2053:, the model of spacetime used in nongravitational physics. One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory. The claim is that this quantum field theory is equivalent to the gravitational theory on the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.
1926:
2241:
1659:, Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle. In this setting, one can imagine the membrane wrapping around the circular dimension. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. In fact, Duff and his collaborators showed that this construction reproduces exactly the strings appearing in type IIA superstring theory.
1168:
1099:
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compactification of string theory in which four of the ten dimensions curl up. If one considers a five-dimensional brane wrapped around these extra dimensions, then the brane looks just like a one-dimensional string. In this way, the conjectured relationship between strings and branes was reduced to a relationship between strings and strings, and the latter could be tested using already established theoretical techniques.
51:
2001:
2398:, and others, is known as heterotic M-theory. In this approach, one imagines that one of the eleven dimensions of M-theory is shaped like a circle. If this circle is very small, then the spacetime becomes effectively ten-dimensional. One then assumes that six of the ten dimensions form a CalabiâYau manifold. If this CalabiâYau manifold is also taken to be small, one is left with a theory in four-dimensions.
1145:
lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.
1473:
1076:. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA and IIB include only closed strings. Each of these five string theories arises as a special limiting case of M-theory. This theory, like its string theory predecessors, is an example of a quantum theory of gravity. It describes a
1187:. This is a relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently.
1014:. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its
863:. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called
1454:), that has never been shown to exist, and in part because it was unable to correctly predict the ratio of an electron's mass to its charge. In addition, these theories were being developed just as other physicists were beginning to discover quantum mechanics, which would ultimately prove successful in describing known forces such as electromagnetism, as well as new
1302:. Roughly speaking, fermions are the constituents of matter, while bosons mediate interactions between particles. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa. When supersymmetry is imposed as a local symmetry, one automatically obtains a quantum mechanical theory that includes gravity. Such a theory is called a
1906:, a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property. This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.
1721:
made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions. Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of two- and five-dimensional branes in
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In string theory, the fundamental objects that give rise to elementary particles are the one-dimensional strings. Although the physical phenomena described by M-theory are still poorly understood, physicists know that the theory describes two- and five-dimensional branes. Much of the current research
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In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can
1110:
In everyday life, there are three familiar dimensions of space: height, width and depth. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a
2266:
Typically, such models are based on the idea of compactification. Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they can construct models roughly similar to the standard model
2125:
factor). In the real world, spacetime is four-dimensional, at least macroscopically, so this version of the correspondence does not provide a realistic model of gravity. Likewise, the dual theory is not a viable model of any real-world system since it describes a world with six spacetime dimensions.
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in 1984, many physicists turned to string theory as a unified theory of particle physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects. Another feature of
1144:
is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles. In the limit where these curled-up dimensions become very small, one obtains a theory in which spacetime has effectively a
2287:
Partly because of theoretical and mathematical difficulties and partly because of the extremely high energies (beyond what is technologically possible for the foreseeable future) needed to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to
1807:
in 1997. This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix
1523:
Initially, many physicists hoped that by compactifying eleven-dimensional supergravity, it might be possible to construct realistic models of our four-dimensional world. The hope was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the
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in three dimensions. In this version of the correspondence, seven of the dimensions of M-theory are curled up, leaving four non-compact dimensions. Since the spacetime of our universe is four-dimensional, this version of the correspondence provides a somewhat more realistic description of gravity.
1960:
in late 1997, the AdS/CFT correspondence is a theoretical result which implies that M-theory is in some cases equivalent to a quantum field theory. In addition to providing insights into the mathematical structure of string and M-theory, the AdS/CFT correspondence has shed light on many aspects of
1499:
General relativity does not place any limits on the possible dimensions of spacetime. Although the theory is typically formulated in four dimensions, one can write down the same equations for the gravitational field in any number of dimensions. Supergravity is more restrictive because it places an
2044:
This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional
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published a similar result which suggested that strongly interacting strings in ten dimensions might have an equivalent description in terms of weakly interacting five-dimensional branes. Initially, physicists were unable to prove this relationship for two important reasons. On the one hand, the
1654:
Supersymmetry severely restricts the possible number of dimensions of a brane. In 1987, Eric
Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes. Intuitively, these objects look like sheets or membranes propagating through the
1347:
is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension
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showed that type IIB string theory with a large coupling constant is equivalent via S-duality to the same theory with small coupling constant. Theorists also found that different string theories may be related by T-duality. This duality implies that strings propagating on completely different
1682:
In spite of this progress, the relationship between strings and five-dimensional branes remained conjectural because theorists were unable to quantize the branes. Starting in 1991, a team of researchers including
Michael Duff, Ramzi Khuri, Jianxin Lu, and Ruben Minasian considered a special
1560:
Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation. However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways.
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should stand for "magic", "mystery", or "membrane" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known. Years later, he would state, "I thought my colleagues would understand that it really stood for membrane.
1945:. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. Quantum field theories are also used throughout condensed matter physics to model particle-like objects called
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String theory extends ordinary particle physics by replacing zero-dimensional point particles by one-dimensional objects called strings. In the late 1980s, it was natural for theorists to attempt to formulate other extensions in which particles are replaced by two-dimensional
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An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-dimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around any given point, it looks just like
1667:
MontonenâOlive duality was still unproven, and so
Strominger's conjecture was even more tenuous. On the other hand, there were many technical issues related to the quantum properties of five-dimensional branes. The first of these problems was solved in 1993 when
1874:
is a branch of mathematics that attempts to generalize this situation. Rather than working with ordinary numbers, one considers some similar objects, such as matrices, whose multiplication does not satisfy the commutative law (that is, objects for which
1733:. Witten and HoĆava studied M-theory on a special spacetime geometry with two ten-dimensional boundary components. Their work shed light on the mathematical structure of M-theory and suggested possible ways of connecting M-theory to real world physics.
1532:, and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as
2271:
partners to analogues of known particles. One popular way of deriving realistic physics from string theory is to start with the heterotic theory in ten dimensions and assume that the six extra dimensions of spacetime are shaped like a six-dimensional
1548:
string theory that many physicists were drawn to in the 1980s and 1990s was its high degree of uniqueness. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary
2221:. The latter theory was popularized by Witten in the late 1980s because of its applications to knot theory. In addition, the ABJM theory serves as a semi-realistic simplified model for solving problems that arise in condensed matter physics.
1996:
of a disk by triangles and squares. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior.
988:. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.
1703:. The shaded region represents a family of different physical scenarios that are possible in M-theory. In certain limiting cases corresponding to the cusps, it is natural to describe the physics using one of the six theories labeled there.
1419:
pioneered the use of four-dimensional geometry for describing the physical world. These efforts culminated in the formulation of
Einstein's general theory of relativity, which relates gravity to the geometry of four-dimensional spacetime.
2284:. CalabiâYau manifolds offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct models with physics resembling to some extent that of our four-dimensional world based on M-theory.
1271:
turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be
2103:
on the six-dimensional boundary. Here "(2,0)" refers to the particular type of supersymmetry that appears in the theory. In this example, the spacetime of the gravitational theory is effectively seven-dimensional (hence the notation
2141:
which relates the physics of this theory to certain physical concepts associated with the surface itself. More recently, theorists have extended these ideas to study the theories obtained by compactifying down to three dimensions.
1746:
As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to
2216:
The ABJM theory appearing in this version of the correspondence is also interesting for a variety of reasons. Introduced by
Aharony, Bergman, Jafferis, and Maldacena, it is closely related to another quantum field theory called
2036:
is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is
1808:
model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.
1783:
is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.
1022:, and other properties determined by the vibrational state of the string. In this way, all of the different elementary particles may be viewed as vibrating strings. One of the vibrational states of a string gives rise to the
1138:(ten spatial dimensions, and one time dimension). In order to describe real physical phenomena using these theories, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.
1722:
string theory. In the months following Witten's announcement, hundreds of new papers appeared on the
Internet confirming that the new theory involved membranes in an important way. Today this flurry of work is known as the
1618:. In other words, a system of strongly interacting particles (large coupling constant) has an equivalent description as a system of weakly interacting particles (small coupling constant) and vice versa by spin-moment.
2288:
any of these models being a correct fundamental description of nature. This has led some in the community to criticize these approaches to unification and question the value of continued research on these problems.
2393:
manifolds, most attempts to construct realistic theories of physics based on M-theory have taken a more indirect approach to compactifying eleven-dimensional spacetime. One approach, pioneered by Witten, HoĆava,
1741:
Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In a paper from 1996, HoĆava and Witten wrote
898:. According to Witten, M should stand for "magic", "mystery" or "membrane" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.
1887:). One imagines that these noncommuting objects are coordinates on some more general notion of "space" and proves theorems about these generalized spaces by exploiting the analogy with ordinary geometry.
2405:
in which the observable universe is thought to exist on a brane in a higher dimensional ambient space. It has also spawned alternative theories of the early universe that do not rely on the theory of
901:
Investigations of the mathematical structure of M-theory have spawned important theoretical results in physics and mathematics. More speculatively, M-theory may provide a framework for developing a
1183:
Theories that arise as different limits of M-theory turn out to be related in highly nontrivial ways. One of the relationships that can exist between these different physical theories is called
2251:
In addition to being an idea of considerable theoretical interest, M-theory provides a framework for constructing models of real world physics that combine general relativity with the
1850:. In ordinary geometry, the coordinates of a point are numbers, so they can be multiplied, and the product of two coordinates does not depend on the order of multiplication. That is,
2129:
Nevertheless, the (2,0)-theory has proven to be important for studying the general properties of quantum field theories. Indeed, this theory subsumes many mathematically interesting
1360:-branes. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They can have mass and other attributes such as charge. A
1552:. In string theory, the possibilities are much more constrained: by the 1990s, physicists had argued that there were only five consistent supersymmetric versions of the theory.
1119:
perform calculations and gain general insights more easily. There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in
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that were being discovered throughout the middle part of the century. Thus it would take almost fifty years for the idea of new dimensions to be taken seriously again.
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1496:. In the mid-1970s, physicists began studying higher-dimensional theories combining general relativity with supersymmetry, the so-called supergravity theories.
2133:
and points to new dualities relating these theories. For example, Luis Alday, Davide
Gaiotto, and Yuji Tachikawa showed that by compactifying this theory on a
1226:
as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the
2149:. For example, the existence of the (2,0)-theory was used by Witten to give a "physical" explanation for a conjectural relationship in mathematics called the
2165:, the branch of mathematics that studies and classifies the different shapes of knots. Another application of the (2,0)-theory in mathematics is the work of
1569:
1123:. Finally, there exist scenarios in which there could actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.
263:
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manifolds are still poorly understood mathematically, and this fact has made it difficult for physicists to fully develop this approach to phenomenology.
1115:, three spatial dimensions and one time dimension. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.
5160: â The "Official String Theory Web Site", created by Patricia Schwarz. References on string theory and M-theory for the layperson and expert.
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in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, a string has
1209:. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius
1492:
New concepts and mathematical tools provided fresh insights into general relativity, giving rise to a period in the 1960sâ70s now known as the
1941:
The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as
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In the 1990s, several theorists generalized
MontonenâOlive duality to the S-duality relationship, which connects different string theories.
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to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.
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to construct candidate models of the four-dimensional world, although so far none has been verified to give rise to physics as observed in
1976:. In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the
1903:
1821:
871:. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a
1504:
showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven. In the same year,
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is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.
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Although a complete formulation of M-theory is not known, such a formulation should describe two- and five-dimensional objects called
197:
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is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas.
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showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.
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had conjectured a special property of certain physical theories. A sharpened version of their conjecture concerns a theory called
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which live on the worldvolume of a brane. The word brane comes from the word "membrane" which refers to a two-dimensional brane.
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The success of general relativity led to efforts to apply higher dimensional geometry to explain other forces. In 1919, work by
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Alday, Luis; Gaiotto, Davide; Tachikawa, Yuji (2010). "Liouville correlation functions from four-dimensional gauge theories".
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is the part of string theory that attempts to construct realistic models of particle physics based on string and M-theory.
1072:). The different theories allow different types of strings, and the particles that arise at low energies exhibit different
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In one approach to M-theory phenomenology, theorists assume that the seven extra dimensions of M-theory are shaped like a
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1313:. There are several different versions of superstring theory which are all subsumed within the M-theory framework. At low
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and therefore cannot be used to make meaningful physical predictions. See Zee 2010, p. 72 for a discussion of this issue.
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969:
594:
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Candelas, Philip; Horowitz, Gary; Strominger, Andrew; Witten, Edward (1985). "Vacuum configurations for superstrings".
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scenarios provide an alternative way of recovering real world physics from string theory. See
Randall and Sundrum 1999.
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5087:
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Banks, Tom; Fischler, Willy; Schenker, Stephen; Susskind, Leonard (1997). "M theory as a matrix model: A conjecture".
2153:. In subsequent work, Witten showed that the (2,0)-theory could be used to understand a concept in mathematics called
1930:
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1595:
1435:, who suggested that the additional dimension proposed by Kaluza could take the form of a circle with radius around
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Sen, Ashoke (1994b). "Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and
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at a given time. The resulting geometric object is three-dimensional anti-de Sitter space. It looks like a solid
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One of the important developments following Witten's announcement was Witten's work in 1996 with string theorist
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The necessity of a quantum mechanical description of gravity follows from the fact that one cannot consistently
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In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the
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2367:. Finally, there are many open questions about the existence, uniqueness, and other mathematical properties of
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and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.
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1294:. This is a mathematical relation that exists in certain physical theories between a class of particles called
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Trespassing on
Einstein's Lawn: A Father, a Daughter, the Meaning of Nothing and the Beginning of Everything
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In addition to its applications in quantum field theory, the (2,0)-theory has spawned important results in
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For an introduction to the applications of quantum field theory to condensed matter physics, see Zee 2010.
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were never completely successful. In part this was because KaluzaâKlein theory predicted a particle (the
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2478:, theorists often formulate and study theories of gravity in unphysical numbers of spacetime dimensions.
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supergravity. Similarly, M-theory is approximated at low energies by supergravity in eleven dimensions.
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1317:, superstring theories are approximated by one of the three supergravities in ten dimensions, known as
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Connes, Alain; Douglas, Michael; Schwarz, Albert (1998). "Noncommutative geometry and matrix theory".
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For example, physicists and mathematicians often assume that space has a mathematical property called
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HoĆava, Petr; Witten, Edward (1996b). "Eleven dimensional supergravity on a manifold with boundary".
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HoĆava, Petr; Witten, Edward (1996a). "Heterotic and Type I string dynamics from eleven dimensions".
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manifold if one wishes to recover the physics of our four-dimensional world. Another problem is that
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or by higher-dimensional objects called branes. Such objects had been considered as early as 1962 by
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in such a way that any point in the interior is actually infinitely far from this boundary surface.
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1590:. The strength with which the particles of this theory interact is measured by a number called the
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1179:. These dualities may be combined to obtain equivalences of any of the five theories with M-theory.
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1106:: At large distances, a two-dimensional surface with one circular dimension looks one-dimensional.
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2012:, each one representing the state of the universe at a given time. One can study theories of
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The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
4964:
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The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory
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2406:
2364:
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1985:
1964:
In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain
1961:
quantum field theory in regimes where traditional calculational techniques are ineffective.
1934:
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176:
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2452:
From a technical point of view, the problem is that the theory one gets in this way is not
2137:, one obtains a four-dimensional quantum field theory, and there is a duality known as the
1651:, and they were reconsidered by a small but enthusiastic group of physicists in the 1980s.
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2013:
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1007:
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944:
6338:
6333:
6293:
6233:
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6143:
6063:
6053:
6048:
6043:
5958:
5953:
5948:
5913:
5898:
5826:
5503:
5366:
4817:
4652:
Sen, Ashoke (1994a). "Strong-weak coupling duality in four-dimensional string theory".
4490:
3607:
3538:
3441:
2453:
2281:
2277:
2174:
2166:
1957:
1899:
1796:
1644:
1587:
1513:
1424:
1415:
In the early 20th century, physicists and mathematicians including Albert Einstein and
1263:
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754:
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644:
634:
554:
474:
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459:
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369:
364:
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324:
309:
212:
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3851:
2376:
manifolds, and mathematicians lack a systematic way of searching for these manifolds.
1671:
established that certain physical theories require the existence of objects with both
1167:
1080:
just like the familiar gravitational force subject to the rules of quantum mechanics.
6378:
6328:
6313:
6288:
6278:
6248:
6193:
6168:
6113:
6098:
6093:
6058:
6033:
5993:
5783:
5435:
5351:
5229:
5210:
4988:
Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law
4976:
4914:
4807:
4778:
4741:
4636:
4558:
Seiberg, Nathan; Witten, Edward (1999). "String Theory and Noncommutative Geometry".
4427:
4347:
4174:
4157:
4110:
4063:
4016:
3898:
3775:
3631:
3528:
3499:
3491:
2318:
2268:
2114:
2084:
2083:
One particular realization of the AdS/CFT correspondence states that M-theory on the
1977:
1946:
1630:
1537:
1509:
1477:
1455:
1291:
1285:
1030:
991:
939:
852:
848:
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704:
689:
659:
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579:
524:
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469:
444:
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404:
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70:
42:
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4118:
4071:
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3822:
3738:
3682:
3432:
3385:
3338:
2240:
2024:
Now imagine a stack of hyperbolic disks where each disk represents the state of the
1952:
One approach to formulating M-theory and studying its properties is provided by the
859:
in 1995. Witten's announcement initiated a flurry of research activity known as the
50:
6358:
6198:
6078:
6028:
5998:
5983:
5791:
5758:
5650:
5576:
5556:
5493:
5356:
5117:
3598:
1993:
1891:
1700:
1481:
1467:
774:
609:
489:
439:
409:
394:
253:
4597:
4149:
1691:
5032:
4356:
3299:
superconformal Chern-Simons-matter theories, M2-branes and their gravity duals".
2317:. This is a special kind of seven-dimensional shape constructed by mathematician
1126:
One notable feature of string theory and M-theory is that these theories require
6353:
6323:
6303:
6163:
6118:
6073:
6038:
5988:
5753:
5722:
5483:
5440:
5163:
4542:
2552:
2306:
2162:
2076:
1565:
1501:
1451:
1432:
1427:
showed that by passing to five-dimensional spacetime, one can unify gravity and
1130:
of spacetime for their mathematical consistency. In string theory, spacetime is
998:
that attempts to reconcile gravity and quantum mechanics. In string theory, the
985:
769:
739:
719:
574:
529:
484:
449:
399:
4256:
3720:
2363:, so theorists are unable to use tools from the branch of mathematics known as
6298:
6238:
6173:
5836:
5816:
5715:
5674:
5488:
5167:
5145:
5107:
4683:
3936:
3907:
3814:
3674:
3377:
2395:
2267:
of particle physics, together with additional undiscovered particles, usually
1668:
1648:
1626:
1622:
714:
649:
584:
276:
3424:
984:, a radically different formalism for describing physical phenomena based on
6213:
6003:
5943:
5589:
5463:
5384:
5379:
5374:
4511:
Randall, Lisa; Sundrum, Raman (1999). "An alternative to compactification".
3686:
2038:
2017:
1309:
A theory of strings that incorporates the idea of supersymmetry is called a
1206:
1184:
1176:
1172:
1162:
1158:
1112:
868:
864:
624:
414:
354:
140:
135:
130:
4300:
3767:
1472:
1134:(nine spatial dimensions, and one time dimension), while in M-theory it is
4472:
2195:
Another realization of the AdS/CFT correspondence states that M-theory on
1902:
showed that some aspects of matrix models and M-theory are described by a
1625:
studied S-duality in the context of heterotic strings in four dimensions.
1390:
in M-theory attempts to better understand the properties of these branes.
5748:
5743:
5473:
5468:
5405:
5321:
5140:
5097:
focuses on the history and emergence of M-theory, and scientists involved
3783:
Duff, Michael (1996). "M-theory (the theory formerly known as strings)".
2594:
2592:
2119:
2025:
1299:
1223:
1023:
909:
of nature. Attempts to connect M-theory to experiment typically focus on
887:
150:
32:
For a more accessible and less technical introduction to this topic, see
4838:
Witten, Edward (1989). "Quantum Field Theory and the Jones Polynomial".
4435:
Nekrasov, Nikita; Schwarz, Albert (1998). "Instantons on noncommutative
4126:
Khovanov, Mikhail (2000). "A categorification of the Jones polynomial".
2276:. This is a special kind of geometric object named after mathematicians
1695:
A schematic illustration of the relationship between M-theory, the five
17:
5478:
5458:
5415:
5410:
4897:
4883:
Witten, Edward (1995). "String theory dynamics in various dimensions".
4859:
4724:
4666:
4619:
4572:
4525:
4455:
4239:
4093:
4046:
3999:
3797:
3711:
3573:
3407:
1343:
In string theory and related theories such as supergravity theories, a
961:
840:
80:
5130:
broadcast dates: October 28, 8â10 p.m. and November 4, 8â9 p.m., 2003)
4196:
4079:
Hull, Chris; Townsend, Paul (1995). "Unity of superstring dualities".
4140:
2430:
For a standard introduction to quantum mechanics, see Griffiths 2004.
1314:
1295:
886:
and should be approximated by eleven-dimensional supergravity at low
883:
4968:
4403:
4323:
3860:
3015:
3013:
1290:
Another important theoretical idea that plays a role in M-theory is
2660:
2658:
5450:
4959:
4936:
4930:
Witten, Edward (2009). "Geometric Langlands from six dimensions".
4605:
Sen, Ashoke (1993). "Electric-magnetic duality in string theory".
3926:
3657:
3360:
3313:
2239:
2066:
1999:
1924:
1868:
of coordinates is the starting point for much of modern geometry.
1690:
1583:
1579:
1471:
1344:
1338:
1166:
1097:
1026:, a quantum mechanical particle that carries gravitational force.
943:
75:
3859:
Duff, Michael; Howe, Paul; Inami, Takeo; Stelle, Kellogg (1987).
3701:
Dine, Michael (2000). "TASI Lectures on M Theory Phenomenology".
2465:
For an accessible introduction to string theory, see Greene 2000.
5179:
5157:
1578:, which describes theoretical particles formally similar to the
1215:
is equivalent to a string propagating around a circle of radius
1015:
956:
One of the deepest problems in modern physics is the problem of
5183:
5170:'s blog on physics in general, and string theory in particular.
2000:
1779:
is a rectangular array of numbers or other data. In physics, a
1679:
charge which were predicted by the work of Montonen and Olive.
890:. Modern attempts to formulate M-theory are typically based on
5127:
3830:
Duff, Michael (1998). "The theory formerly known as strings".
1954:
anti-de Sitter/conformal field theory (AdS/CFT) correspondence
1846:
as the distances between any point in the plane and a pair of
3270:
3268:
3266:
3264:
1171:
A diagram of string theory dualities. Yellow arrows indicate
948:
The fundamental objects of string theory are open and closed
2711:
2709:
1655:
eleven-dimensional spacetime. Shortly after this discovery,
3908:"Wall-crossing, Hitchin systems, and the WKB approximation"
3746:
Dirac, Paul (1962). "An extensible model of the electron".
2443:
a classical system to a quantum one. See Wald 1984, p. 382.
4225:
limit of superconformal field theories and supergravity".
2947:
2945:
1500:
upper limit on the number of dimensions. In 1978, work by
1205:
Another relationship between different string theories is
3906:
Gaiotto, Davide; Moore, Gregory; Neitzke, Andrew (2013).
1431:
into a single force. This idea was improved by physicist
3463:
Bergshoeff, Eric; Sezgin, Ergin; Townsend, Paul (1987).
3107:
3105:
3095:
3093:
3091:
2499:
This analogy is used for example in Greene 2000, p. 186.
2401:
Heterotic M-theory has been used to construct models of
1611:
is equivalent to the same theory with coupling constant
1605:
supersymmetric YangâMills theory with coupling constant
1484:
theories. In 1995, he introduced M-theory, sparking the
3639:
Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei (2010).
2341:, but this property cannot be assumed in the case of a
1980:) is different from the notion of distance in ordinary
1202:
is related to itself in a nontrivial way by S-duality.
4167:"Solving Quantum Field Theories via Curved Spacetimes"
3191:
3189:
851:
first conjectured the existence of such a theory at a
2490:
is another way of modifying the number of dimensions.
2177:, which used physical ideas to derive new results in
1830:. For example, in order to study the geometry of the
5176: - Witten's 1995 lecture introducing M-Theory.
3610:(1978). "Supergravity theory in eleven dimensions".
3465:"Supermembranes and eleven-dimensional supergravity"
3003:
3001:
1267:
refers to a situation where two seemingly different
5881:
5858:
5835:
5782:
5667:
5575:
5517:
5449:
5398:
5365:
5260:
5217:
2854:
2852:
2815:
2813:
2785:
2783:
2781:
2779:
2777:
2775:
2681:
2679:
2573:
2571:
2208:is equivalent to a quantum field theory called the
1634:spacetime geometries may be physically equivalent.
5031:
4985:
4816:
4489:
4441:and (2,0) superconformal six dimensional theory".
3537:
2161:around 2000, Khovanov homology provides a tool in
1860:. This property of multiplication is known as the
4786:van Nieuwenhuizen, Peter (1981). "Supergravity".
4757:Strominger, Andrew (1990). "Heterotic solitons".
3446:String theory and M-theory: A modern introduction
2543:For a review of the (2,0)-theory, see Moore 2012.
1864:, and this relationship between geometry and the
1594:. The result of Montonen and Olive, now known as
4313:: CS1 maint: bot: original URL status unknown (
4227:Advances in Theoretical and Mathematical Physics
4210:: CS1 maint: bot: original URL status unknown (
1713:Speaking at the string theory conference at the
4945:Witten, Edward (2012). "Fivebranes and knots".
1992:as illustrated on the left. This image shows a
1787:One important example of a matrix model is the
1744:
1446:and subsequent attempts by Einstein to develop
1191:turns out to be equivalent by S-duality to the
3063:
3061:
972:, which is formulated within the framework of
5195:
2652:Becker, Becker, and Schwarz 2007, pp. 339â347
2525:A standard text is Peskin and Schroeder 1995.
2075:has been used to understand results from the
1826:In geometry, it is often useful to introduce
1029:There are several versions of string theory:
817:
8:
5038:(2nd ed.). Princeton University Press.
3641:"Gauge theories labelled by three-manifolds"
1374:-dimensional volume in spacetime called its
4488:Peskin, Michael; Schroeder, Daniel (1995).
4404:"Supersymmetries and their representations"
4303:. Archived from the original on 2013-11-10.
4200:. Archived from the original on 2010-06-10.
5202:
5188:
5180:
3748:Proceedings of the Royal Society of London
1242:in one description, it will have momentum
824:
810:
49:
38:
4958:
4935:
4896:
4723:
4665:
4654:International Journal of Modern Physics A
4618:
4571:
4524:
4454:
4238:
4195:
4139:
4092:
4045:
3998:
3935:
3925:
3796:
3785:International Journal of Modern Physics A
3750:. A. Mathematical and Physical Sciences.
3710:
3656:
3572:
3406:
3359:
3312:
4385:"Lecture Notes for Felix Klein Lectures"
4324:"Magnetic monopoles as gauge particles?"
4165:Klebanov, Igor; Maldacena, Juan (2009).
2939:Becker, Becker, and Schwarz 2007, p. 296
2508:For example, see the subsections on the
1756:Unfortunately, it got people confused."
843:that unifies all consistent versions of
4492:An Introduction to Quantum Field Theory
2598:Becker, Becker, and Schwarz 2007, p. 12
2567:
2423:
189:
163:
122:
88:
57:
41:
5011:Yau, Shing-Tung; Nadis, Steve (2010).
4840:Communications in Mathematical Physics
4443:Communications in Mathematical Physics
4322:Montonen, Claus; Olive, David (1977).
4306:
4203:
3645:Communications in Mathematical Physics
1564:In the late 1970s, Claus Montonen and
1254:in the dual description. For example,
980:are described within the framework of
2876:Bergshoeff, Sezgin, and Townsend 1987
1556:Relationships between string theories
7:
5600:Bogomol'nyiâPrasadâSommerfield bound
5086: â 2002 feature documentary by
2510:6D (2,0) superconformal field theory
2063:6D (2,0) superconformal field theory
2057:6D (2,0) superconformal field theory
1480:contributed to the understanding of
1198:heterotic string theory. Similarly,
267:= 4 supersymmetric YangâMills theory
5110:-winning, three-hour miniseries by
4219:Maldacena, Juan (1998). "The Large
2474:For example, in the context of the
1904:noncommutative quantum field theory
1822:Noncommutative quantum field theory
5034:Quantum Field Theory in a Nutshell
3138:Alday, Gaiotto, and Tachikawa 2010
3085:Klebanov and Maldacena 2009, p. 28
2253:standard model of particle physics
2151:geometric Langlands correspondence
2113:), and there are four additional "
2016:such as M-theory in the resulting
198:Geometric Langlands correspondence
25:
4293:10.1038/scientificamerican1105-56
3968:Introduction to Quantum Mechanics
3852:10.1038/scientificamerican0298-64
3606:Cremmer, Eugene; Julia, Bernard;
3019:Connes, Douglas, and Schwarz 1998
2384:Because of the difficulties with
1715:University of Southern California
857:University of Southern California
3183:Gaiotto, Moore, and Neitzke 2013
3147:Dimofte, Gaiotto, and Gukov 2010
2664:Becker, Becker, and Schwarz 2007
2514:ABJM superconformal field theory
2191:ABJM superconformal field theory
2185:ABJM superconformal field theory
2131:effective quantum field theories
2045:models of anti-de Sitter space.
1576:supersymmetric YangâMills theory
1494:golden age of general relativity
1298:and a class of particles called
5797:Eleven-dimensional supergravity
5174:M-Theory - Edward Witten (1995)
5055:A First Course in String Theory
4823:. University of Chicago Press.
3348:Letters in Mathematical Physics
3240:Yau and Nadis 2010, pp. 147â150
2769:Cremmer, Julia, and Scherk 1978
2099:is equivalent to the so-called
960:. The current understanding of
877:eleven-dimensional supergravity
27:Framework of superstring theory
5057:. Cambridge University Press.
4710:invariance in string theory".
4560:Journal of High Energy Physics
3561:Journal of High Energy Physics
3448:. Cambridge University Press.
3301:Journal of High Energy Physics
3288:Aharony, Ofer; Bergman, Oren;
1834:, one defines the coordinates
1717:in 1995, Edward Witten of the
1:
5245:Second superstring revolution
4590:10.1088/1126-6708/1999/09/032
4150:10.1215/S0012-7094-00-10131-7
3591:10.1088/1126-6708/1998/02/003
3331:10.1088/1126-6708/2008/10/091
2117:" dimensions (encoded by the
1724:second superstring revolution
1709:Second superstring revolution
1687:Second superstring revolution
1486:second superstring revolution
861:second superstring revolution
5739:Generalized complex manifold
5240:First superstring revolution
4915:10.1016/0550-3213(95)00158-O
4808:10.1016/0370-1573(81)90157-5
4779:10.1016/0550-3213(90)90599-9
4742:10.1016/0370-2693(94)90763-3
4637:10.1016/0550-3213(93)90475-5
4428:10.1016/0550-3213(78)90218-3
4348:10.1016/0370-2693(77)90076-4
4111:10.1016/0550-3213(94)00559-W
4064:10.1016/0550-3213(96)00308-2
4017:10.1016/0550-3213(95)00621-4
3899:10.1016/0370-2693(87)91323-2
3632:10.1016/0370-2693(78)90894-8
3529:10.1016/0550-3213(85)90602-9
3492:10.1016/0370-2693(87)91272-X
2077:mathematical theory of knots
1881:is not necessarily equal to
1719:Institute for Advanced Study
1545:first superstring revolution
970:general theory of relativity
4543:10.1103/PhysRevLett.83.4690
4357:"What is ... a Brane?"
3703:Strings, Branes and Gravity
3292:; Maldacena, Juan (2008). "
3076:Klebanov and Maldacena 2009
1988:, which can be viewed as a
1984:. It is closely related to
1230:. If a string has momentum
930:Quantum gravity and strings
6406:
5337:Non-critical string theory
4257:10.4310/ATMP.1998.V2.N2.A1
3721:10.1142/9789812799630_0006
3274:Yau and Nadis 2010, p. 150
3258:Yau and Nadis 2010, p. 149
2233:
2188:
2060:
1918:
1815:
1768:
1706:
1465:
1462:Early work on supergravity
1408:
1397:
1336:
1283:
1152:
1094:Compactification (physics)
1087:
933:
31:
5053:Zwiebach, Barton (2009).
4684:10.1142/S0217751X94001497
4266:"The Illusion of Gravity"
4128:Duke Mathematical Journal
3970:. Pearson Prentice Hall.
3966:Griffiths, David (2004).
3937:10.1016/j.aim.2012.09.027
3815:10.1142/S0217751X96002583
3675:10.1007/s00220-013-1863-2
3378:10.1007/s11005-010-0369-5
3231:Yau and Nadis 2010, p. ix
3129:Maldacena 2005, pp. 61â62
3046:Nekrasov and Schwarz 1998
2733:Yau and Nadis 2010, p. 13
2724:Yau and Nadis 2010, p. 12
2715:Yau and Nadis 2010, p. 10
2643:Yau and Nadis 2010, Ch. 6
1699:, and eleven-dimensional
1378:. Physicists often study
5873:Introduction to M-theory
5567:WessâZuminoâWitten model
5509:HananyâWitten transition
5235:History of string theory
4264:Maldacena, Juan (2005).
3444:; Schwarz, John (2007).
3425:10.1103/physrevd.55.5112
2703:Yau and Nadis 2010, p. 9
2616:Zee 2010, Parts V and VI
1937:by triangles and squares
1638:Membranes and fivebranes
1518:Ăcole Normale SupĂ©rieure
1400:History of string theory
1121:condensed matter physics
123:Non-perturbative results
34:Introduction to M-theory
5552:Vertex operator algebra
5252:String theory landscape
4992:. Basic Books. p.
4513:Physical Review Letters
4383:Moore, Gregory (2012).
4355:Moore, Gregory (2005).
3913:Advances in Mathematics
3868:from supermembranes in
3540:Noncommutative Geometry
3055:Seiberg and Witten 1999
2979:Gefter, Amanda (2014).
2960:HoĆava and Witten 1996b
2951:HoĆava and Witten 1996a
2807:Montonen and Olive 1977
1872:Noncommutative geometry
1818:Noncommutative geometry
1812:Noncommutative geometry
1771:Matrix theory (physics)
1394:History and development
1175:. Blue arrows indicate
978:nongravitational forces
5850:AdS/CFT correspondence
5605:Exceptional Lie groups
5547:Superconformal algebra
5519:Conformal field theory
5390:MontonenâOlive duality
5342:Non-linear sigma model
5084:: "Parallel Universes"
3945:Greene, Brian (2000).
3768:10.1098/rspa.1962.0124
3536:Connes, Alain (1994).
3290:Jafferis, Daniel Louis
2846:Hull and Townsend 1995
2751:van Nieuwenhuizen 1981
2476:AdS/CFT correspondence
2248:
2080:
2021:
1938:
1921:AdS/CFT correspondence
1910:AdS/CFT correspondence
1890:In a paper from 1998,
1749:
1704:
1596:MontonenâOlive duality
1489:
1256:type IIA string theory
1200:type IIB string theory
1180:
1107:
953:
896:AdS/CFT correspondence
239:Conformal field theory
156:AdS/CFT correspondence
5845:Holographic principle
5812:Type IIB supergravity
5807:Type IIA supergravity
5659:-form electrodynamics
5278:Bosonic string theory
5158:Superstringtheory.com
5030:Zee, Anthony (2010).
4815:Wald, Robert (1984).
4473:10.1007/s002200050490
4402:Nahm, Walter (1978).
3120:Zwiebach 2009, p. 552
3111:Maldacena 2005, p. 61
3099:Maldacena 2005, p. 60
2673:Zwiebach 2009, p. 376
2586:Zwiebach 2009, p. 324
2488:Dimensional reduction
2244:A cross section of a
2243:
2070:
2003:
1928:
1694:
1475:
1384:electromagnetic field
1261:In general, the term
1170:
1101:
1041:, and two flavors of
996:theoretical framework
947:
282:Holographic principle
259:Twistor string theory
5764:HoĆavaâWitten theory
5711:HyperkÀhler manifold
5399:Particles and fields
5347:Tachyon condensation
5332:Matrix string theory
5123:The Elegant Universe
5104:The Elegant Universe
5095:"Parallel Universes"
4984:Woit, Peter (2006).
3222:Candelas et al. 1985
2930:Duff 1998, pp. 67â68
2323:University of Oxford
2292:Compactification on
2261:String phenomenology
2236:String phenomenology
2179:hyperkÀhler geometry
2071:The six-dimensional
2006:anti-de Sitter space
1974:anti-de Sitter space
1943:quantum field theory
1697:superstring theories
1448:unified field theory
1366:-brane sweeps out a
1189:Type I string theory
1084:Number of dimensions
1000:point-like particles
234:Theory of everything
5802:Type I supergravity
5706:CalabiâYau manifold
5701:Ricci-flat manifold
5680:KaluzaâKlein theory
5421:RamondâRamond field
5327:String field theory
5120:, adapted from his
4907:1995NuPhB.443...85W
4852:1989CMaPh.121..351W
4800:1981PhR....68..189V
4771:1990NuPhB.343..167S
4734:1994PhLB..329..217S
4676:1994IJMPA...9.3707S
4629:1993NuPhB.404..109S
4582:1999JHEP...09..032S
4535:1999PhRvL..83.4690R
4465:1998CMaPh.198..689N
4420:1978NuPhB.135..149N
4340:1977PhLB...72..117M
4285:2005SciAm.293e..56M
4273:Scientific American
4249:1998AdTMP...2..231M
4188:2009PhT....62a..28K
4103:1995NuPhB.438..109H
4056:1996NuPhB.475...94H
4009:1996NuPhB.460..506H
3891:1987PhLB..191...70D
3844:1998SciAm.278b..64D
3832:Scientific American
3807:1996IJMPA..11.5623D
3760:1962RSPSA.268...57D
3667:2014CMaPh.325..367D
3624:1978PhLB...76..409C
3583:1998JHEP...02..003C
3521:1985NuPhB.258...46C
3484:1987PhLB..189...75B
3417:1997PhRvD..55.5112B
3370:2010LMaPh..91..167A
3323:2008JHEP...10..091A
3195:Aharony et al. 2008
2903:Duff 1998, pp 66â67
2634:Zwiebach 2009, p. 8
2625:Zwiebach 2009, p. 9
2274:CalabiâYau manifold
2246:CalabiâYau manifold
2219:ChernâSimons theory
2008:is like a stack of
1970:Einstein's equation
1866:commutative algebra
1530:weak nuclear forces
1444:KaluzaâKlein theory
1411:KaluzaâKlein theory
1405:KaluzaâKlein theory
1354:, these are called
1304:supergravity theory
1248:and winding number
1236:and winding number
919:high-energy physics
272:KaluzaâKlein theory
208:Monstrous moonshine
89:Perturbative theory
58:Fundamental objects
6390:1995 introductions
5769:K-theory (physics)
5646:ADE classification
5283:Superstring theory
4860:10.1007/BF01217730
4819:General Relativity
4496:. Westview Press.
4364:Notices of the AMS
3544:. Academic Press.
2380:Heterotic M-theory
2359:manifolds are not
2249:
2139:AGT correspondence
2081:
2022:
2004:Three-dimensional
1982:Euclidean geometry
1939:
1896:Michael R. Douglas
1775:In mathematics, a
1737:Origin of the term
1705:
1490:
1311:superstring theory
1181:
1136:eleven-dimensional
1108:
954:
907:fundamental forces
855:conference at the
845:superstring theory
6372:
6371:
6154:van Nieuwenhuizen
5690:Why 10 dimensions
5595:ChernâSimons form
5562:KacâMoody algebra
5542:Conformal algebra
5537:Conformal anomaly
5431:Magnetic monopole
5426:KalbâRamond field
5268:NambuâGoto action
5064:978-0-521-88032-9
5045:978-0-691-14034-6
5022:978-0-465-02023-2
4885:Nuclear Physics B
4830:978-0-226-87033-5
4759:Nuclear Physics B
4712:Physics Letters B
4660:(21): 3707â3750.
4607:Nuclear Physics B
4519:(23): 4690â4693.
4503:978-0-201-50397-5
4408:Nuclear Physics B
4328:Physics Letters B
4197:10.1063/1.3074260
4081:Nuclear Physics B
4034:Nuclear Physics B
3987:Nuclear Physics B
3977:978-0-13-111892-8
3958:978-0-9650888-0-0
3879:Nuclear Physics B
3861:"Superstrings in
3730:978-981-02-4774-4
3612:Physics Letters B
3551:978-0-12-185860-5
3509:Nuclear Physics B
3472:Physics Letters B
3455:978-0-521-86069-7
3395:Physical Review D
3028:Connes 1994, p. 1
3007:Banks et al. 1997
2577:Duff 1996, sec. 1
2361:complex manifolds
2155:Khovanov homology
1789:BFSS matrix model
1765:BFSS matrix model
1664:Andrew Strominger
1592:coupling constant
1417:Hermann Minkowski
1382:analogous to the
1111:four-dimensional
982:quantum mechanics
974:classical physics
834:
833:
565:van Nieuwenhuizen
16:(Redirected from
6397:
5882:String theorists
5822:Lie superalgebra
5774:Twisted K-theory
5732:Spin(7)-manifold
5685:Compactification
5527:Virasoro algebra
5310:Heterotic string
5204:
5197:
5190:
5181:
5068:
5049:
5037:
5026:
5007:
4991:
4980:
4962:
4947:Quantum Topology
4941:
4939:
4926:
4900:
4879:
4834:
4822:
4811:
4782:
4753:
4727:
4709:
4695:
4669:
4648:
4622:
4601:
4575:
4554:
4528:
4507:
4495:
4484:
4458:
4440:
4431:
4398:
4396:
4394:
4389:
4379:
4377:
4375:
4361:
4351:
4318:
4312:
4304:
4270:
4260:
4242:
4224:
4215:
4209:
4201:
4199:
4171:
4161:
4143:
4122:
4096:
4075:
4049:
4028:
4002:
3981:
3962:
3951:. Random House.
3941:
3939:
3929:
3902:
3874:
3867:
3855:
3826:
3800:
3779:
3742:
3714:
3697:
3695:
3694:
3685:. Archived from
3660:
3635:
3602:
3576:
3555:
3543:
3532:
3503:
3469:
3459:
3440:Becker, Katrin;
3436:
3410:
3401:(8): 5112â5128.
3389:
3363:
3342:
3316:
3298:
3275:
3272:
3259:
3256:
3250:
3247:
3241:
3238:
3232:
3229:
3223:
3220:
3214:
3211:
3205:
3202:
3196:
3193:
3184:
3181:
3175:
3172:
3166:
3163:
3157:
3154:
3148:
3145:
3139:
3136:
3130:
3127:
3121:
3118:
3112:
3109:
3100:
3097:
3086:
3083:
3077:
3074:
3068:
3065:
3056:
3053:
3047:
3044:
3038:
3035:
3029:
3026:
3020:
3017:
3008:
3005:
2996:
2994:
2990:978-0-345-531438
2983:. Random House.
2976:
2970:
2969:Duff 1998, p. 68
2967:
2961:
2958:
2952:
2949:
2940:
2937:
2931:
2928:
2922:
2919:
2913:
2910:
2904:
2901:
2895:
2892:
2886:
2885:Duff et al. 1987
2883:
2877:
2874:
2868:
2865:
2859:
2858:Duff 1998, p. 67
2856:
2847:
2844:
2838:
2835:
2829:
2826:
2820:
2819:Duff 1998, p. 66
2817:
2808:
2805:
2799:
2796:
2790:
2789:Duff 1998, p. 65
2787:
2770:
2767:
2761:
2758:
2752:
2749:
2743:
2740:
2734:
2731:
2725:
2722:
2716:
2713:
2704:
2701:
2695:
2692:
2686:
2685:Duff 1998, p. 64
2683:
2674:
2671:
2665:
2662:
2653:
2650:
2644:
2641:
2635:
2632:
2626:
2623:
2617:
2614:
2608:
2605:
2599:
2596:
2587:
2584:
2578:
2575:
2556:
2550:
2544:
2541:
2535:
2532:
2526:
2523:
2517:
2506:
2500:
2497:
2491:
2485:
2479:
2472:
2466:
2463:
2457:
2450:
2444:
2437:
2431:
2428:
2407:cosmic inflation
2392:
2375:
2365:complex analysis
2358:
2349:
2333:
2314:
2300:
2207:
2159:Mikhail Khovanov
2147:pure mathematics
2124:
2112:
2098:
2010:hyperbolic disks
1986:hyperbolic space
1935:hyperbolic plane
1886:
1880:
1859:
1845:
1839:
1805:Leonard Susskind
1617:
1610:
1604:
1575:
1438:
1429:electromagnetism
1373:
1365:
1359:
1353:
1269:physical systems
1253:
1247:
1241:
1235:
1221:
1214:
1197:
1142:Compactification
1128:extra dimensions
1104:compactification
1090:Extra dimensions
1071:
1053:
1043:heterotic string
1006:are replaced by
1004:particle physics
915:extra dimensions
826:
819:
812:
228:Related concepts
53:
39:
21:
6405:
6404:
6400:
6399:
6398:
6396:
6395:
6394:
6375:
6374:
6373:
6368:
5877:
5854:
5831:
5778:
5726:
5696:KĂ€hler manifold
5663:
5640:
5633:
5626:
5619:
5612:
5571:
5532:Mirror symmetry
5513:
5499:Brane cosmology
5445:
5394:
5361:
5317:N=2 superstring
5303:Type IIB string
5298:Type IIA string
5273:Polyakov action
5256:
5213:
5208:
5154:
5137:
5126:book (original
5076:
5071:
5065:
5052:
5046:
5029:
5023:
5015:. Basic Books.
5010:
5004:
4983:
4944:
4929:
4882:
4837:
4831:
4814:
4788:Physics Reports
4785:
4756:
4700:
4698:
4651:
4604:
4557:
4510:
4504:
4487:
4436:
4434:
4401:
4392:
4390:
4387:
4382:
4373:
4371:
4359:
4354:
4321:
4305:
4268:
4263:
4220:
4218:
4202:
4169:
4164:
4125:
4078:
4031:
3984:
3978:
3965:
3959:
3944:
3905:
3869:
3862:
3858:
3829:
3791:(32): 6523â41.
3782:
3754:(1332): 57â67.
3745:
3731:
3700:
3692:
3690:
3638:
3605:
3558:
3552:
3535:
3506:
3467:
3462:
3456:
3442:Becker, Melanie
3439:
3392:
3345:
3293:
3287:
3283:
3278:
3273:
3262:
3257:
3253:
3248:
3244:
3239:
3235:
3230:
3226:
3221:
3217:
3212:
3208:
3203:
3199:
3194:
3187:
3182:
3178:
3173:
3169:
3164:
3160:
3155:
3151:
3146:
3142:
3137:
3133:
3128:
3124:
3119:
3115:
3110:
3103:
3098:
3089:
3084:
3080:
3075:
3071:
3066:
3059:
3054:
3050:
3045:
3041:
3036:
3032:
3027:
3023:
3018:
3011:
3006:
2999:
2991:
2978:
2977:
2973:
2968:
2964:
2959:
2955:
2950:
2943:
2938:
2934:
2929:
2925:
2920:
2916:
2911:
2907:
2902:
2898:
2894:Strominger 1990
2893:
2889:
2884:
2880:
2875:
2871:
2866:
2862:
2857:
2850:
2845:
2841:
2836:
2832:
2827:
2823:
2818:
2811:
2806:
2802:
2797:
2793:
2788:
2773:
2768:
2764:
2759:
2755:
2750:
2746:
2742:Wald 1984, p. 3
2741:
2737:
2732:
2728:
2723:
2719:
2714:
2707:
2702:
2698:
2693:
2689:
2684:
2677:
2672:
2668:
2663:
2656:
2651:
2647:
2642:
2638:
2633:
2629:
2624:
2620:
2615:
2611:
2607:Wald 1984, p. 4
2606:
2602:
2597:
2590:
2585:
2581:
2576:
2569:
2565:
2560:
2559:
2551:
2547:
2542:
2538:
2533:
2529:
2524:
2520:
2507:
2503:
2498:
2494:
2486:
2482:
2473:
2469:
2464:
2460:
2451:
2447:
2438:
2434:
2429:
2425:
2420:
2415:
2403:brane cosmology
2391:
2385:
2382:
2374:
2368:
2357:
2351:
2348:
2342:
2332:
2326:
2313:
2307:
2303:
2299:
2293:
2238:
2232:
2227:
2202:
2196:
2193:
2187:
2157:. Developed by
2118:
2111:
2105:
2093:
2087:
2065:
2059:
2051:Minkowski space
2014:quantum gravity
1966:vacuum solution
1923:
1917:
1912:
1882:
1876:
1862:commutative law
1851:
1841:
1835:
1832:Euclidean plane
1824:
1816:Main articles:
1814:
1801:Stephen Shenker
1773:
1767:
1762:
1739:
1711:
1689:
1640:
1612:
1606:
1599:
1570:
1558:
1470:
1464:
1436:
1413:
1407:
1402:
1396:
1372: + 1)
1367:
1361:
1355:
1349:
1341:
1335:
1288:
1282:
1249:
1243:
1237:
1231:
1216:
1210:
1192:
1165:
1153:Main articles:
1151:
1132:ten-dimensional
1096:
1088:Main articles:
1086:
1069:
1062:
1055:
1046:
1010:objects called
1008:one-dimensional
966:Albert Einstein
958:quantum gravity
942:
936:Quantum gravity
934:Main articles:
932:
927:
839:is a theory in
830:
785:
784:
295:
287:
286:
244:Quantum gravity
229:
203:Mirror symmetry
37:
28:
23:
22:
15:
12:
11:
5:
6403:
6401:
6393:
6392:
6387:
6377:
6376:
6370:
6369:
6367:
6366:
6361:
6356:
6351:
6346:
6341:
6336:
6331:
6326:
6321:
6316:
6311:
6306:
6301:
6296:
6291:
6286:
6281:
6276:
6271:
6266:
6261:
6256:
6251:
6246:
6241:
6236:
6231:
6226:
6221:
6216:
6211:
6206:
6204:Randjbar-Daemi
6201:
6196:
6191:
6186:
6181:
6176:
6171:
6166:
6161:
6156:
6151:
6146:
6141:
6136:
6131:
6126:
6121:
6116:
6111:
6106:
6101:
6096:
6091:
6086:
6081:
6076:
6071:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
6026:
6021:
6016:
6011:
6006:
6001:
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5961:
5956:
5951:
5946:
5941:
5936:
5931:
5926:
5921:
5916:
5911:
5906:
5901:
5896:
5891:
5885:
5883:
5879:
5878:
5876:
5875:
5870:
5864:
5862:
5856:
5855:
5853:
5852:
5847:
5841:
5839:
5833:
5832:
5830:
5829:
5827:Lie supergroup
5824:
5819:
5814:
5809:
5804:
5799:
5794:
5788:
5786:
5780:
5779:
5777:
5776:
5771:
5766:
5761:
5756:
5751:
5746:
5741:
5736:
5735:
5734:
5729:
5724:
5720:
5719:
5718:
5708:
5698:
5693:
5687:
5682:
5677:
5671:
5669:
5665:
5664:
5662:
5661:
5653:
5648:
5643:
5638:
5631:
5624:
5617:
5610:
5602:
5597:
5592:
5587:
5581:
5579:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5534:
5529:
5523:
5521:
5515:
5514:
5512:
5511:
5506:
5504:Quiver diagram
5501:
5496:
5491:
5486:
5481:
5476:
5471:
5466:
5461:
5455:
5453:
5447:
5446:
5444:
5443:
5438:
5433:
5428:
5423:
5418:
5413:
5408:
5402:
5400:
5396:
5395:
5393:
5392:
5387:
5382:
5377:
5371:
5369:
5367:String duality
5363:
5362:
5360:
5359:
5354:
5349:
5344:
5339:
5334:
5329:
5324:
5319:
5314:
5313:
5312:
5307:
5306:
5305:
5300:
5293:Type II string
5290:
5280:
5275:
5270:
5264:
5262:
5258:
5257:
5255:
5254:
5249:
5248:
5247:
5242:
5232:
5230:Cosmic strings
5227:
5221:
5219:
5215:
5214:
5209:
5207:
5206:
5199:
5192:
5184:
5178:
5177:
5171:
5164:Not Even Wrong
5161:
5153:
5152:External links
5150:
5149:
5148:
5143:
5136:
5133:
5132:
5131:
5106:] â 2003
5102:PBS.org-NOVA:
5098:
5075:
5074:Popularization
5072:
5070:
5069:
5063:
5050:
5044:
5027:
5021:
5008:
5002:
4981:
4942:
4927:
4898:hep-th/9503124
4880:
4846:(3): 351â399.
4835:
4829:
4812:
4794:(4): 189â398.
4783:
4765:(1): 167â184.
4754:
4725:hep-th/9402032
4718:(2): 217â221.
4696:
4667:hep-th/9402002
4649:
4620:hep-th/9207053
4613:(1): 109â126.
4602:
4573:hep-th/9908142
4555:
4526:hep-th/9906064
4508:
4502:
4485:
4456:hep-th/9802068
4449:(3): 689â703.
4432:
4414:(1): 149â166.
4399:
4380:
4352:
4334:(1): 117â120.
4319:
4261:
4240:hep-th/9711200
4233:(2): 231â252.
4216:
4162:
4134:(3): 359â426.
4123:
4094:hep-th/9410167
4087:(1): 109â137.
4076:
4047:hep-th/9603142
4029:
4000:hep-th/9510209
3993:(3): 506â524.
3982:
3976:
3963:
3957:
3942:
3903:
3856:
3827:
3798:hep-th/9608117
3780:
3743:
3729:
3712:hep-th/0003175
3698:
3651:(2): 367â419.
3636:
3618:(4): 409â412.
3603:
3574:hep-th/9711162
3556:
3550:
3533:
3504:
3460:
3454:
3437:
3408:hep-th/9610043
3390:
3354:(2): 167â197.
3343:
3284:
3282:
3279:
3277:
3276:
3260:
3251:
3242:
3233:
3224:
3215:
3206:
3197:
3185:
3176:
3167:
3158:
3149:
3140:
3131:
3122:
3113:
3101:
3087:
3078:
3069:
3067:Maldacena 1998
3057:
3048:
3039:
3030:
3021:
3009:
2997:
2989:
2971:
2962:
2953:
2941:
2932:
2923:
2914:
2905:
2896:
2887:
2878:
2869:
2860:
2848:
2839:
2830:
2821:
2809:
2800:
2791:
2771:
2762:
2753:
2744:
2735:
2726:
2717:
2705:
2696:
2687:
2675:
2666:
2654:
2645:
2636:
2627:
2618:
2609:
2600:
2588:
2579:
2566:
2564:
2561:
2558:
2557:
2545:
2536:
2527:
2518:
2501:
2492:
2480:
2467:
2458:
2454:renormalizable
2445:
2432:
2422:
2421:
2419:
2416:
2414:
2411:
2389:
2381:
2378:
2372:
2355:
2346:
2330:
2311:
2302:
2297:
2290:
2282:Shing-Tung Yau
2278:Eugenio Calabi
2269:supersymmetric
2234:Main article:
2231:
2228:
2226:
2223:
2200:
2189:Main article:
2186:
2183:
2175:Andrew Neitzke
2167:Davide Gaiotto
2109:
2091:
2061:Main article:
2058:
2055:
1958:Juan Maldacena
1956:. Proposed by
1947:quasiparticles
1919:Main article:
1916:
1913:
1911:
1908:
1900:Albert Schwarz
1813:
1810:
1797:Willy Fischler
1769:Main article:
1766:
1763:
1761:
1758:
1738:
1735:
1707:Main article:
1688:
1685:
1645:supermembranes
1639:
1636:
1598:, states that
1557:
1554:
1506:EugĂšne Cremmer
1476:In the 1980s,
1466:Main article:
1463:
1460:
1456:nuclear forces
1425:Theodor Kaluza
1409:Main article:
1406:
1403:
1398:Main article:
1395:
1392:
1337:Main article:
1334:
1331:
1284:Main article:
1281:
1278:
1228:winding number
1155:String duality
1150:
1147:
1102:An example of
1085:
1082:
1067:
1060:
931:
928:
926:
923:
905:of all of the
903:unified theory
832:
831:
829:
828:
821:
814:
806:
803:
802:
801:
800:
795:
787:
786:
783:
782:
777:
772:
767:
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:
615:Randjbar-Daemi
612:
607:
602:
597:
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:
296:
293:
292:
289:
288:
285:
284:
279:
274:
269:
261:
256:
251:
246:
241:
236:
230:
227:
226:
223:
222:
221:
220:
215:
213:Vertex algebra
210:
205:
200:
192:
191:
187:
186:
185:
184:
179:
174:
166:
165:
161:
160:
159:
158:
153:
148:
143:
138:
133:
125:
124:
120:
119:
118:
117:
99:
91:
90:
86:
85:
84:
83:
78:
73:
68:
60:
59:
55:
54:
46:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6402:
6391:
6388:
6386:
6385:String theory
6383:
6382:
6380:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6349:Zamolodchikov
6347:
6345:
6344:Zamolodchikov
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6315:
6312:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6290:
6287:
6285:
6282:
6280:
6277:
6275:
6272:
6270:
6267:
6265:
6262:
6260:
6257:
6255:
6252:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6180:
6177:
6175:
6172:
6170:
6167:
6165:
6162:
6160:
6157:
6155:
6152:
6150:
6147:
6145:
6142:
6140:
6137:
6135:
6132:
6130:
6127:
6125:
6122:
6120:
6117:
6115:
6112:
6110:
6107:
6105:
6102:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
6000:
5997:
5995:
5992:
5990:
5987:
5985:
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5915:
5912:
5910:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5886:
5884:
5880:
5874:
5871:
5869:
5868:Matrix theory
5866:
5865:
5863:
5861:
5857:
5851:
5848:
5846:
5843:
5842:
5840:
5838:
5834:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5789:
5787:
5785:
5784:Supersymmetry
5781:
5775:
5772:
5770:
5767:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5740:
5737:
5733:
5730:
5728:
5721:
5717:
5714:
5713:
5712:
5709:
5707:
5704:
5703:
5702:
5699:
5697:
5694:
5691:
5688:
5686:
5683:
5681:
5678:
5676:
5673:
5672:
5670:
5666:
5660:
5658:
5654:
5652:
5649:
5647:
5644:
5641:
5634:
5627:
5620:
5613:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5582:
5580:
5578:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5524:
5522:
5520:
5516:
5510:
5507:
5505:
5502:
5500:
5497:
5495:
5492:
5490:
5487:
5485:
5482:
5480:
5477:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5456:
5454:
5452:
5448:
5442:
5439:
5437:
5436:Dual graviton
5434:
5432:
5429:
5427:
5424:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5403:
5401:
5397:
5391:
5388:
5386:
5383:
5381:
5378:
5376:
5373:
5372:
5370:
5368:
5364:
5358:
5355:
5353:
5352:RNS formalism
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5311:
5308:
5304:
5301:
5299:
5296:
5295:
5294:
5291:
5289:
5288:Type I string
5286:
5285:
5284:
5281:
5279:
5276:
5274:
5271:
5269:
5266:
5265:
5263:
5259:
5253:
5250:
5246:
5243:
5241:
5238:
5237:
5236:
5233:
5231:
5228:
5226:
5223:
5222:
5220:
5216:
5212:
5211:String theory
5205:
5200:
5198:
5193:
5191:
5186:
5185:
5182:
5175:
5172:
5169:
5165:
5162:
5159:
5156:
5155:
5151:
5147:
5144:
5142:
5139:
5138:
5134:
5129:
5125:
5124:
5119:
5115:
5114:
5109:
5105:
5101:
5099:
5096:
5092:
5091:
5085:
5083:
5078:
5077:
5073:
5066:
5060:
5056:
5051:
5047:
5041:
5036:
5035:
5028:
5024:
5018:
5014:
5009:
5005:
5003:0-465-09275-6
4999:
4995:
4990:
4989:
4982:
4978:
4974:
4970:
4969:10.4171/QT/26
4966:
4961:
4956:
4952:
4948:
4943:
4938:
4933:
4928:
4924:
4920:
4916:
4912:
4908:
4904:
4899:
4894:
4891:(1): 85â126.
4890:
4886:
4881:
4877:
4873:
4869:
4865:
4861:
4857:
4853:
4849:
4845:
4841:
4836:
4832:
4826:
4821:
4820:
4813:
4809:
4805:
4801:
4797:
4793:
4789:
4784:
4780:
4776:
4772:
4768:
4764:
4760:
4755:
4751:
4747:
4743:
4739:
4735:
4731:
4726:
4721:
4717:
4713:
4707:
4703:
4697:
4693:
4689:
4685:
4681:
4677:
4673:
4668:
4663:
4659:
4655:
4650:
4646:
4642:
4638:
4634:
4630:
4626:
4621:
4616:
4612:
4608:
4603:
4599:
4595:
4591:
4587:
4583:
4579:
4574:
4569:
4565:
4561:
4556:
4552:
4548:
4544:
4540:
4536:
4532:
4527:
4522:
4518:
4514:
4509:
4505:
4499:
4494:
4493:
4486:
4482:
4478:
4474:
4470:
4466:
4462:
4457:
4452:
4448:
4444:
4439:
4433:
4429:
4425:
4421:
4417:
4413:
4409:
4405:
4400:
4386:
4381:
4369:
4365:
4358:
4353:
4349:
4345:
4341:
4337:
4333:
4329:
4325:
4320:
4316:
4310:
4302:
4298:
4294:
4290:
4286:
4282:
4278:
4274:
4267:
4262:
4258:
4254:
4250:
4246:
4241:
4236:
4232:
4228:
4223:
4217:
4213:
4207:
4198:
4193:
4189:
4185:
4181:
4177:
4176:
4175:Physics Today
4168:
4163:
4159:
4155:
4151:
4147:
4142:
4137:
4133:
4129:
4124:
4120:
4116:
4112:
4108:
4104:
4100:
4095:
4090:
4086:
4082:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4048:
4043:
4040:(1): 94â114.
4039:
4035:
4030:
4026:
4022:
4018:
4014:
4010:
4006:
4001:
3996:
3992:
3988:
3983:
3979:
3973:
3969:
3964:
3960:
3954:
3950:
3949:
3943:
3938:
3933:
3928:
3923:
3919:
3915:
3914:
3909:
3904:
3900:
3896:
3892:
3888:
3884:
3880:
3876:
3872:
3865:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3828:
3824:
3820:
3816:
3812:
3808:
3804:
3799:
3794:
3790:
3786:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3744:
3740:
3736:
3732:
3726:
3722:
3718:
3713:
3708:
3704:
3699:
3689:on 2020-09-18
3688:
3684:
3680:
3676:
3672:
3668:
3664:
3659:
3654:
3650:
3646:
3642:
3637:
3633:
3629:
3625:
3621:
3617:
3613:
3609:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3575:
3570:
3566:
3562:
3557:
3553:
3547:
3542:
3541:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3505:
3501:
3497:
3493:
3489:
3485:
3481:
3477:
3473:
3466:
3461:
3457:
3451:
3447:
3443:
3438:
3434:
3430:
3426:
3422:
3418:
3414:
3409:
3404:
3400:
3396:
3391:
3387:
3383:
3379:
3375:
3371:
3367:
3362:
3357:
3353:
3349:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3315:
3310:
3306:
3302:
3296:
3291:
3286:
3285:
3280:
3271:
3269:
3267:
3265:
3261:
3255:
3252:
3246:
3243:
3237:
3234:
3228:
3225:
3219:
3216:
3210:
3207:
3201:
3198:
3192:
3190:
3186:
3180:
3177:
3174:Khovanov 2000
3171:
3168:
3162:
3159:
3153:
3150:
3144:
3141:
3135:
3132:
3126:
3123:
3117:
3114:
3108:
3106:
3102:
3096:
3094:
3092:
3088:
3082:
3079:
3073:
3070:
3064:
3062:
3058:
3052:
3049:
3043:
3040:
3034:
3031:
3025:
3022:
3016:
3014:
3010:
3004:
3002:
2998:
2992:
2986:
2982:
2975:
2972:
2966:
2963:
2957:
2954:
2948:
2946:
2942:
2936:
2933:
2927:
2924:
2918:
2915:
2909:
2906:
2900:
2897:
2891:
2888:
2882:
2879:
2873:
2870:
2864:
2861:
2855:
2853:
2849:
2843:
2840:
2834:
2831:
2825:
2822:
2816:
2814:
2810:
2804:
2801:
2795:
2792:
2786:
2784:
2782:
2780:
2778:
2776:
2772:
2766:
2763:
2757:
2754:
2748:
2745:
2739:
2736:
2730:
2727:
2721:
2718:
2712:
2710:
2706:
2700:
2697:
2691:
2688:
2682:
2680:
2676:
2670:
2667:
2661:
2659:
2655:
2649:
2646:
2640:
2637:
2631:
2628:
2622:
2619:
2613:
2610:
2604:
2601:
2595:
2593:
2589:
2583:
2580:
2574:
2572:
2568:
2562:
2554:
2549:
2546:
2540:
2537:
2531:
2528:
2522:
2519:
2515:
2511:
2505:
2502:
2496:
2493:
2489:
2484:
2481:
2477:
2471:
2468:
2462:
2459:
2455:
2449:
2446:
2442:
2436:
2433:
2427:
2424:
2417:
2412:
2410:
2408:
2404:
2399:
2397:
2388:
2379:
2377:
2371:
2366:
2362:
2354:
2345:
2340:
2335:
2329:
2324:
2320:
2319:Dominic Joyce
2316:
2310:
2296:
2291:
2289:
2285:
2283:
2279:
2275:
2270:
2264:
2262:
2258:
2257:Phenomenology
2254:
2247:
2242:
2237:
2229:
2225:Phenomenology
2224:
2222:
2220:
2214:
2211:
2206:
2199:
2192:
2184:
2182:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
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2143:
2140:
2136:
2132:
2127:
2123:
2122:
2116:
2108:
2102:
2097:
2090:
2086:
2085:product space
2078:
2074:
2069:
2064:
2056:
2054:
2052:
2046:
2042:
2040:
2035:
2034:cross section
2032:in which any
2031:
2027:
2019:
2015:
2011:
2007:
2002:
1998:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1962:
1959:
1955:
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1948:
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1932:
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1901:
1897:
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1863:
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1854:
1849:
1844:
1838:
1833:
1829:
1823:
1819:
1811:
1809:
1806:
1802:
1798:
1794:
1790:
1785:
1782:
1778:
1772:
1764:
1760:Matrix theory
1759:
1757:
1754:
1748:
1743:
1736:
1734:
1732:
1727:
1725:
1720:
1716:
1710:
1702:
1698:
1693:
1686:
1684:
1680:
1678:
1674:
1670:
1665:
1660:
1658:
1652:
1650:
1646:
1637:
1635:
1632:
1631:Paul Townsend
1628:
1624:
1619:
1616:
1609:
1602:
1597:
1593:
1589:
1588:atomic nuclei
1586:that make up
1585:
1581:
1577:
1573:
1567:
1562:
1555:
1553:
1551:
1546:
1541:
1539:
1538:Edward Witten
1535:
1531:
1527:
1521:
1519:
1515:
1511:
1510:Bernard Julia
1507:
1503:
1497:
1495:
1487:
1483:
1479:
1478:Edward Witten
1474:
1469:
1461:
1459:
1457:
1453:
1449:
1445:
1440:
1434:
1430:
1426:
1421:
1418:
1412:
1404:
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1393:
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1387:
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1371:
1364:
1358:
1352:
1346:
1340:
1332:
1330:
1328:
1324:
1320:
1316:
1312:
1307:
1305:
1301:
1297:
1293:
1292:supersymmetry
1287:
1286:Supersymmetry
1280:Supersymmetry
1279:
1277:
1275:
1270:
1266:
1265:
1259:
1257:
1252:
1246:
1240:
1234:
1229:
1225:
1220:
1213:
1208:
1203:
1201:
1195:
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1186:
1178:
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1129:
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1116:
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1095:
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1066:
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1044:
1040:
1036:
1032:
1027:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
992:String theory
989:
987:
983:
979:
975:
971:
967:
963:
959:
951:
946:
941:
940:String theory
937:
929:
924:
922:
921:experiments.
920:
916:
912:
911:compactifying
908:
904:
899:
897:
893:
892:matrix theory
889:
885:
880:
878:
874:
870:
866:
862:
858:
854:
853:string theory
850:
849:Edward Witten
846:
842:
838:
827:
822:
820:
815:
813:
808:
807:
805:
804:
799:
796:
794:
791:
790:
789:
788:
781:
778:
776:
773:
771:
768:
766:
765:Zamolodchikov
763:
761:
760:Zamolodchikov
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
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723:
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631:
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626:
623:
621:
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616:
613:
611:
608:
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603:
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596:
593:
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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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478:
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451:
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431:
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348:
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321:
318:
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301:
298:
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291:
290:
283:
280:
278:
275:
273:
270:
268:
266:
262:
260:
257:
255:
252:
250:
249:Supersymmetry
247:
245:
242:
240:
237:
235:
232:
231:
225:
224:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
195:
194:
193:
188:
183:
180:
178:
175:
173:
172:Phenomenology
170:
169:
168:
167:
164:Phenomenology
162:
157:
154:
152:
149:
147:
144:
142:
139:
137:
134:
132:
129:
128:
127:
126:
121:
115:
111:
107:
103:
100:
98:
95:
94:
93:
92:
87:
82:
79:
77:
74:
72:
71:Cosmic string
69:
67:
64:
63:
62:
61:
56:
52:
48:
47:
44:
43:String theory
40:
35:
30:
19:
5894:Arkani-Hamed
5859:
5792:Supergravity
5759:Moduli space
5656:
5651:Dirac string
5577:Gauge theory
5557:Loop algebra
5494:Black string
5357:GS formalism
5121:
5118:Brian Greene
5112:
5103:
5089:
5081:
5054:
5033:
5012:
4987:
4953:(1): 1â137.
4950:
4946:
4888:
4884:
4843:
4839:
4818:
4791:
4787:
4762:
4758:
4715:
4711:
4705:
4701:
4657:
4653:
4610:
4606:
4563:
4559:
4516:
4512:
4491:
4446:
4442:
4437:
4411:
4407:
4391:. Retrieved
4372:. Retrieved
4367:
4363:
4331:
4327:
4309:cite journal
4279:(5): 56â63.
4276:
4272:
4230:
4226:
4221:
4206:cite journal
4179:
4173:
4141:math/9908171
4131:
4127:
4084:
4080:
4037:
4033:
3990:
3986:
3967:
3947:
3917:
3911:
3885:(1): 70â74.
3882:
3878:
3870:
3863:
3835:
3831:
3788:
3784:
3751:
3747:
3702:
3691:. Retrieved
3687:the original
3648:
3644:
3615:
3611:
3608:Scherk, Joël
3564:
3560:
3539:
3512:
3508:
3478:(1): 75â78.
3475:
3471:
3445:
3398:
3394:
3351:
3347:
3304:
3300:
3294:
3281:Bibliography
3254:
3245:
3236:
3227:
3218:
3209:
3200:
3179:
3170:
3161:
3152:
3143:
3134:
3125:
3116:
3081:
3072:
3051:
3042:
3033:
3024:
2980:
2974:
2965:
2956:
2935:
2926:
2917:
2908:
2899:
2890:
2881:
2872:
2863:
2842:
2833:
2824:
2803:
2794:
2765:
2756:
2747:
2738:
2729:
2720:
2699:
2690:
2669:
2648:
2639:
2630:
2621:
2612:
2603:
2582:
2548:
2539:
2530:
2521:
2504:
2495:
2483:
2470:
2461:
2448:
2435:
2426:
2400:
2386:
2383:
2369:
2352:
2343:
2336:
2327:
2308:
2304:
2294:
2286:
2265:
2250:
2215:
2204:
2197:
2194:
2144:
2128:
2120:
2106:
2101:(2,0)-theory
2095:
2088:
2082:
2073:(2,0)-theory
2047:
2043:
2023:
1994:tessellation
1963:
1951:
1940:
1931:tessellation
1892:Alain Connes
1889:
1883:
1877:
1870:
1856:
1852:
1842:
1836:
1825:
1791:proposed by
1786:
1781:matrix model
1774:
1752:
1750:
1745:
1740:
1728:
1712:
1701:supergravity
1681:
1661:
1657:Michael Duff
1653:
1641:
1620:
1614:
1607:
1600:
1571:
1563:
1559:
1542:
1522:
1498:
1491:
1482:supergravity
1468:Supergravity
1441:
1422:
1414:
1388:
1375:
1369:
1362:
1356:
1350:
1342:
1308:
1289:
1273:
1262:
1260:
1250:
1244:
1238:
1232:
1218:
1211:
1204:
1193:
1182:
1140:
1135:
1131:
1125:
1117:
1109:
1064:
1057:
1048:
1028:
990:
964:is based on
955:
900:
881:
873:field theory
836:
835:
305:Arkani-Hamed
264:
254:Supergravity
145:
29:
6254:Silverstein
5754:Orientifold
5489:Black holes
5484:Black brane
5441:Dual photon
3920:: 239â403.
3838:(2): 64â9.
3705:: 545â612.
3307:(10): 091.
3204:Witten 1989
3165:Witten 2012
3156:Witten 2009
3037:Connes 1994
2921:Witten 1995
2553:Brane world
2210:ABJM theory
2163:knot theory
1828:coordinates
1731:Petr HoĆava
1566:David Olive
1514:Joël Scherk
1502:Werner Nahm
1433:Oskar Klein
1376:worldvolume
986:probability
976:. However,
665:Silverstein
190:Mathematics
102:Superstring
6379:Categories
6274:Strominger
6269:Steinhardt
6264:Staudacher
6179:Polchinski
6129:Nanopoulos
6089:Mandelstam
6069:Kontsevich
5909:Berenstein
5837:Holography
5817:Superspace
5716:K3 surface
5675:Worldsheet
5590:Instantons
5218:Background
5168:Peter Woit
5146:Multiverse
5108:Emmy Award
5093:, episode
4566:(9): 032.
3693:2017-07-04
3567:(2): 003.
2867:Dirac 1962
2694:Moore 2005
2413:References
2396:Burt Ovrut
2339:smoothness
2171:Greg Moore
1747:membranes.
1669:Ashoke Sen
1649:Paul Dirac
1627:Chris Hull
1623:Ashoke Sen
1550:Lagrangian
1074:symmetries
925:Background
685:Strominger
680:Steinhardt
675:Staudacher
590:Polchinski
540:Nanopoulos
500:Mandelstam
480:Kontsevich
320:Berenstein
277:Multiverse
6309:Veneziano
6189:Rajaraman
6084:Maldacena
5974:Gopakumar
5924:Dijkgraaf
5919:Curtright
5585:Anomalies
5464:NS5-brane
5385:U-duality
5380:S-duality
5375:T-duality
4977:119248828
4960:1101.3216
4937:0905.2720
4393:14 August
4182:(1): 28.
4158:119585149
3927:0907.3987
3776:122728729
3658:1108.4389
3515:: 46â74.
3500:123289423
3361:0906.3219
3314:0806.1218
3249:Woit 2006
3213:Dine 2000
2837:Sen 1994b
2828:Sen 1994a
2798:Duff 1998
2760:Nahm 1978
2563:Citations
2301:manifolds
2018:spacetime
1793:Tom Banks
1662:In 1990,
1534:chirality
1207:T-duality
1185:S-duality
1177:T-duality
1173:S-duality
1163:T-duality
1159:S-duality
1149:Dualities
1113:spacetime
869:T-duality
865:S-duality
725:Veneziano
600:Rajaraman
495:Maldacena
385:Gopakumar
335:Dijkgraaf
330:Curtright
294:Theorists
182:Landscape
177:Cosmology
141:U-duality
136:T-duality
131:S-duality
114:Heterotic
6364:Zwiebach
6319:Verlinde
6314:Verlinde
6289:Townsend
6284:Susskind
6219:Sagnotti
6184:Polyakov
6139:Nekrasov
6104:Minwalla
6099:Martinec
6064:Knizhnik
6059:Klebanov
6054:Kapustin
6019:'t Hooft
5954:Fischler
5889:AganagiÄ
5860:M-theory
5749:Conifold
5744:Orbifold
5727:manifold
5668:Geometry
5474:M5-brane
5469:M2-brane
5406:Graviton
5322:F-theory
5166: â
5141:F-theory
5135:See also
4923:16790997
4876:14951363
4750:17534677
4692:16706816
4645:18887335
4551:18530420
4481:14125789
4374:6 August
4301:16318027
4119:13889163
4072:16122181
4025:17028835
3823:17432791
3739:17851652
3683:10882599
3433:13073785
3386:15459761
3339:16987793
2912:Sen 1993
2325:. These
2315:manifold
2230:Overview
2030:cylinder
2026:universe
1915:Overview
1677:magnetic
1673:electric
1327:type IIB
1323:type IIA
1315:energies
1300:fermions
1224:momentum
1045:theory (
1039:type IIB
1035:type IIA
1024:graviton
888:energies
837:M-theory
798:Glossary
780:Zwiebach
735:Verlinde
730:Verlinde
705:Townsend
700:'t Hooft
695:Susskind
630:Sagnotti
595:Polyakov
550:Nekrasov
515:Minwalla
510:Martinec
475:Knizhnik
470:Klebanov
465:Kapustin
435:Horowitz
365:Fischler
300:AganagiÄ
218:K-theory
151:F-theory
146:M-theory
18:M Theory
6294:Trivedi
6279:Sundrum
6244:Shenker
6234:Seiberg
6229:Schwarz
6199:Randall
6159:Novikov
6149:Nielsen
6134:NÄstase
6044:Kallosh
6029:Gibbons
5969:Gliozzi
5959:Friedan
5949:Ferrara
5934:Douglas
5929:Distler
5479:S-brane
5459:D-brane
5416:Tachyon
5411:Dilaton
5225:Strings
5090:Horizon
5082:Horizon
4903:Bibcode
4868:0990772
4848:Bibcode
4796:Bibcode
4767:Bibcode
4730:Bibcode
4672:Bibcode
4625:Bibcode
4578:Bibcode
4531:Bibcode
4461:Bibcode
4416:Bibcode
4336:Bibcode
4281:Bibcode
4245:Bibcode
4184:Bibcode
4099:Bibcode
4052:Bibcode
4005:Bibcode
3887:Bibcode
3840:Bibcode
3803:Bibcode
3756:Bibcode
3663:Bibcode
3620:Bibcode
3599:7562354
3579:Bibcode
3517:Bibcode
3480:Bibcode
3413:Bibcode
3366:Bibcode
3319:Bibcode
2321:of the
2135:surface
2115:compact
1972:called
1933:of the
1543:In the
1516:of the
1264:duality
1012:strings
962:gravity
950:strings
894:or the
875:called
841:physics
793:History
710:Trivedi
690:Sundrum
655:Shenker
645:Seiberg
640:Schwarz
610:Randall
570:Novikov
560:Nielsen
545:NÄstase
455:Kallosh
440:Gibbons
380:Gliozzi
370:Friedan
360:Ferrara
345:Douglas
340:Distler
110:Type II
97:Bosonic
81:D-brane
6359:Zumino
6354:Zaslow
6339:Yoneya
6329:Witten
6249:Siegel
6224:Scherk
6194:Ramond
6169:Ooguri
6094:Marolf
6049:Kaluza
6034:Kachru
6024:HoĆava
6014:Harvey
6009:Hanson
5994:Gubser
5984:Greene
5914:Bousso
5899:Atiyah
5451:Branes
5261:Theory
5061:
5042:
5019:
5000:
4975:
4921:
4874:
4866:
4827:
4748:
4690:
4643:
4598:668885
4596:
4549:
4500:
4479:
4299:
4156:
4117:
4070:
4023:
3974:
3955:
3821:
3774:
3737:
3727:
3681:
3597:
3548:
3498:
3452:
3431:
3384:
3337:
2995:at 345
2987:
2441:couple
2173:, and
2039:curved
1978:metric
1898:, and
1803:, and
1777:matrix
1584:gluons
1580:quarks
1526:strong
1512:, and
1452:radion
1380:fields
1333:Branes
1325:, and
1319:type I
1296:bosons
1161:, and
1031:type I
1020:charge
884:branes
775:Zumino
770:Zaslow
755:Yoneya
745:Witten
660:Siegel
635:Scherk
605:Ramond
580:Ooguri
505:Marolf
460:Kaluza
445:Kachru
430:HoĆava
425:Harvey
420:Hanson
405:Gubser
395:Greene
325:Bousso
310:Atiyah
106:Type I
66:String
6299:Turok
6209:RoÄek
6174:Ovrut
6164:Olive
6144:Neveu
6124:Myers
6119:Mukhi
6109:Moore
6079:Linde
6074:Klein
5999:Gukov
5989:Gross
5979:Green
5964:Gates
5944:Dvali
5904:Banks
5116:with
4973:S2CID
4955:arXiv
4932:arXiv
4919:S2CID
4893:arXiv
4872:S2CID
4746:S2CID
4720:arXiv
4688:S2CID
4662:arXiv
4641:S2CID
4615:arXiv
4594:S2CID
4568:arXiv
4547:S2CID
4521:arXiv
4477:S2CID
4451:arXiv
4388:(PDF)
4370:: 214
4360:(PDF)
4269:(PDF)
4235:arXiv
4170:(PDF)
4154:S2CID
4136:arXiv
4115:S2CID
4089:arXiv
4068:S2CID
4042:arXiv
4021:S2CID
3995:arXiv
3922:arXiv
3819:S2CID
3793:arXiv
3772:S2CID
3735:S2CID
3707:arXiv
3679:S2CID
3653:arXiv
3595:S2CID
3569:arXiv
3565:19981
3496:S2CID
3468:(PDF)
3429:S2CID
3403:arXiv
3382:S2CID
3356:arXiv
3335:S2CID
3309:arXiv
2418:Notes
1345:brane
1339:Brane
1078:force
994:is a
715:Turok
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