968:
1144:
38:
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states that the derived category of coherent sheaves on one CalabiâYau manifold is equivalent in a certain sense to the Fukaya category of a completely different CalabiâYau manifold. This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic
1170:
at the endpoints of strings. Intuitively, one can think of a submanifold as a surface embedded inside of a CalabiâYau manifold, although submanifolds can also exist in dimensions different from two. In mathematical language, the category having these branes as its objects is known as the
1187:. This means, among other things, that they have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing. The category having these branes as its objects is called the
990:
are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to the
2142:
Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balåzs; Wilson, P.M.H., eds. (2009).
1118:
1074:
1138:
1094:
1018:, a theoretical tool that physicists use to translate difficult problems in gauge theory into more mathematically tractable problems in string theory.
2682:
2284:
1975:
1964:
1960:
250:
1956:
1700:
1056:
between these structures. One can likewise consider categories where the objects are D-branes and the morphisms between two branes
1183:, the D-branes can again be viewed as submanifolds of a CalabiâYau manifold. Roughly speaking, they are what mathematicians call
810:
2244:
1446:
2253:
2232:
2178:
2159:
1002:, a kind of highly symmetric physical theory which is also used to describe the behavior of elementary particles in the
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One crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by a
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The Shape of Inner Space: String Theory and the
Geometry of the Universe's Hidden Dimensions
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may be open (forming a segment with two endpoints) or closed (forming a closed loop).
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A point particle is a 0-brane, of dimension zero; a string, named after vibrating
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between them. In most examples, the objects are mathematical structures (such as
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846:, or a two-dimensional membrane to higher-dimensional objects. Branes are
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3201:
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2805:
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1733:
1729:
1501:
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1194:
The derived category of coherent sheaves is constructed using tools from
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1769:
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1497:
1370:
1268:
1263:
912:
889:
137:
132:
2242:
Zaslow, Eric (2008). "Mirror
Symmetry". In Gowers, Timothy (ed.).
1179:
on the CalabiâYau. In another version of string theory called the
1166:, together with additional data that arise physically from having
2561:
2541:
2498:
2493:
1741:
1659:
1383:
1199:
987:
972:
962:
67:
2262:
915:, is a 2-brane. The corresponding object of arbitrary dimension
1841:
1753:
1737:
1725:
1611:
1595:
1489:
1467:
2141:
1206:. On the other hand, the Fukaya category is constructed using
1026:
Mathematically, branes can be described using the notion of a
1599:
1509:
1198:, a branch of mathematics that describes geometric shapes in
1143:
1761:
1493:
1219:
859:
835:
is a physical object that generalizes the notion of a zero-
1591:
888:
The word "brane" originated in 1987 as a contraction of "
2030:
A basic reference on category theory is Mac Lane 1998.
1978:, "Semiclassical quantization of the supermembrane",
1214:. Symplectic geometry studies spaces equipped with a
1210:, a branch of mathematics that arose from studies of
1126:
1106:
1082:
1062:
1006:. This connection has led to important insights into
1910:
1218:, a mathematical tool that can be used to compute
1132:
1112:
1088:
1068:
1030:. This is a mathematical structure consisting of
869:Mathematically, branes can be represented within
3459:
2135:
1909:
941:+1)-dimensional volume in spacetime called its
1014:. For example, it led to the discovery of the
2278:
1154:In one version of string theory known as the
804:
953:, which live on the worldvolume of a brane.
2285:
2271:
1021:
811:
797:
36:
1162:of certain six-dimensional shapes called
16:Extended physical object in string theory
2218:
2171:Categories for the Working Mathematician
2168:
1142:
1034:, and for any pair of objects, a set of
966:
907:, is a 1-brane; a membrane, named after
3460:
2245:The Princeton Companion to Mathematics
2241:
2219:Yau, Shing-Tung; Nadis, Steve (2010).
1447:Atomic, molecular, and optical physics
862:and can have other attributes such as
2266:
2187:
1701:Newton's law of universal gravitation
2683:Bogomol'nyiâPrasadâSommerfield bound
2144:Dirichlet Branes and Mirror Symmetry
1931:participating institution membership
1202:and solves geometric problems using
850:objects which can propagate through
254:= 4 supersymmetric YangâMills theory
971:Open strings attached to a pair of
13:
1100:of open strings stretched between
1004:standard model of particle physics
185:Geometric Langlands correspondence
14:
3479:
2292:
1460:Atomic and molecular astrophysics
1550:Nanoscale and mesoscopic physics
2880:Eleven-dimensional supergravity
2123:
2114:
2105:
2096:
2087:
2078:
2069:
2060:
2051:
2042:
1185:special Lagrangian submanifolds
995:, which the D-brane satisfies.
2033:
2024:
2015:
2006:
1997:
1988:
1946:
1937:
1903:
1322:Particle physics phenomenology
827:and related theories (such as
1:
2328:Second superstring revolution
2152:American Mathematical Society
2120:Aspinwall et al. 2009, p. 616
2102:Yau and Nadis 2010, pp. 180â1
1412:Spontaneous symmetry breaking
1222:in two-dimensional examples.
2822:Generalized complex manifold
2323:First superstring revolution
2148:Clay Mathematics Monographs
2136:General and cited references
2093:Aspinwal et al. 2009, p. 575
2075:Aspinwal et al. 2009, p. 575
2066:Aspinwal et al. 2009, p. 575
1897:
993:Dirichlet boundary condition
7:
2193:"What is ... a Brane?"
2169:Mac Lane, Saunders (1998).
1886:Quantum information science
1718:Cosmic background radiation
1485:Quantum information science
1237:
1227:homological mirror symmetry
956:
895:
879:homological mirror symmetry
10:
3484:
2420:Non-critical string theory
2129:Yau and Nadis 2010, p. 181
2084:Yau and Nadis 2010, p. 175
2057:Yau and Nadis 2010, p. 165
960:
854:according to the rules of
18:
2964:
2941:
2918:
2865:
2750:
2658:
2600:
2532:
2481:
2448:
2343:
2300:
1918:Oxford English Dictionary
1780:
1750:Gravitational singularity
1567:Density functional theory
1506:Electromagnetic radiation
945:. Physicists often study
923:-brane, a term coined by
2956:Introduction to M-theory
2650:WessâZuminoâWitten model
2592:HananyâWitten transition
2318:History of string theory
1664:High-energy astrophysics
1604:BoseâEinstein condensate
1529:Condensed matter physics
1052:) and the morphisms are
110:Non-perturbative results
2635:Vertex operator algebra
2335:String theory landscape
1923:Oxford University Press
1746:Gravitational radiation
1542:Low-temperature physics
1380:Fundamental interaction
1335:Quantum electrodynamics
1113:{\displaystyle \alpha }
1069:{\displaystyle \alpha }
1022:Categorical description
883:noncommutative geometry
2933:AdS/CFT correspondence
2688:Exceptional Lie groups
2630:Superconformal algebra
2602:Conformal field theory
2473:MontonenâOlive duality
2425:Non-linear sigma model
1866:Physics of computation
1668:Planetary astrophysics
1375:AdS/CFT correspondence
1347:Effective field theory
1339:Quantum chromodynamics
1259:Lagrangian submanifold
1151:
1134:
1133:{\displaystyle \beta }
1114:
1090:
1089:{\displaystyle \beta }
1070:
1016:AdS/CFT correspondence
975:
226:Conformal field theory
143:AdS/CFT correspondence
21:Brane (disambiguation)
2928:Holographic principle
2895:Type IIB supergravity
2890:Type IIA supergravity
2742:-form electrodynamics
2361:Bosonic string theory
2021:Aspinwall et al. 2009
1862:Physical oceanography
1806:Communication physics
1608:Electrical conduction
1583:Statistical mechanics
1538:High-pressure physics
1318:Astroparticle physics
1147:A cross section of a
1146:
1135:
1115:
1091:
1071:
970:
951:electromagnetic field
937:-brane sweeps out a (
873:, and are studied in
829:supergravity theories
269:Holographic principle
246:Twistor string theory
2847:HoĆavaâWitten theory
2794:HyperkÀhler manifold
2482:Particles and fields
2430:Tachyon condensation
2415:Matrix string theory
1709:Magnetohydrodynamics
1684:Stellar astrophysics
1514:Polarization (waves)
1436:Theory of everything
1416:Neutrino oscillation
1363:Grand Unified Theory
1351:Lattice field theory
1331:Quantum field theory
1310:Nuclear astrophysics
1164:CalabiâYau manifolds
1160:complex submanifolds
1124:
1104:
1080:
1060:
1012:quantum field theory
842:, a one-dimensional
221:Theory of everything
19:For other uses, see
2885:Type I supergravity
2789:CalabiâYau manifold
2784:Ricci-flat manifold
2763:KaluzaâKlein theory
2504:RamondâRamond field
2410:String field theory
2111:Zaslow 2008, p. 531
2048:Zaslow 2008, p. 536
2039:Zaslow 2008, p. 536
1921:(Online ed.).
1882:Quantum electronics
1874:Solid-state devices
1814:Engineering physics
1794:Atmospheric physics
1782:Accelerator physics
1660:Gravitation physics
1575:Fermi liquid theory
1534:Solid-state physics
1400:Elementary particle
1208:symplectic geometry
1204:algebraic equations
1181:topological A-model
1158:, the D-branes are
1156:topological B-model
1149:CalabiâYau manifold
909:vibrating membranes
259:KaluzaâKlein theory
195:Monstrous moonshine
76:Perturbative theory
45:Fundamental objects
2852:K-theory (physics)
2729:ADE classification
2366:Superstring theory
2200:Notices of the AMS
2012:Moore 2005, p. 215
2003:Moore 2005, p. 215
1994:Moore 2005, p. 214
1943:Moore 2005, p. 214
1858:Physical chemistry
1697:General relativity
1367:Superstring theory
1343:Electroweak theory
1152:
1130:
1110:
1086:
1066:
1050:topological spaces
976:
3455:
3454:
3237:van Nieuwenhuizen
2773:Why 10 dimensions
2678:ChernâSimons form
2645:KacâMoody algebra
2625:Conformal algebra
2620:Conformal anomaly
2514:Magnetic monopole
2509:KalbâRamond field
2351:NambuâGoto action
2255:978-0-691-11880-2
2234:978-0-465-02023-2
2180:978-0-387-98403-2
2161:978-0-8218-3848-8
1929:(Subscription or
1895:
1894:
1878:Quantum chemistry
1830:Materials physics
1620:Self-organization
1481:Quantum chemistry
1456:Molecular physics
1212:classical physics
949:analogous to the
877:for insight into
856:quantum mechanics
821:
820:
552:van Nieuwenhuizen
3475:
2965:String theorists
2905:Lie superalgebra
2857:Twisted K-theory
2815:Spin(7)-manifold
2768:Compactification
2610:Virasoro algebra
2393:Heterotic string
2287:
2280:
2273:
2264:
2263:
2259:
2238:
2215:
2213:
2211:
2197:
2184:
2165:
2130:
2127:
2121:
2118:
2112:
2109:
2103:
2100:
2094:
2091:
2085:
2082:
2076:
2073:
2067:
2064:
2058:
2055:
2049:
2046:
2040:
2037:
2031:
2028:
2022:
2019:
2013:
2010:
2004:
2001:
1995:
1992:
1986:
1973:
1950:
1944:
1941:
1935:
1934:
1926:
1914:
1907:
1890:Vehicle dynamics
1802:Chemical physics
1705:Lambda-CDM model
1693:Cosmic inflation
1579:Many-body theory
1464:Chemical physics
1420:Seesaw mechanism
1314:Particle physics
1301:particle physics
1279:
1278:
1231:Maxim Kontsevich
1196:complex geometry
1177:coherent sheaves
1173:derived category
1139:
1137:
1136:
1131:
1119:
1117:
1116:
1111:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
875:pure mathematics
813:
806:
799:
215:Related concepts
40:
26:
25:
3483:
3482:
3478:
3477:
3476:
3474:
3473:
3472:
3458:
3457:
3456:
3451:
2960:
2937:
2914:
2861:
2809:
2779:KĂ€hler manifold
2746:
2723:
2716:
2709:
2702:
2695:
2654:
2615:Mirror symmetry
2596:
2582:Brane cosmology
2528:
2477:
2444:
2400:N=2 superstring
2386:Type IIB string
2381:Type IIA string
2356:Polyakov action
2339:
2296:
2291:
2256:
2235:
2209:
2207:
2195:
2181:
2162:
2150:. Vol. 4.
2138:
2133:
2128:
2124:
2119:
2115:
2110:
2106:
2101:
2097:
2092:
2088:
2083:
2079:
2074:
2070:
2065:
2061:
2056:
2052:
2047:
2043:
2038:
2034:
2029:
2025:
2020:
2016:
2011:
2007:
2002:
1998:
1993:
1989:
1967:
1951:
1947:
1942:
1938:
1928:
1908:
1904:
1900:
1846:Optoelectronics
1834:Medical physics
1777:Applied physics
1563:Bloch's theorem
1554:Polymer physics
1546:Surface physics
1432:Quantum gravity
1388:electromagnetic
1306:Nuclear physics
1249:Brane cosmology
1240:
1216:symplectic form
1200:algebraic terms
1189:Fukaya category
1125:
1122:
1121:
1105:
1102:
1101:
1081:
1078:
1077:
1061:
1058:
1057:
1024:
965:
959:
905:musical strings
901:
817:
772:
771:
282:
274:
273:
231:Quantum gravity
216:
190:Mirror symmetry
24:
17:
12:
11:
5:
3481:
3471:
3470:
3453:
3452:
3450:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3287:Randjbar-Daemi
3284:
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3169:
3164:
3159:
3154:
3149:
3144:
3139:
3134:
3129:
3124:
3119:
3114:
3109:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3019:
3014:
3009:
3004:
2999:
2994:
2989:
2984:
2979:
2974:
2968:
2966:
2962:
2961:
2959:
2958:
2953:
2947:
2945:
2939:
2938:
2936:
2935:
2930:
2924:
2922:
2916:
2915:
2913:
2912:
2910:Lie supergroup
2907:
2902:
2897:
2892:
2887:
2882:
2877:
2871:
2869:
2863:
2862:
2860:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2818:
2817:
2812:
2807:
2803:
2802:
2801:
2791:
2781:
2776:
2770:
2765:
2760:
2754:
2752:
2748:
2747:
2745:
2744:
2736:
2731:
2726:
2721:
2714:
2707:
2700:
2693:
2685:
2680:
2675:
2670:
2664:
2662:
2656:
2655:
2653:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2606:
2604:
2598:
2597:
2595:
2594:
2589:
2587:Quiver diagram
2584:
2579:
2574:
2569:
2564:
2559:
2554:
2549:
2544:
2538:
2536:
2530:
2529:
2527:
2526:
2521:
2516:
2511:
2506:
2501:
2496:
2491:
2485:
2483:
2479:
2478:
2476:
2475:
2470:
2465:
2460:
2454:
2452:
2450:String duality
2446:
2445:
2443:
2442:
2437:
2432:
2427:
2422:
2417:
2412:
2407:
2402:
2397:
2396:
2395:
2390:
2389:
2388:
2383:
2376:Type II string
2373:
2363:
2358:
2353:
2347:
2345:
2341:
2340:
2338:
2337:
2332:
2331:
2330:
2325:
2315:
2313:Cosmic strings
2310:
2304:
2302:
2298:
2297:
2290:
2289:
2282:
2275:
2267:
2261:
2260:
2254:
2239:
2233:
2216:
2189:Moore, Gregory
2185:
2179:
2166:
2160:
2137:
2134:
2132:
2131:
2122:
2113:
2104:
2095:
2086:
2077:
2068:
2059:
2050:
2041:
2032:
2023:
2014:
2005:
1996:
1987:
1945:
1936:
1901:
1899:
1896:
1893:
1892:
1870:Plasma physics
1838:Nanotechnology
1818:Fluid dynamics
1779:
1773:
1772:
1711:
1686:
1672:Plasma physics
1645:
1639:
1638:
1628:superconductor
1585:
1556:
1531:
1525:
1524:
1522:Casimir effect
1487:
1477:Quantum optics
1474:
1452:Atomic physics
1449:
1443:
1442:
1377:
1327:Standard Model
1324:
1303:
1293:
1292:
1289:
1288:Major theories
1286:
1283:
1277:
1276:
1271:
1266:
1261:
1256:
1254:Dirac membrane
1251:
1246:
1239:
1236:
1229:conjecture of
1129:
1109:
1085:
1065:
1023:
1020:
961:Main article:
958:
955:
900:
894:
840:point particle
819:
818:
816:
815:
808:
801:
793:
790:
789:
788:
787:
782:
774:
773:
770:
769:
764:
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:
602:Randjbar-Daemi
599:
594:
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:
283:
280:
279:
276:
275:
272:
271:
266:
261:
256:
248:
243:
238:
233:
228:
223:
217:
214:
213:
210:
209:
208:
207:
202:
200:Vertex algebra
197:
192:
187:
179:
178:
174:
173:
172:
171:
166:
161:
153:
152:
148:
147:
146:
145:
140:
135:
130:
125:
120:
112:
111:
107:
106:
105:
104:
86:
78:
77:
73:
72:
71:
70:
65:
60:
55:
47:
46:
42:
41:
33:
32:
15:
9:
6:
4:
3:
2:
3480:
3469:
3468:String theory
3466:
3465:
3463:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3432:Zamolodchikov
3430:
3428:
3427:Zamolodchikov
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
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:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2969:
2967:
2963:
2957:
2954:
2952:
2951:Matrix theory
2949:
2948:
2946:
2944:
2940:
2934:
2931:
2929:
2926:
2925:
2923:
2921:
2917:
2911:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2876:
2873:
2872:
2870:
2868:
2867:Supersymmetry
2864:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2816:
2813:
2811:
2804:
2800:
2797:
2796:
2795:
2792:
2790:
2787:
2786:
2785:
2782:
2780:
2777:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2755:
2753:
2749:
2743:
2741:
2737:
2735:
2732:
2730:
2727:
2724:
2717:
2710:
2703:
2696:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2665:
2663:
2661:
2657:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2607:
2605:
2603:
2599:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2539:
2537:
2535:
2531:
2525:
2522:
2520:
2519:Dual graviton
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2486:
2484:
2480:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2455:
2453:
2451:
2447:
2441:
2438:
2436:
2435:RNS formalism
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2406:
2403:
2401:
2398:
2394:
2391:
2387:
2384:
2382:
2379:
2378:
2377:
2374:
2372:
2371:Type I string
2369:
2368:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2348:
2346:
2342:
2336:
2333:
2329:
2326:
2324:
2321:
2320:
2319:
2316:
2314:
2311:
2309:
2306:
2305:
2303:
2299:
2295:
2294:String theory
2288:
2283:
2281:
2276:
2274:
2269:
2268:
2265:
2257:
2251:
2247:
2246:
2240:
2236:
2230:
2226:
2222:
2217:
2205:
2201:
2194:
2190:
2186:
2182:
2176:
2172:
2167:
2163:
2157:
2153:
2149:
2145:
2140:
2139:
2126:
2117:
2108:
2099:
2090:
2081:
2072:
2063:
2054:
2045:
2036:
2027:
2018:
2009:
2000:
1991:
1984:
1981:
1977:
1971:
1966:
1962:
1958:
1954:
1949:
1940:
1932:
1924:
1920:
1919:
1913:
1906:
1902:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1855:
1854:Photovoltaics
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1826:Laser physics
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1778:
1775:
1774:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1722:Cosmic string
1719:
1715:
1712:
1710:
1706:
1702:
1698:
1694:
1690:
1687:
1685:
1681:
1680:Space physics
1677:
1676:Solar physics
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1646:
1644:
1641:
1640:
1637:
1633:
1632:superfluidity
1629:
1625:
1624:Semiconductor
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1586:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1557:
1555:
1551:
1547:
1543:
1539:
1535:
1532:
1530:
1527:
1526:
1523:
1519:
1518:Spectral line
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1488:
1486:
1482:
1478:
1475:
1473:
1469:
1465:
1461:
1457:
1453:
1450:
1448:
1445:
1444:
1441:
1440:Vacuum energy
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1384:gravitational
1381:
1378:
1376:
1372:
1368:
1364:
1360:
1359:Supersymmetry
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1325:
1323:
1319:
1315:
1311:
1307:
1304:
1302:
1298:
1295:
1294:
1290:
1287:
1284:
1281:
1280:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1241:
1235:
1232:
1228:
1223:
1221:
1217:
1213:
1209:
1205:
1201:
1197:
1192:
1190:
1186:
1182:
1178:
1174:
1169:
1165:
1161:
1157:
1150:
1145:
1141:
1127:
1107:
1099:
1083:
1063:
1055:
1051:
1047:
1046:vector spaces
1043:
1039:
1038:
1033:
1029:
1019:
1017:
1013:
1009:
1005:
1001:
996:
994:
989:
985:
981:
980:string theory
974:
969:
964:
954:
952:
948:
944:
940:
936:
931:
929:
926:
922:
918:
914:
910:
906:
898:
893:
891:
886:
884:
880:
876:
872:
867:
865:
861:
857:
853:
849:
845:
841:
838:
834:
830:
826:
825:string theory
814:
809:
807:
802:
800:
795:
794:
792:
791:
786:
783:
781:
778:
777:
776:
775:
768:
765:
763:
760:
758:
755:
753:
752:Zamolodchikov
750:
748:
747:Zamolodchikov
745:
743:
740:
738:
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:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
278:
277:
270:
267:
265:
262:
260:
257:
255:
253:
249:
247:
244:
242:
239:
237:
236:Supersymmetry
234:
232:
229:
227:
224:
222:
219:
218:
212:
211:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
182:
181:
180:
176:
175:
170:
167:
165:
162:
160:
159:Phenomenology
157:
156:
155:
154:
151:Phenomenology
150:
149:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
119:
116:
115:
114:
113:
109:
108:
102:
98:
94:
90:
87:
85:
82:
81:
80:
79:
75:
74:
69:
66:
64:
61:
59:
58:Cosmic string
56:
54:
51:
50:
49:
48:
44:
43:
39:
35:
34:
31:
30:String theory
28:
27:
22:
2977:Arkani-Hamed
2875:Supergravity
2842:Moduli space
2739:
2734:Dirac string
2660:Gauge theory
2640:Loop algebra
2577:Black string
2533:
2440:GS formalism
2243:
2220:
2208:. Retrieved
2203:
2199:
2170:
2143:
2125:
2116:
2107:
2098:
2089:
2080:
2071:
2062:
2053:
2044:
2035:
2026:
2017:
2008:
1999:
1990:
1985:(1988), 515.
1982:
1979:
1976:K. S. Stelle
1948:
1939:
1916:
1905:
1810:Econophysics
1758:Solar System
1643:Astrophysics
1423:
1355:Gauge theory
1224:
1193:
1153:
1035:
1031:
1025:
1008:gauge theory
1000:gauge theory
997:
977:
942:
938:
934:
932:
927:
920:
919:is called a
916:
902:
896:
887:
868:
858:. They have
832:
822:
292:Arkani-Hamed
251:
241:Supergravity
62:
3337:Silverstein
2837:Orientifold
2572:Black holes
2567:Black brane
2524:Dual photon
2225:Basic Books
1980:Nucl. Phys.
1968: [
1790:Agrophysics
1734:Dark matter
1730:Dark energy
1502:Diffraction
1244:Black brane
943:worldvolume
837:dimensional
652:Silverstein
177:Mathematics
89:Superstring
3357:Strominger
3352:Steinhardt
3347:Staudacher
3262:Polchinski
3212:Nanopoulos
3172:Mandelstam
3152:Kontsevich
2992:Berenstein
2920:Holography
2900:Superspace
2799:K3 surface
2758:Worldsheet
2673:Instantons
2301:Background
1961:C. N. Pope
1953:M. J. Duff
1933:required.)
1822:Geophysics
1798:Biophysics
1714:Black hole
1652:Astrometry
1559:BCS theory
1408:Antimatter
1234:geometry.
925:M. J. Duff
871:categories
672:Strominger
667:Steinhardt
662:Staudacher
577:Polchinski
527:Nanopoulos
487:Mandelstam
467:Kontsevich
307:Berenstein
264:Multiverse
3392:Veneziano
3272:Rajaraman
3167:Maldacena
3057:Gopakumar
3007:Dijkgraaf
3002:Curtright
2668:Anomalies
2547:NS5-brane
2468:U-duality
2463:S-duality
2458:T-duality
1965:E. Sezgin
1898:Citations
1850:Photonics
1786:Acoustics
1766:Supernova
1656:Cosmology
1648:Astronomy
1616:Magnetism
1571:Fermi gas
1472:Photonics
1291:Concepts
1285:Subfields
1274:NS5-brane
1128:β
1108:α
1084:β
1064:α
1054:functions
1037:morphisms
930:in 1988.
913:drumheads
852:spacetime
848:dynamical
712:Veneziano
587:Rajaraman
482:Maldacena
372:Gopakumar
322:Dijkgraaf
317:Curtright
281:Theorists
169:Landscape
164:Cosmology
128:U-duality
123:T-duality
118:S-duality
101:Heterotic
3462:Category
3447:Zwiebach
3402:Verlinde
3397:Verlinde
3372:Townsend
3367:Susskind
3302:Sagnotti
3267:Polyakov
3222:Nekrasov
3187:Minwalla
3182:Martinec
3147:Knizhnik
3142:Klebanov
3137:Kapustin
3102:'t Hooft
3037:Fischler
2972:AganagiÄ
2943:M-theory
2832:Conifold
2827:Orbifold
2810:manifold
2751:Geometry
2557:M5-brane
2552:M2-brane
2489:Graviton
2405:F-theory
2191:(2005).
1957:T. Inami
1770:Universe
1689:Big Bang
1498:Molecule
1371:M-theory
1269:M5-brane
1264:M2-brane
1238:See also
1028:category
988:D-branes
973:D-branes
957:D-branes
911:such as
890:membrane
785:Glossary
767:Zwiebach
722:Verlinde
717:Verlinde
692:Townsend
687:'t Hooft
682:Susskind
617:Sagnotti
582:Polyakov
537:Nekrasov
502:Minwalla
497:Martinec
462:Knizhnik
457:Klebanov
452:Kapustin
422:Horowitz
352:Fischler
287:AganagiÄ
205:K-theory
138:F-theory
133:M-theory
3377:Trivedi
3362:Sundrum
3327:Shenker
3317:Seiberg
3312:Schwarz
3282:Randall
3242:Novikov
3232:Nielsen
3217:NÄstase
3127:Kallosh
3112:Gibbons
3052:Gliozzi
3042:Friedan
3032:Ferrara
3017:Douglas
3012:Distler
2562:S-brane
2542:D-brane
2499:Tachyon
2494:Dilaton
2308:Strings
2210:June 7,
1912:"brane"
1742:Gravity
1297:Nuclear
1168:charges
1032:objects
963:D-brane
899:-branes
780:History
697:Trivedi
677:Sundrum
642:Shenker
632:Seiberg
627:Schwarz
597:Randall
557:Novikov
547:Nielsen
532:NÄstase
442:Kallosh
427:Gibbons
367:Gliozzi
357:Friedan
347:Ferrara
332:Douglas
327:Distler
97:Type II
84:Bosonic
68:D-brane
3442:Zumino
3437:Zaslow
3422:Yoneya
3412:Witten
3332:Siegel
3307:Scherk
3277:Ramond
3252:Ooguri
3177:Marolf
3132:Kaluza
3117:Kachru
3107:HoĆava
3097:Harvey
3092:Hanson
3077:Gubser
3067:Greene
2997:Bousso
2982:Atiyah
2534:Branes
2344:Theory
2252:
2231:
2177:
2158:
1974:, and
1842:Optics
1754:Planet
1738:Galaxy
1726:Cosmos
1612:Phonon
1596:liquid
1588:Phases
1490:Photon
1468:Optics
1428:String
1396:strong
1098:states
984:string
947:fields
928:et al.
864:charge
844:string
762:Zumino
757:Zaslow
742:Yoneya
732:Witten
647:Siegel
622:Scherk
592:Ramond
567:Ooguri
492:Marolf
447:Kaluza
432:Kachru
417:HoĆava
412:Harvey
407:Hanson
392:Gubser
382:Greene
312:Bousso
297:Atiyah
93:Type I
53:String
3382:Turok
3292:RoÄek
3257:Ovrut
3247:Olive
3227:Neveu
3207:Myers
3202:Mukhi
3192:Moore
3162:Linde
3157:Klein
3082:Gukov
3072:Gross
3062:Green
3047:Gates
3027:Dvali
2987:Banks
2206:: 214
2196:(PDF)
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1927:
1600:solid
1510:Laser
1424:Brane
1282:Field
1048:, or
1010:and
833:brane
831:), a
702:Turok
607:RoÄek
572:Ovrut
562:Olive
542:Neveu
522:Myers
517:Mukhi
507:Moore
477:Linde
472:Klein
397:Gukov
387:Gross
377:Green
362:Gates
342:Dvali
302:Banks
63:Brane
3407:Wess
3387:Vafa
3297:Rohm
3197:Motl
3122:Kaku
3087:Guth
3022:Duff
2250:ISBN
2229:ISBN
2212:2018
2175:ISBN
2156:ISBN
1983:B297
1762:Star
1636:Spin
1494:Atom
1404:Spin
1392:weak
1299:and
1225:The
1220:area
1120:and
1096:are
1076:and
1042:sets
982:, a
881:and
860:mass
727:Wess
707:Vafa
612:Rohm
512:Motl
437:Kaku
402:Guth
337:Duff
3417:Yau
3342:SÆĄn
3322:Sen
1602:),
1592:gas
1398:),
1175:of
978:In
892:".
823:In
737:Yau
657:SÆĄn
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