952:
1949:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1885:
1879:
1875:
1871:
1866:
1862:
1857:
1854:
1851:
1846:
1842:
1838:
1835:
1829:
1825:
1821:
1816:
1812:
1807:
1804:
1801:
1798:
1793:
1789:
1785:
1780:
1776:
1772:
1745:
1725:
1724:
1715:
1713:
1699:
1696:
1691:
1688:
1682:
1679:
1663:central charge
1650:
1628:
1599:
1588:
1582:
1563:
1560:
1543:
1539:
1535:
1530:
1526:
1521:
1517:
1510:
1506:
1502:
1497:
1493:
1488:
1465:
1461:
1457:
1452:
1448:
1432:
1431:
1417:
1413:
1390:
1386:
1375:
1357:
1353:
1342:
1330:
1308:
1304:
1278:
1274:
1253:
1233:
1230:
1227:
1224:
1202:
1198:
1184:
1183:
1174:
1172:
1158:
1155:
1148:
1144:
1132:
1127:
1123:
1097:
1085:
1082:
1065:
1062:
1059:
1039:
1015:
991:Main article:
988:
985:
955:
943:
942:
926:
917:
913:
908:
899:
892:
883:
859:
856:
854:form in 2007.
831:anti-de Sitter
809:
808:
806:
805:
798:
791:
783:
780:
779:
778:
777:
772:
764:
763:
760:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
609:
604:
599:
594:
592:Randjbar-Daemi
589:
584:
579:
574:
569:
564:
559:
554:
549:
544:
539:
534:
529:
524:
519:
514:
509:
504:
499:
494:
489:
484:
479:
474:
469:
464:
459:
454:
449:
444:
439:
434:
429:
424:
419:
414:
409:
404:
399:
394:
389:
384:
379:
374:
369:
364:
359:
354:
349:
344:
339:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
284:
279:
273:
270:
269:
266:
265:
262:
261:
256:
251:
246:
238:
233:
228:
223:
218:
213:
207:
204:
203:
200:
199:
198:
197:
192:
190:Vertex algebra
187:
182:
177:
169:
168:
164:
163:
162:
161:
156:
151:
143:
142:
138:
137:
136:
135:
130:
125:
120:
115:
110:
102:
101:
97:
96:
95:
94:
76:
68:
67:
63:
62:
61:
60:
55:
50:
45:
37:
36:
32:
31:
23:
22:
13:
10:
9:
6:
4:
3:
2:
3613:
3602:
3601:String theory
3599:
3597:
3594:
3593:
3591:
3575:
3567:
3564:
3563:
3559:
3551:
3548:
3546:
3543:
3541:
3538:
3537:
3536:
3533:
3532:
3530:
3526:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3490:Dual graviton
3488:
3486:
3483:
3481:
3478:
3477:
3475:
3471:
3465:
3462:
3461:
3459:
3457:
3453:
3447:
3444:
3442:
3439:
3438:
3436:
3434:
3430:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3405:
3403:
3401:
3400:String theory
3397:
3394:
3390:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3340:
3338:
3336:
3332:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3307:
3305:
3303:
3297:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3262:
3260:
3258:
3254:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3179:
3177:
3173:
3169:
3162:
3157:
3155:
3150:
3148:
3143:
3142:
3139:
3128:
3124:
3120:
3116:
3112:
3108:
3103:
3098:
3094:
3087:
3084:
3079:
3075:
3071:
3065:
3061:
3057:
3052:
3047:
3043:
3036:
3034:
3032:
3028:
3023:
3019:
3015:
3011:
3007:
3003:
2998:
2993:
2989:
2985:
2978:
2975:
2963:
2957:
2954:
2942:
2938:
2932:
2929:
2924:
2920:
2916:
2912:
2908:
2904:
2899:
2894:
2890:
2886:
2879:
2876:
2868:
2867:
2859:
2856:
2846:
2845:
2837:
2834:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2801:
2796:
2791:
2787:
2783:
2776:
2774:
2772:
2768:
2761:
2759:
2752:
2745:
2743:
2723:
2719:
2716:
2710:
2707:
2700:
2696:
2691:
2683:
2680:
2675:
2672:
2667:
2658:
2654:
2651:
2645:
2640:
2636:
2628:
2624:
2620:
2616:
2608:
2605:
2599:
2596:
2592:
2587:
2582:
2578:
2570:
2569:
2566:
2564:
2563:
2558:
2557:
2540:
2537:
2530:
2526:
2521:
2507:
2500:
2498:
2482:
2478:
2475:
2469:
2464:
2460:
2454:
2450:
2446:
2441:
2437:
2433:
2430:
2427:
2424:
2417:
2413:
2405:
2401:
2396:
2388:
2385:
2378:
2377:
2374:
2360:
2356:
2352:
2349:
2346:
2343:
2340:
2320:
2317:
2314:
2294:
2274:
2251:
2247:
2243:
2240:
2237:
2234:
2231:
2225:
2222:
2202:
2179:
2176:
2173:
2167:
2161:
2158:
2153:
2149:
2145:
2142:
2130:
2116:
2112:
2108:
2105:
2098:
2094:
2089:
2080:
2066:
2046:
2026:
2004:
2000:
1996:
1993:
1973:
1947:
1939:
1920:
1917:
1914:
1911:
1908:
1896:
1877:
1873:
1869:
1864:
1860:
1855:
1852:
1849:
1844:
1840:
1836:
1833:
1827:
1823:
1819:
1814:
1810:
1805:
1802:
1799:
1791:
1787:
1783:
1778:
1774:
1770:
1762:
1761:
1743:
1729:Furthermore,
1723:
1716:
1714:
1697:
1694:
1689:
1686:
1680:
1677:
1670:
1669:
1666:
1664:
1648:
1626:
1597:
1586:
1580:
1561:
1559:
1541:
1537:
1533:
1528:
1524:
1519:
1515:
1508:
1504:
1500:
1495:
1491:
1486:
1463:
1459:
1455:
1450:
1446:
1437:
1415:
1411:
1388:
1384:
1376:
1373:
1355:
1351:
1343:
1328:
1306:
1302:
1294:
1293:
1292:
1276:
1272:
1251:
1225:
1222:
1200:
1196:
1182:
1175:
1173:
1156:
1153:
1146:
1142:
1138:Area of
1130:
1125:
1121:
1113:
1112:
1109:
1083:
1081:
1077:
1063:
1060:
1057:
1037:
1029:
1013:
1004:
1000:
994:
986:
984:
982:
977:
975:
953:
924:
915:
911:
906:
897:
890:
881:
873:
872:
871:
869:
865:
857:
855:
853:
849:
845:
841:
837:
832:
828:
824:
820:
816:
804:
799:
797:
792:
790:
785:
784:
782:
781:
776:
773:
771:
768:
767:
766:
765:
758:
755:
753:
750:
748:
745:
743:
742:Zamolodchikov
740:
738:
737:Zamolodchikov
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
523:
520:
518:
515:
513:
510:
508:
505:
503:
500:
498:
495:
493:
490:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
375:
373:
370:
368:
365:
363:
360:
358:
355:
353:
350:
348:
345:
343:
340:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
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260:
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234:
232:
229:
227:
226:Supersymmetry
224:
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217:
214:
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202:
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196:
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149:Phenomenology
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141:Phenomenology
139:
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51:
49:
48:Cosmic string
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41:
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38:
33:
29:
25:
24:
21:
20:String theory
17:
3565:
3528:Applications
3418:Supergravity
3325:Unruh effect
3236:
3232:Quantum foam
3227:Planck units
3222:IR/UV mixing
3197:Causal patch
3092:
3086:
3041:
2987:
2983:
2977:
2967:10 September
2965:. Retrieved
2956:
2944:. Retrieved
2940:
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1078:
1026:dimensional
999:gauge theory
996:
978:
944:
861:
814:
812:
282:Arkani-Hamed
241:
231:Supergravity
3596:Conjectures
3485:Causal sets
3335:Black holes
2872:, p. 2
836:Shinsei Ryu
642:Silverstein
167:Mathematics
79:Superstring
3590:Categories
3550:Multiverse
3392:Approaches
3270:CGHS model
3257:Toy models
3192:CA-duality
3051:1609.00026
2990:(7): 062.
2850:2017-05-07
2788:(8): 045.
2762:References
1372:homologous
1084:Conjecture
858:Motivation
819:holography
662:Strominger
657:Steinhardt
652:Staudacher
567:Polchinski
517:Nanopoulos
477:Mandelstam
457:Kontsevich
297:Berenstein
254:Multiverse
3566:See also:
3505:Spin foam
3285:RST model
3078:119273886
2997:0705.0016
2923:119441463
2820:1029-8479
2717:π
2711:
2697:ρ
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2646:
2625:ρ
2609:
2527:ρ
2476:π
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2402:γ
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2241:π
2226:∈
2223:θ
2180:θ
2162:θ
2150:ρ
2106:∼
2095:ρ
2027:ρ
2001:ρ
1997:≤
1994:ρ
1974:ρ
1954:∞
1951:→
1948:ρ
1921:θ
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1874:θ
1861:ρ
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1811:ρ
1806:
1800:−
1665:given by
1534:∪
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1412:γ
1385:γ
1352:γ
1303:γ
1273:γ
1232:Σ
1229:∂
1226:⊂
1143:γ
1096:Σ
974:spacetime
916:ℓ
852:covariant
702:Veneziano
577:Rajaraman
472:Maldacena
362:Gopakumar
312:Dijkgraaf
307:Curtright
271:Theorists
159:Landscape
154:Cosmology
118:U-duality
113:T-duality
108:S-duality
91:Heterotic
3413:M-theory
3373:ER = EPR
3212:Graviton
3127:15945690
3022:15206042
2915:16712357
2828:14858887
2538:>>
1758:has the
775:Glossary
757:Zwiebach
712:Verlinde
707:Verlinde
682:Townsend
677:'t Hooft
672:Susskind
607:Sagnotti
572:Polyakov
527:Nekrasov
492:Minwalla
487:Martinec
452:Knizhnik
447:Klebanov
442:Kapustin
412:Horowitz
342:Fischler
277:Aganagić
195:K-theory
128:F-theory
123:M-theory
3107:Bibcode
3002:Bibcode
2800:Bibcode
1562:Example
868:entropy
770:History
687:Trivedi
667:Sundrum
632:Shenker
622:Seiberg
617:Schwarz
587:Randall
547:Novikov
537:Nielsen
522:Năstase
432:Kallosh
417:Gibbons
357:Gliozzi
347:Friedan
337:Ferrara
322:Douglas
317:Distler
87:Type II
74:Bosonic
58:D-brane
3473:Others
3125:
3076:
3066:
3020:
2921:
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2818:
2267:where
1760:metric
1264:, and
1188:where
752:Zumino
747:Zaslow
732:Yoneya
722:Witten
637:Siegel
612:Scherk
582:Ramond
557:Ooguri
482:Marolf
437:Kaluza
422:Kachru
407:Hořava
402:Harvey
397:Hanson
382:Gubser
372:Greene
302:Bousso
287:Atiyah
83:Type I
43:String
3123:S2CID
3097:arXiv
3074:S2CID
3046:arXiv
3018:S2CID
2992:arXiv
2946:3 Aug
2919:S2CID
2893:arXiv
2870:(PDF)
2824:S2CID
2790:arXiv
1374:to A.
825:of a
692:Turok
597:Roček
562:Ovrut
552:Olive
532:Neveu
512:Myers
507:Mukhi
497:Moore
467:Linde
462:Klein
387:Gukov
377:Gross
367:Green
352:Gates
332:Dvali
292:Banks
53:Brane
3064:ISBN
2988:2007
2984:JHEP
2969:2024
2948:2018
2911:PMID
2816:ISSN
2786:2006
2438:sinh
2386:cosh
2333:and
1853:sinh
1803:cosh
1478:and
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838:and
813:The
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602:Rohm
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