1820:. In certain cases, the entropy calculation done for zero string coupling remains valid when the strings interact. The challenge for a string theorist is to devise a situation in which a black hole can exist which does not "break" supersymmetry. In recent years, this has been done by building black holes out of D-branes. Calculating the entropies of these hypothetical holes gives results which agree with the expected Bekenstein entropy. Unfortunately, the cases studied so far all involve higher-dimensional spaces â D5-branes in nine-dimensional space, for example. They do not directly apply to the familiar case, the Schwarzschild black holes observed in our own universe.
1312:"daughter" strings). Since endpoints are restricted to lie on D-branes, it is evident that a string may interact with a string, but not with a or a one. The masses of these strings will be influenced by the separation between the branes, as discussed above, so for simplicity's sake, we can imagine the branes squeezed closer and closer together until they lie atop one another. If we regard two overlapping branes as distinct objects, then we still have all the sectors we had before, but without the effects due to the brane separations.
1327:) gauge theory. (The string theory does contain other interactions, but they are only detectable at very high energies.) Gauge theories were not invented starting with bosonic or fermionic strings; they originated from a different area of physics, and have become quite useful in their own right. If nothing else, the relation between D-brane geometry and gauge theory offers a useful pedagogical tool for explaining gauge interactions, even if string theory fails to be the "theory of everything".
916:, which pin the string endpoint. Each coordinate of the string must satisfy one or the other of these conditions. There can also exist strings with mixed boundary conditions, where the two endpoints satisfy NN, DD, ND and DN boundary conditions. If p spatial dimensions satisfy the Neumann boundary condition, then the string endpoint is confined to move within a p-dimensional hyperplane. This hyperplane provides one description of a Dp-brane.
31:
1068:
each other by some angle. The annulus amplitude yields singularities that correspond to the on-shell production of open strings stretched between the two branes. This is true irrespective of the charge of the D-branes. At non-relativistic scattering velocities the open strings may be described by a low-energy effective action that contains two complex scalar fields that are coupled via a term
1039:—we have to find a reason why the extra dimensions are not apparent. One possibility would be that the visible Universe is in fact a very large D-brane extending over three spatial dimensions. Material objects, made of open strings, are bound to the D-brane, and cannot move "at right angles to reality" to explore the Universe outside the brane. This scenario is called a
1163:, but they are really just labels identifying the branes.) A string in either the or the sector has a minimum length: it cannot be shorter than the separation between the branes. All strings have some tension, against which one must pull to lengthen the object; this pull does work on the string, adding to its energy. Because string theories are by nature
1813:) is more difficult. However, a world without interactions is an uninteresting place: most significantly for the black hole problem, gravity is an interaction, and so if the "string coupling" is turned off, no black hole could ever arise. Therefore, calculating black hole entropy requires working in a regime where string interactions exist.
1834:
boundary conditions). This paper, though prescient, was little-noted in its time (a 1985 parody by Siegel, "The Super-g String", contains an almost dead-on description of braneworlds). Dirichlet conditions for all coordinates including
Euclidean time (defining what are now known as D-instantons) were introduced by
1784:
The concept of black hole entropy poses some interesting conundra. In an ordinary situation, a system has entropy when a large number of different "microstates" can satisfy the same macroscopic condition. For example, given a box full of gas, many different arrangements of the gas atoms can have the
1178:
Furthermore, affixing a string's endpoint to a brane influences the way the string can move and vibrate. Because particle states "emerge" from the string theory as the different vibrational states the string can experience, the arrangement of D-branes controls the types of particles present in the
1067:
When two D-branes approach each other the interaction is captured by the one loop annulus amplitude of strings between the two branes. The scenario of two parallel branes approaching each other at a constant velocity can be mapped to the problem of two stationary branes that are rotated relative to
1833:
in 1976 as a means of lowering the critical dimension of open string theory from 26 or 10 to 4 (Siegel also cites unpublished work by
Halpern, and a 1974 paper by Chodos and Thorn, but a reading of the latter paper shows that it is actually concerned with linear dilation backgrounds, not Dirichlet
1828:
Dirichlet boundary conditions and D-branes had a long "pre-history" before their full significance was recognized. A series of 1975â76 papers by
Bardeen, Bars, Hanson and Peccei dealt with an early concrete proposal of interacting particles at the ends of strings (quarks interacting with QCD flux
1796:
String theorists have constructed models in which a black hole is a very long (and hence very massive) string. This model gives rough agreement with the expected entropy of a
Schwarzschild black hole, but an exact proof has yet to be found one way or the other. The chief difficulty is that it is
1311:
have more intriguing properties. For starters, it is worthwhile to ask which sectors of strings can interact with one another. One straightforward mechanism for a string interaction is for two strings to join endpoints (or, conversely, for one string to "split down the middle" and make two
1158:
objects: each one carries an "arrow" defining a direction along its length.) The open strings permissible in this situation then fall into two categories, or "sectors": those originating on brane 1 and terminating on brane 2, and those originating on brane 2 and terminating on brane 1.
1257:
of the brane, corresponding to the different ways the symmetry of empty space can be broken. Placing a D-brane in a universe breaks the symmetry among locations, because it defines a particular place, assigning a special meaning to a particular location along each of the
1488:
1153:
The arrangement of D-branes constricts the types of string states which can exist in a system. For example, if we have two parallel D2-branes, we can easily imagine strings stretching from brane 1 to brane 2 or vice versa. (In most theories, strings are
1222:
In this sense, then, one can say that string theory "predicts" electromagnetism: D-branes are a necessary part of the theory if we permit open strings to exist, and all D-branes carry an electromagnetic field on their volume.
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of any open string theory. The Dai et al. paper also notes that the locus of the
Dirichlet boundary conditions is dynamical, and coins the term Dirichlet-brane (D-brane) for the resulting object (this paper also coins
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1226:
Other particle states originate from strings beginning and ending on the same D-brane. Some correspond to massless particles like the photon; also in this group are a set of massless scalar particles. If a
1744:
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1347:, dropping an amount of hot gas into a black hole. Since the gas cannot escape from the hole's gravitational pull, its entropy would seem to have vanished from the universe. In order to maintain the
939:
D-branes are nearly coincident, the spectrum of strings stretching between them becomes very rich. One set of modes produce a non-abelian gauge theory on the world-volume. Another set of modes is an
1829:
tubes), with dynamical boundary conditions for string endpoints where the
Dirichlet conditions were dynamical rather than static. Mixed Dirichlet/Neumann boundary conditions were first considered by
919:
Although rigid in the limit of zero coupling, the spectrum of open strings ending on a D-brane contains modes associated with its fluctuations, implying that D-branes are dynamical objects. When
1243:(particles which do not have polarizations like the photons making up light). Intriguingly, there are just as many massless scalars as there are directions perpendicular to the brane; the
1861:
interchanges the usual
Neumann boundary conditions with Dirichlet boundary conditions. This result implies that such boundary conditions must necessarily appear in regions of the
1159:
Symbolically, we say we have the and the sectors. In addition, a string may begin and end on the same brane, giving and sectors. (The numbers inside the brackets are called
1103:
844:
A D0-brane is a single point, a D1-brane is a line (sometimes called a "D-string"), a D2-brane is a plane, and a D25-brane fills the highest-dimensional space considered in
1019:. Tachyon condensation is still very poorly understood. This is due to the lack of an exact string field theory that would describe the off-shell evolution of the tachyon.
1775:
963:
2812:
1901:
908:
The equations of motion of string theory require that the endpoints of an open string (a string with endpoints) satisfy one of two types of boundary conditions: The
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1143:
1123:
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dimensional matrix for each transverse dimension of the brane. If these matrices commute, they may be diagonalized, and the eigenvalues define the position of the
1842:. String compactifications studied by Harvey and Minahan, Ishibashi and Onogi, and Pradisi and Sagnotti in 1987â1989 also employed Dirichlet boundary conditions.
2414:
1574:
Using this expression for the
Hawking temperature, and assuming that a zero-mass black hole has zero entropy, one can use thermodynamic arguments to derive the "
1538:
1510:
983:
937:
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strings. Because closed strings do not have to be attached to D-branes, gravitational effects could depend upon the extra dimensions orthogonal to the brane.
243:
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studied in introductory thermodynamics: the easiest situation to model is when the gas atoms do not have interactions among themselves. Developing the
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803:
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discovered that a hole should emit energy with the characteristic spectrum of thermal radiation. The characteristic temperature of this
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177:
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D-branes in space. More generally, the branes are described by non-commutative geometry, which allows exotic behavior such as the
1870:
for another object that arises under string T-duality). A 1989 paper by Leigh showed that D-brane dynamics are governed by the
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182:
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in 1977 as a means of introducing point-like structure into string theory, in an attempt to construct a string theory of the
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for convenience. Open strings in this system exist in one of many sectors: the strings beginning and ending on some brane
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2457:
2400:
1874:. D-instantons were extensively studied by Green in the early 1990s, and were shown by Polchinski in 1994 to produce the
913:
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1351:, one must postulate that the black hole gained whatever entropy the infalling gas originally had. Attempting to apply
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833:
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1483:{\displaystyle T_{\rm {H}}={\frac {\hbar c^{3}}{8\pi GMk_{B}}}\;\quad (\approx {1.227\times 10^{23}\;kg \over M}\;K),}
796:
2602:
2155:
Bars, I. (1976). "A quantum string theory of hadrons and its relation to quantum chromodynamics in two dimensions".
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give that brane a
Maxwell field and some massless scalar fields on its volume. The strings stretching from brane
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Extending the simpler case of non-interacting strings to the regime where a black hole could exist requires
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changes. This induces open string production and as a result the two scattering branes will be trapped.
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85:
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Dai, Jin; Leigh, R.G.; Polchinski, Joseph (1989-10-20). "New connections between string theories".
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of order 1. D-branes can be used to generate gauge theories of higher order, in the following way:
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1031:. Because string theory implies that the Universe has more dimensions than we expect—26 for
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flux) an arbitrary D-brane configuration to be obtained from a stack of D9 and anti D9-branes.
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1994:
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1175:. Therefore, the separation between D-branes controls the minimum mass open strings may have.
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319:
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912:, corresponding to free endpoints moving through spacetime at the speed of light, or the
599:
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2226:
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2132:
2095:
2050:
1982:
1939:
1809:
in the case where the gas atoms or molecules experience inter-particle forces (like the
1793:"). What, then, are the "degrees of freedom" which can give rise to black hole entropy?
1664:
is proportional to the mass, the
Bekenstein entropy is proportional to the black hole's
1203:, the fundamental quantum of the electromagnetic field. The resemblance is precise: a
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3446:
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3256:
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2648:
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2442:
2423:
2250:
2176:
2119:
Bars, Itzhak (1976-06-28). "Exact Equivalence of Chromodynamics to a String Theory".
1990:
1881:
1830:
1817:
1785:
same total energy. However, a black hole was believed to be a featureless object (in
1778:
1270:
1008:
881:
817:
724:
709:
684:
669:
639:
584:
559:
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489:
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50:
22:
1884:. In 1995 Polchinski showed that D-branes are the sources of electric and magnetic
1183:-brane, that is to say the strings which begin and end on any particular D-brane of
30:
3571:
3411:
3291:
3241:
3211:
3196:
3004:
2971:
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2789:
2769:
2706:
2569:
2369:
1892:, leading to rapid progress in the nonperturbative understanding of string theory.
1862:
1660:
The Bekenstein entropy is proportional to the black hole mass squared; because the
1513:
1266:
885:
836:, after which they are named. D-branes are typically classified by their spatial
754:
589:
469:
419:
389:
374:
233:
2082:
Bars, Itzhak; Hanson, Andrew J. (1976-03-15). "Quarks at the ends of the string".
1319:
coincident D-branes yields a set of interacting quantum fields which is exactly a
1235:
spatial dimensions, the brane carries (in addition to its Maxwell field) a set of
1199:
the string), one finds that among the spectrum of particles is one resembling the
1167:, adding energy to a string is equivalent to adding mass, by Einstein's relation
3566:
3536:
3516:
3376:
3331:
3286:
3251:
3201:
2966:
2935:
2696:
2653:
2345:
2140:
1867:
749:
719:
699:
554:
509:
464:
429:
379:
2314:
Polchinski, Joseph (1995-12-25). "Dirichlet Branes and Ramond-Ramond Charges".
3511:
3451:
3386:
3049:
3029:
2928:
2887:
2701:
2234:
2203:
Giveon, Amit; Kutasov, David (1999-07-01). "Brane dynamics and gauge theory".
1947:
1339:. Since the 1970s, scientists have debated the problem of black holes having
1336:
996:
694:
629:
564:
256:
2353:
2242:
2184:
2148:
2111:
2103:
2066:
2058:
2033:
Bardeen, W. A.; Bars, Itzhak; Hanson, Andrew J.; Peccei, R. D. (1976-04-15).
1998:
1955:
1295:-branes, arranged in parallel for simplicity. The branes are labeled 1,2,...,
3426:
3216:
3156:
2802:
2676:
2597:
2592:
2587:
1858:
1802:
1016:
849:
837:
604:
394:
334:
120:
115:
110:
2361:
1315:
The zero-mass states in the open-string particle spectrum for a system of
3072:
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2956:
2686:
2681:
2618:
2534:
1906:
1052:
897:
130:
125:
1797:
relatively easy to count the degrees of freedom quantum strings possess
2691:
2628:
2623:
2374:. An article which established D-branes' significance in string theory.
2328:
2303:
2217:
2196:
1340:
1044:
2074:
2034:
1265:
The quantum version of Maxwell's electromagnetism is only one kind of
2011:
Polchinski, J. (1995). "Dirichlet branes and Ramond-Ramond charges".
1253:
of the particles existing on it. In fact, these massless scalars are
1200:
2663:
853:
55:
1969:
HoĆava, Petr (1989). "Background duality of open-string models".
1003:, tachyon condensation allows (in the absence of Neveu-Schwarz 3-
989:, in which a collection of Dp-branes expand into a D(p+2)-brane.
2392:
1651:{\displaystyle S_{\rm {B}}={\frac {k_{B}4\pi G}{\hbar c}}M^{2}.}
857:
2396:
1739:{\displaystyle S_{\rm {B}}={\frac {Ak_{B}}{4l_{\rm {P}}^{2}}},}
1207:-dimensional version of the electromagnetic field, obeying a
1011:
has shown that such configurations will be classified by the
1335:
Another important use of D-branes has been in the study of
1055:
which carry gravitational forces are vibrational states of
1125:(separation of the branes) changes, the mass of the field
880:, in 1989. In 1995, Polchinski identified D-branes with
2259:
D-Brane: Superstrings and New Perspective of Our World.
1754:
1676:
1586:
1546:
1526:
1498:
1371:
1131:
1111:
1074:
971:
945:
925:
2035:"Study of the longitudinal kink modes of the string"
3094:
3071:
3048:
2995:
2880:
2788:
2730:
2662:
2611:
2578:
2473:
2430:
1247:of the brane arrangement is closely related to the
840:, which is indicated by a number written after the
2127:(26). American Physical Society (APS): 1521â1525.
1769:
1738:
1650:
1559:
1532:
1504:
1482:
1137:
1117:
1097:
977:
957:
931:
828:, are a class of extended objects upon which open
2090:(6). American Physical Society (APS): 1744â1760.
2045:(8). American Physical Society (APS): 2364â2382.
1934:(21). World Scientific Pub Co Pte Lt: 2073â2083.
1179:theory. The simplest case is the sector for a D
1187:dimensions. Examining the consequences of the
1880:nonperturbative string effects anticipated by
2408:
2191:Bachas, C. P. "Lectures on D-branes" (1998).
797:
8:
2415:
2401:
2393:
1470:
1457:
1430:
852:D(â1)-branes, which are localized in both
804:
790:
29:
18:
2327:
2216:
1799:if they do not interact with one another.
1760:
1759:
1753:
1724:
1718:
1717:
1702:
1692:
1682:
1681:
1675:
1639:
1609:
1602:
1592:
1591:
1585:
1551:
1545:
1525:
1497:
1451:
1438:
1421:
1397:
1387:
1377:
1376:
1370:
1130:
1110:
1089:
1079:
1073:
970:
944:
924:
1918:
1626:
1390:
1262:directions perpendicular to the brane.
169:
143:
102:
68:
37:
21:
16:Extended objects found in string theory
1231:-brane is embedded in a spacetime of
868:D-branes were discovered by Jin Dai,
7:
2813:Bogomol'nyiâPrasadâSommerfield bound
1902:Bogomol'nyiâPrasadâSommerfield bound
995:is a central concept in this field.
247:= 4 supersymmetric YangâMills theory
2381:Cambridge University Press (2004).
1761:
1719:
1683:
1593:
1378:
1098:{\displaystyle \phi ^{2}\chi ^{2}}
178:Geometric Langlands correspondence
14:
888:, a discovery that triggered the
2379:A First Course in String Theory.
3010:Eleven-dimensional supergravity
1857:independently, discovered that
1431:
1474:
1432:
1:
2458:Second superstring revolution
1540:is the black hole's mass and
1355:to the study of black holes,
1276:gauge theory where the gauge
914:Dirichlet boundary conditions
890:Second Superstring Revolution
834:Dirichlet boundary conditions
2952:Generalized complex manifold
2453:First superstring revolution
2177:10.1016/0550-3213(76)90327-8
1991:10.1016/0370-2693(89)90209-8
1349:second law of thermodynamics
2346:10.1103/physrevlett.75.4724
2163:(3). Elsevier BV: 413â440.
2141:10.1103/physrevlett.36.1521
1977:(3). Elsevier BV: 251â257.
1770:{\displaystyle l_{\rm {P}}}
1191:(and applying the rules of
3614:
2550:Non-critical string theory
2278:Cambridge University Press
2272:Johnson, Clifford (2003).
1027:This has implications for
910:Neumann boundary condition
2296:TASI Lectures on D-branes
2235:10.1103/revmodphys.71.983
2205:Reviews of Modern Physics
1948:10.1142/s0217732389002331
1801:This is analogous to the
1211:-dimensional analogue of
1051:due to open strings; the
958:{\displaystyle N\times N}
3086:Introduction to M-theory
2780:WessâZuminoâWitten model
2722:HananyâWitten transition
2448:History of string theory
2104:10.1103/physrevd.13.1744
2059:10.1103/physrevd.13.2364
1928:Modern Physics Letters A
1872:DiracâBornâInfeld action
1791:Black holes have no hair
103:Non-perturbative results
2765:Vertex operator algebra
2465:String theory landscape
2316:Physical Review Letters
2121:Physical Review Letters
1807:kinetic theory of gases
1033:bosonic string theories
876:, and independently by
3063:AdS/CFT correspondence
2818:Exceptional Lie groups
2760:Superconformal algebra
2732:Conformal field theory
2603:MontonenâOlive duality
2555:Non-linear sigma model
1771:
1740:
1652:
1561:
1534:
1518:gravitational constant
1506:
1484:
1139:
1119:
1099:
1001:Type IIB string theory
979:
959:
933:
904:Theoretical background
219:Conformal field theory
136:AdS/CFT correspondence
3058:Holographic principle
3025:Type IIB supergravity
3020:Type IIA supergravity
2872:-form electrodynamics
2491:Bosonic string theory
2306:. Lectures given at
1888:that are required by
1772:
1741:
1653:
1562:
1560:{\displaystyle k_{B}}
1535:
1507:
1485:
1255:Goldstone excitations
1140:
1138:{\displaystyle \chi }
1120:
1118:{\displaystyle \phi }
1105:. Thus, as the field
1100:
980:
960:
934:
846:bosonic string theory
262:Holographic principle
239:Twistor string theory
2977:HoĆavaâWitten theory
2924:HyperkÀhler manifold
2612:Particles and fields
2560:Tachyon condensation
2545:Matrix string theory
2294:Polchinski, Joseph,
1886:RamondâRamond fields
1752:
1674:
1662:Schwarzschild radius
1584:
1569:Boltzmann's constant
1544:
1524:
1496:
1369:
1287:Consider a group of
1250:quantum field theory
1129:
1109:
1072:
1037:superstring theories
1023:Braneworld cosmology
993:Tachyon condensation
969:
943:
923:
214:Theory of everything
3015:Type I supergravity
2919:CalabiâYau manifold
2914:Ricci-flat manifold
2893:KaluzaâKlein theory
2634:RamondâRamond field
2540:String field theory
2377:Zwiebach, Barton.
2338:1995PhRvL..75.4724P
2227:1999RvMP...71..983G
2169:1976NuPhB.111..413B
2133:1976PhRvL..36.1521B
2096:1976PhRvD..13.1744B
2051:1976PhRvD..13.2364B
1983:1989PhLB..231..251H
1940:1989MPLA....4.2073D
1811:van der Waals force
1729:
1280:is made of unitary
1215:, exists on every D
1213:Maxwell's equations
999:has argued that in
252:KaluzaâKlein theory
188:Monstrous moonshine
69:Perturbative theory
38:Fundamental objects
2982:K-theory (physics)
2859:ADE classification
2496:Superstring theory
2019:(10): R6041âR6045.
1840:strong interaction
1767:
1736:
1713:
1648:
1557:
1530:
1502:
1480:
1345:thought experiment
1343:. Consider, as a
1161:ChanâPaton indices
1135:
1115:
1095:
1063:D-brane scattering
1029:physical cosmology
975:
955:
929:
848:. There are also
826:Dirichlet membrane
3585:
3584:
3367:van Nieuwenhuizen
2903:Why 10 dimensions
2808:ChernâSimons form
2775:KacâMoody algebra
2755:Conformal algebra
2750:Conformal anomaly
2644:Magnetic monopole
2639:KalbâRamond field
2481:NambuâGoto action
2322:(26): 4724â4727.
2267:978-3-642-23573-3
2261:Springer (2012).
2257:Hashimoto, Koji,
2157:Nuclear Physics B
2084:Physical Review D
2039:Physical Review D
2013:Physical Review D
1971:Physics Letters B
1789:'s catchphrase, "
1731:
1633:
1533:{\displaystyle M}
1505:{\displaystyle G}
1468:
1428:
1361:Hawking radiation
1353:quantum mechanics
1307:to another brane
1193:quantum mechanics
1189:NambuâGoto action
978:{\displaystyle N}
932:{\displaystyle N}
814:
813:
545:van Nieuwenhuizen
3605:
3095:String theorists
3035:Lie superalgebra
2987:Twisted K-theory
2945:Spin(7)-manifold
2898:Compactification
2740:Virasoro algebra
2523:Heterotic string
2417:
2410:
2403:
2394:
2373:
2331:
2291:
2254:
2220:
2188:
2152:
2115:
2078:
2020:
2009:
2003:
2002:
1966:
1960:
1959:
1923:
1879:
1776:
1774:
1773:
1768:
1766:
1765:
1764:
1745:
1743:
1742:
1737:
1732:
1730:
1728:
1723:
1722:
1708:
1707:
1706:
1693:
1688:
1687:
1686:
1657:
1655:
1654:
1649:
1644:
1643:
1634:
1632:
1624:
1614:
1613:
1603:
1598:
1597:
1596:
1566:
1564:
1563:
1558:
1556:
1555:
1539:
1537:
1536:
1531:
1511:
1509:
1508:
1503:
1489:
1487:
1486:
1481:
1469:
1464:
1456:
1455:
1439:
1429:
1427:
1426:
1425:
1403:
1402:
1401:
1388:
1383:
1382:
1381:
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1094:
1093:
1084:
1083:
984:
982:
981:
976:
964:
962:
961:
956:
938:
936:
935:
930:
892:and led to both
806:
799:
792:
208:Related concepts
33:
19:
3613:
3612:
3608:
3607:
3606:
3604:
3603:
3602:
3588:
3587:
3586:
3581:
3090:
3067:
3044:
2991:
2939:
2909:KĂ€hler manifold
2876:
2853:
2846:
2839:
2832:
2825:
2784:
2745:Mirror symmetry
2726:
2712:Brane cosmology
2658:
2607:
2574:
2530:N=2 superstring
2516:Type IIB string
2511:Type IIA string
2486:Polyakov action
2469:
2426:
2421:
2313:
2288:
2271:
2211:(4): 983â1084.
2202:
2154:
2118:
2081:
2032:
2029:
2024:
2023:
2010:
2006:
1968:
1967:
1963:
1925:
1924:
1920:
1915:
1898:
1875:
1826:
1755:
1750:
1749:
1709:
1698:
1694:
1677:
1672:
1671:
1635:
1625:
1605:
1604:
1587:
1582:
1581:
1547:
1542:
1541:
1522:
1521:
1494:
1493:
1447:
1440:
1417:
1404:
1393:
1389:
1372:
1367:
1366:
1357:Stephen Hawking
1333:
1151:
1127:
1126:
1107:
1106:
1085:
1075:
1070:
1069:
1065:
1043:. The force of
1041:brane cosmology
1025:
967:
966:
941:
940:
921:
920:
906:
866:
810:
765:
764:
275:
267:
266:
224:Quantum gravity
209:
183:Mirror symmetry
17:
12:
11:
5:
3611:
3609:
3601:
3600:
3590:
3589:
3583:
3582:
3580:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3417:Randjbar-Daemi
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:
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:
3098:
3096:
3092:
3091:
3089:
3088:
3083:
3077:
3075:
3069:
3068:
3066:
3065:
3060:
3054:
3052:
3046:
3045:
3043:
3042:
3040:Lie supergroup
3037:
3032:
3027:
3022:
3017:
3012:
3007:
3001:
2999:
2993:
2992:
2990:
2989:
2984:
2979:
2974:
2969:
2964:
2959:
2954:
2949:
2948:
2947:
2942:
2937:
2933:
2932:
2931:
2921:
2911:
2906:
2900:
2895:
2890:
2884:
2882:
2878:
2877:
2875:
2874:
2866:
2861:
2856:
2851:
2844:
2837:
2830:
2823:
2815:
2810:
2805:
2800:
2794:
2792:
2786:
2785:
2783:
2782:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2736:
2734:
2728:
2727:
2725:
2724:
2719:
2717:Quiver diagram
2714:
2709:
2704:
2699:
2694:
2689:
2684:
2679:
2674:
2668:
2666:
2660:
2659:
2657:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2615:
2613:
2609:
2608:
2606:
2605:
2600:
2595:
2590:
2584:
2582:
2580:String duality
2576:
2575:
2573:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2526:
2525:
2520:
2519:
2518:
2513:
2506:Type II string
2503:
2493:
2488:
2483:
2477:
2475:
2471:
2470:
2468:
2467:
2462:
2461:
2460:
2455:
2445:
2443:Cosmic strings
2440:
2434:
2432:
2428:
2427:
2422:
2420:
2419:
2412:
2405:
2397:
2391:
2390:
2375:
2329:hep-th/9510017
2311:
2304:hep-th/9611050
2292:
2286:
2269:
2255:
2218:hep-th/9802067
2200:
2197:hep-th/9806199
2189:
2116:
2079:
2028:
2025:
2022:
2021:
2004:
1961:
1917:
1916:
1914:
1911:
1910:
1909:
1904:
1897:
1894:
1890:string duality
1845:In 1989, Dai,
1825:
1822:
1763:
1758:
1735:
1727:
1721:
1716:
1712:
1705:
1701:
1697:
1691:
1685:
1680:
1647:
1642:
1638:
1631:
1628:
1623:
1620:
1617:
1612:
1608:
1601:
1595:
1590:
1554:
1550:
1529:
1501:
1479:
1476:
1473:
1467:
1463:
1460:
1454:
1450:
1446:
1443:
1437:
1434:
1424:
1420:
1416:
1413:
1410:
1407:
1400:
1396:
1392:
1386:
1380:
1375:
1332:
1329:
1150:
1149:Gauge theories
1147:
1134:
1114:
1092:
1088:
1082:
1078:
1064:
1061:
1024:
1021:
974:
954:
951:
948:
928:
905:
902:
865:
862:
812:
811:
809:
808:
801:
794:
786:
783:
782:
781:
780:
775:
767:
766:
763:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
602:
597:
595:Randjbar-Daemi
592:
587:
582:
577:
572:
567:
562:
557:
552:
547:
542:
537:
532:
527:
522:
517:
512:
507:
502:
497:
492:
487:
482:
477:
472:
467:
462:
457:
452:
447:
442:
437:
432:
427:
422:
417:
412:
407:
402:
397:
392:
387:
382:
377:
372:
367:
362:
357:
352:
347:
342:
337:
332:
327:
322:
317:
312:
307:
302:
297:
292:
287:
282:
276:
273:
272:
269:
268:
265:
264:
259:
254:
249:
241:
236:
231:
226:
221:
216:
210:
207:
206:
203:
202:
201:
200:
195:
193:Vertex algebra
190:
185:
180:
172:
171:
167:
166:
165:
164:
159:
154:
146:
145:
141:
140:
139:
138:
133:
128:
123:
118:
113:
105:
104:
100:
99:
98:
97:
79:
71:
70:
66:
65:
64:
63:
58:
53:
48:
40:
39:
35:
34:
26:
25:
15:
13:
10:
9:
6:
4:
3:
2:
3610:
3599:
3598:String theory
3596:
3595:
3593:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3562:Zamolodchikov
3560:
3558:
3557:Zamolodchikov
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
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:
3099:
3097:
3093:
3087:
3084:
3082:
3081:Matrix theory
3079:
3078:
3076:
3074:
3070:
3064:
3061:
3059:
3056:
3055:
3053:
3051:
3047:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3002:
3000:
2998:
2997:Supersymmetry
2994:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2946:
2943:
2941:
2934:
2930:
2927:
2926:
2925:
2922:
2920:
2917:
2916:
2915:
2912:
2910:
2907:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2885:
2883:
2879:
2873:
2871:
2867:
2865:
2862:
2860:
2857:
2854:
2847:
2840:
2833:
2826:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2795:
2793:
2791:
2787:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2737:
2735:
2733:
2729:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2669:
2667:
2665:
2661:
2655:
2652:
2650:
2649:Dual graviton
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2616:
2614:
2610:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2585:
2583:
2581:
2577:
2571:
2568:
2566:
2565:RNS formalism
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2524:
2521:
2517:
2514:
2512:
2509:
2508:
2507:
2504:
2502:
2501:Type I string
2499:
2498:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2478:
2476:
2472:
2466:
2463:
2459:
2456:
2454:
2451:
2450:
2449:
2446:
2444:
2441:
2439:
2436:
2435:
2433:
2429:
2425:
2424:String theory
2418:
2413:
2411:
2406:
2404:
2399:
2398:
2395:
2388:
2387:0-521-83143-1
2384:
2380:
2376:
2371:
2367:
2363:
2359:
2355:
2351:
2347:
2343:
2339:
2335:
2330:
2325:
2321:
2317:
2312:
2309:
2305:
2301:
2297:
2293:
2289:
2287:0-521-80912-6
2283:
2279:
2276:. Cambridge:
2275:
2270:
2268:
2264:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2219:
2214:
2210:
2206:
2201:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2080:
2076:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2036:
2031:
2030:
2026:
2018:
2014:
2008:
2005:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1965:
1962:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1922:
1919:
1912:
1908:
1905:
1903:
1900:
1899:
1895:
1893:
1891:
1887:
1883:
1878:
1873:
1869:
1864:
1860:
1856:
1852:
1848:
1843:
1841:
1837:
1836:Michael Green
1832:
1831:Warren Siegel
1823:
1821:
1819:
1818:supersymmetry
1814:
1812:
1808:
1804:
1800:
1794:
1792:
1788:
1782:
1780:
1779:Planck length
1756:
1746:
1733:
1725:
1714:
1710:
1703:
1699:
1695:
1689:
1678:
1669:
1667:
1666:surface area.
1663:
1658:
1645:
1640:
1636:
1629:
1621:
1618:
1615:
1610:
1606:
1599:
1588:
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1572:
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1548:
1527:
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1422:
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1414:
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1408:
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1373:
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1342:
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1330:
1328:
1326:
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1279:
1275:
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1234:
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1224:
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1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1176:
1174:
1170:
1166:
1162:
1157:
1148:
1146:
1132:
1112:
1090:
1086:
1080:
1076:
1062:
1060:
1058:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1022:
1020:
1018:
1014:
1010:
1009:Edward Witten
1006:
1002:
998:
994:
990:
988:
972:
952:
949:
946:
926:
917:
915:
911:
903:
901:
899:
895:
891:
887:
884:solutions of
883:
882:black p-brane
879:
875:
871:
863:
861:
859:
855:
851:
847:
843:
839:
835:
832:can end with
831:
827:
823:
819:
818:string theory
807:
802:
800:
795:
793:
788:
787:
785:
784:
779:
776:
774:
771:
770:
769:
768:
761:
758:
756:
753:
751:
748:
746:
745:Zamolodchikov
743:
741:
740:Zamolodchikov
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
536:
533:
531:
528:
526:
523:
521:
518:
516:
513:
511:
508:
506:
503:
501:
498:
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493:
491:
488:
486:
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481:
478:
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473:
471:
468:
466:
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461:
458:
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453:
451:
448:
446:
443:
441:
438:
436:
433:
431:
428:
426:
423:
421:
418:
416:
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411:
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406:
403:
401:
398:
396:
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388:
386:
383:
381:
378:
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371:
368:
366:
363:
361:
358:
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353:
351:
348:
346:
343:
341:
338:
336:
333:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
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301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
277:
271:
270:
263:
260:
258:
255:
253:
250:
248:
246:
242:
240:
237:
235:
232:
230:
229:Supersymmetry
227:
225:
222:
220:
217:
215:
212:
211:
205:
204:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
175:
174:
173:
168:
163:
160:
158:
155:
153:
152:Phenomenology
150:
149:
148:
147:
144:Phenomenology
142:
137:
134:
132:
129:
127:
124:
122:
119:
117:
114:
112:
109:
108:
107:
106:
101:
95:
91:
87:
83:
80:
78:
75:
74:
73:
72:
67:
62:
59:
57:
54:
52:
51:Cosmic string
49:
47:
44:
43:
42:
41:
36:
32:
28:
27:
24:
23:String theory
20:
3107:Arkani-Hamed
3005:Supergravity
2972:Moduli space
2869:
2864:Dirac string
2790:Gauge theory
2770:Loop algebra
2707:Black string
2671:
2570:GS formalism
2378:
2319:
2315:
2295:
2273:
2258:
2208:
2204:
2160:
2156:
2124:
2120:
2087:
2083:
2042:
2038:
2016:
2012:
2007:
1974:
1970:
1964:
1931:
1927:
1921:
1876:
1863:moduli space
1844:
1827:
1815:
1798:
1795:
1787:John Wheeler
1783:
1747:
1670:
1665:
1659:
1580:
1573:
1491:
1365:
1363:is given by
1334:
1324:
1320:
1316:
1314:
1308:
1304:
1300:
1296:
1292:
1288:
1286:
1271:
1267:gauge theory
1264:
1259:
1248:
1244:
1236:
1232:
1228:
1225:
1221:
1216:
1208:
1204:
1184:
1180:
1177:
1172:
1168:
1165:relativistic
1155:
1152:
1066:
1056:
1048:
1026:
991:
987:Myers effect
918:
907:
886:supergravity
867:
841:
825:
824:, short for
821:
815:
285:Arkani-Hamed
244:
234:Supergravity
60:
3467:Silverstein
2967:Orientifold
2702:Black holes
2697:Black brane
2654:Dual photon
1868:orientifold
1337:black holes
1331:Black holes
1035:and 10 for
900:dualities.
894:holographic
850:instantonic
645:Silverstein
170:Mathematics
82:Superstring
3487:Strominger
3482:Steinhardt
3477:Staudacher
3392:Polchinski
3342:Nanopoulos
3302:Mandelstam
3282:Kontsevich
3122:Berenstein
3050:Holography
3030:Superspace
2929:K3 surface
2888:Worldsheet
2803:Instantons
2431:Background
2027:References
1851:Polchinski
1578:entropy":
1576:Bekenstein
1291:separate D
997:Ashoke Sen
874:Polchinski
665:Strominger
660:Steinhardt
655:Staudacher
570:Polchinski
520:Nanopoulos
480:Mandelstam
460:Kontsevich
300:Berenstein
257:Multiverse
3522:Veneziano
3402:Rajaraman
3297:Maldacena
3187:Gopakumar
3137:Dijkgraaf
3132:Curtright
2798:Anomalies
2677:NS5-brane
2598:U-duality
2593:S-duality
2588:T-duality
2354:0031-9007
2251:119460857
2243:0034-6861
2185:0550-3213
2149:0031-9007
2112:0556-2821
2067:0556-2821
1999:0370-2693
1956:0217-7323
1859:T-duality
1803:ideal gas
1668:In fact,
1627:ℏ
1619:π
1445:×
1436:≈
1409:π
1391:ℏ
1239:massless
1133:χ
1113:ϕ
1087:χ
1077:ϕ
1053:gravitons
1017:spacetime
950:×
864:Discovery
838:dimension
705:Veneziano
580:Rajaraman
475:Maldacena
365:Gopakumar
315:Dijkgraaf
310:Curtright
274:Theorists
162:Landscape
157:Cosmology
121:U-duality
116:T-duality
111:S-duality
94:Heterotic
3592:Category
3577:Zwiebach
3532:Verlinde
3527:Verlinde
3502:Townsend
3497:Susskind
3432:Sagnotti
3397:Polyakov
3352:Nekrasov
3317:Minwalla
3312:Martinec
3277:Knizhnik
3272:Klebanov
3267:Kapustin
3232:'t Hooft
3167:Fischler
3102:AganagiÄ
3073:M-theory
2962:Conifold
2957:Orbifold
2940:manifold
2881:Geometry
2687:M5-brane
2682:M2-brane
2619:Graviton
2535:F-theory
2362:10059981
2308:TASI '96
2274:D-branes
2153:; ibid.
1907:M-theory
1896:See also
1282:matrices
1245:geometry
1219:-brane.
1197:quantize
1156:oriented
1013:K-theory
898:M-theory
822:D-branes
778:Glossary
760:Zwiebach
715:Verlinde
710:Verlinde
685:Townsend
680:'t Hooft
675:Susskind
610:Sagnotti
575:Polyakov
530:Nekrasov
495:Minwalla
490:Martinec
455:Knizhnik
450:Klebanov
445:Kapustin
415:Horowitz
345:Fischler
280:AganagiÄ
198:K-theory
131:F-theory
126:M-theory
3507:Trivedi
3492:Sundrum
3457:Shenker
3447:Seiberg
3442:Schwarz
3412:Randall
3372:Novikov
3362:Nielsen
3347:NÄstase
3257:Kallosh
3242:Gibbons
3182:Gliozzi
3172:Friedan
3162:Ferrara
3147:Douglas
3142:Distler
2692:S-brane
2672:D-brane
2629:Tachyon
2624:Dilaton
2438:Strings
2370:4671529
2334:Bibcode
2223:Bibcode
2165:Bibcode
2129:Bibcode
2092:Bibcode
2075:1443701
2047:Bibcode
1979:Bibcode
1936:Bibcode
1882:Shenker
1824:History
1777:is the
1341:entropy
1241:scalars
1045:gravity
1015:of the
830:strings
773:History
690:Trivedi
670:Sundrum
635:Shenker
625:Seiberg
620:Schwarz
590:Randall
550:Novikov
540:Nielsen
525:NÄstase
435:Kallosh
420:Gibbons
360:Gliozzi
350:Friedan
340:Ferrara
325:Douglas
320:Distler
90:Type II
77:Bosonic
61:D-brane
3572:Zumino
3567:Zaslow
3552:Yoneya
3542:Witten
3462:Siegel
3437:Scherk
3407:Ramond
3382:Ooguri
3307:Marolf
3262:Kaluza
3247:Kachru
3237:HoĆava
3227:Harvey
3222:Hanson
3207:Gubser
3197:Greene
3127:Bousso
3112:Atiyah
2664:Branes
2474:Theory
2385:
2368:
2360:
2352:
2284:
2265:
2249:
2241:
2183:
2147:
2110:
2073:
2065:
1997:
1954:
1855:HoĆava
1853:, and
1748:where
1514:Newton
1492:where
1201:photon
1057:closed
878:HoĆava
872:, and
755:Zumino
750:Zaslow
735:Yoneya
725:Witten
640:Siegel
615:Scherk
585:Ramond
560:Ooguri
485:Marolf
440:Kaluza
425:Kachru
410:HoĆava
405:Harvey
400:Hanson
385:Gubser
375:Greene
305:Bousso
290:Atiyah
86:Type I
46:String
3512:Turok
3422:RoÄek
3387:Ovrut
3377:Olive
3357:Neveu
3337:Myers
3332:Mukhi
3322:Moore
3292:Linde
3287:Klein
3212:Gukov
3202:Gross
3192:Green
3177:Gates
3157:Dvali
3117:Banks
2366:S2CID
2324:arXiv
2300:arXiv
2247:S2CID
2213:arXiv
2193:arXiv
1913:Notes
1847:Leigh
1442:1.227
1278:group
1260:d - p
1237:d - p
870:Leigh
854:space
695:Turok
600:RoÄek
565:Ovrut
555:Olive
535:Neveu
515:Myers
510:Mukhi
500:Moore
470:Linde
465:Klein
390:Gukov
380:Gross
370:Green
355:Gates
335:Dvali
295:Banks
56:Brane
3537:Wess
3517:Vafa
3427:Rohm
3327:Motl
3252:Kaku
3217:Guth
3152:Duff
2383:ISBN
2358:PMID
2350:ISSN
2282:ISBN
2263:ISBN
2239:ISSN
2181:ISSN
2145:ISSN
2108:ISSN
2071:OSTI
2063:ISSN
1995:ISSN
1952:ISSN
1269:, a
1005:form
896:and
858:time
856:and
720:Wess
700:Vafa
605:Rohm
505:Motl
430:Kaku
395:Guth
330:Duff
3547:Yau
3472:SÆĄn
3452:Sen
2342:doi
2231:doi
2173:doi
2161:111
2137:doi
2100:doi
2055:doi
1987:doi
1975:231
1944:doi
1567:is
1516:'s
1512:is
1274:(1)
1195:to
1049:not
1047:is
816:In
730:Yau
650:SÆĄn
630:Sen
3594::
2848:,
2841:,
2834:,
2827:,
2364:.
2356:.
2348:.
2340:.
2332:.
2320:75
2318:.
2298:,
2280:.
2245:.
2237:.
2229:.
2221:.
2209:71
2207:.
2179:.
2171:.
2159:.
2143:.
2135:.
2125:36
2123:.
2106:.
2098:.
2088:13
2086:.
2069:.
2061:.
2053:.
2043:13
2041:.
2037:.
2017:50
2015:,
1993:.
1985:.
1973:.
1950:.
1942:.
1932:04
1930:.
1849:,
1781:.
1571:.
1520:,
1453:23
1449:10
1173:mc
1171:=
860:.
842:D.
820:,
92:,
88:,
2938:2
2936:G
2905:?
2870:p
2855:)
2852:8
2850:E
2845:7
2843:E
2838:6
2836:E
2831:4
2829:F
2824:2
2822:G
2820:(
2416:e
2409:t
2402:v
2389:.
2372:.
2344::
2336::
2326::
2310:.
2302::
2290:.
2253:.
2233::
2225::
2215::
2199:.
2195::
2187:.
2175::
2167::
2151:.
2139::
2131::
2114:.
2102::
2094::
2077:.
2057::
2049::
2001:.
1989::
1981::
1958:.
1946::
1938::
1877:e
1762:P
1757:l
1734:,
1726:2
1720:P
1715:l
1711:4
1704:B
1700:k
1696:A
1690:=
1684:B
1679:S
1646:.
1641:2
1637:M
1630:c
1622:G
1616:4
1611:B
1607:k
1600:=
1594:B
1589:S
1553:B
1549:k
1528:M
1500:G
1478:,
1475:)
1472:K
1466:M
1462:g
1459:k
1433:(
1423:B
1419:k
1415:M
1412:G
1406:8
1399:3
1395:c
1385:=
1379:H
1374:T
1325:N
1323:(
1321:U
1317:N
1309:j
1305:i
1301:i
1297:N
1293:p
1289:N
1272:U
1233:d
1229:p
1227:D
1217:p
1209:p
1205:p
1185:p
1181:p
1169:E
1091:2
1081:2
973:N
953:N
947:N
927:N
805:e
798:t
791:v
245:N
96:)
84:(
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