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{{short description|Framework of superstring theory}}
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{{For introduction|Introduction to M-theory}}
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In [[physics]], '''M-theory''' is a theory that unifies all [[Consistency|consistent]] versions of [[superstring theory]]. [[Edward Witten]] first conjectured the existence of such a theory at a [[string theory]] conference at the [[University of Southern California]] in 1995. Witten's announcement initiated a flurry of research activity known as the [[second superstring revolution]]. 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 [[S-duality]] and [[T-duality]]. 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 [[field theory (physics)|field theory]] called [[eleven-dimensional supergravity]].

Although a complete formulation of M-theory is not known, such a formulation should describe two- and five-dimensional objects called [[branes]] and should be approximated by eleven-dimensional supergravity at low [[energies]]. Modern attempts to formulate M-theory are typically based on [[matrix theory (physics)|matrix theory]] or the [[AdS/CFT correspondence]]. 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.<ref name="Duff 1996, sec. 1">Duff 1996, sec. 1</ref>

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 [[theory of everything|unified theory]] of all of the [[fundamental force]]s of nature. Attempts to connect M-theory to experiment typically focus on [[compactification (physics)|compactifying]] its [[extra dimension]]s to construct candidate models of the four-dimensional world, although so far none have been verified to give rise to physics as observed in [[high-energy physics]] experiments.

==Background==

===Quantum gravity and strings===

{{main article|Quantum gravity|String theory}}

[[Image:Open and closed strings.svg|right|thumb|alt=A wavy open segment and closed loop of string.|The fundamental objects of string theory are open and closed [[string (physics)|strings]]. ]]

One of the deepest problems in modern physics is the problem of [[quantum gravity]]. The current understanding of [[gravity]] is based on [[Albert Einstein]]'s [[general theory of relativity]], which is formulated within the framework of [[classical physics]]. However, [[fundamental interaction|nongravitational forces]] are described within the framework of [[quantum mechanics]], a radically different formalism for describing physical phenomena based on [[probability]].{{efn|For a standard introduction to quantum mechanics, see Griffiths 2004.}} A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics,{{efn|The necessity of a quantum mechanical description of gravity follows from the fact that one cannot consistently [[coupling (physics)|couple]] a classical system to a quantum one. See Wald 1984, p. 382.}} but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.{{efn|From a technical point of view, the problem is that the theory one gets in this way is not [[renormalizable]] and therefore cannot be used to make meaningful physical predictions. See Zee 2010, p. 72 for a discussion of this issue.}}

[[String theory]] is a [[mathematical theory|theoretical framework]] that attempts to reconcile gravity and quantum mechanics. In string theory, the [[point particle|point-like particles]] of [[particle physics]] are replaced by [[one-dimensional]] objects called [[string (physics)|strings]]. 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 [[mass]], [[charge (physics)|charge]], 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 [[graviton]], a quantum mechanical particle that carries gravitational force.{{efn|For an accessible introduction to string theory, see Greene 2000.}}

There are several versions of string theory: [[type I string|type I]], [[type IIA string|type IIA]], [[type IIB string|type IIB]], and two flavors of [[heterotic string]] theory ({{math|[[special orthogonal group|''SO''(32)]]}} and {{math|[[E8 (mathematics)|''E''<sub>8</sub>×''E''<sub>8</sub>]]}}). The different theories allow different types of strings, and the particles that arise at low energies exhibit different [[symmetry (physics)|symmetries]]. 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.<ref>Zwiebach 2009, p. 324</ref> 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 [[force (physics)|force]] just like the familiar gravitational force subject to the rules of quantum mechanics.<ref name="Becker, Becker, and Schwarz 2007, p. 12">Becker, Becker, and Schwarz 2007, p. 12</ref>

===Number of dimensions===

{{main article|Extra dimensions|Compactification (physics)}}

[[File:Compactification example.svg|left|thumb|alt=A tubular surface and corresponding one-dimensional curve.|An example of [[compactification (physics)|compactification]]: At large distances, a two-dimensional surface with one circular dimension looks one-dimensional.]]

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 four-dimensional [[spacetime]], three spatial dimensions and one time dimension. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.<ref>Wald 1984, p. 4</ref>

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 perform calculations and gain general insights more easily.{{efn|For example, in the context of the [[AdS/CFT correspondence]], theorists often formulate and study theories of gravity in unphysical numbers of spacetime dimensions.}} There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in [[condensed matter physics]].<ref>Zee 2010, Parts V and VI</ref> Finally, there exist scenarios in which there could actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.<ref>Zwiebach 2009, p. 9</ref>

One notable feature of string theory and M-theory is that these theories require [[extra dimensions]] of spacetime for their mathematical consistency. In string theory, spacetime is ''ten-dimensional'' (nine spatial dimensions, and one time dimension), {{anchor|11-dimensional spacetime}}while in M-theory it is ''eleven-dimensional'' (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.<ref>Zwiebach 2009, p. 8</ref>

[[Compactification (physics)|Compactification]] is one way of modifying the number of dimensions in a physical theory.{{efn|[[Dimensional reduction]] is another way of modifying the number of dimensions.}} In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.<ref name=autogenerated1>Yau and Nadis 2010, Ch. 6</ref> In the limit where these curled-up dimensions become very small, one obtains a theory in which spacetime has effectively a 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.{{efn|This analogy is used for example in Greene 2000, p. 186.}}

===Dualities===

{{main article|String duality|S-duality|T-duality}}

[[File:StringTheoryDualities.svg|right|thumb|alt=A diagram indicating the relationships between M-theory and the five string theories.|upright=1.2|A diagram of string theory dualities. Yellow arrows indicate [[S-duality]]. Blue arrows indicate [[T-duality]]. These dualities may be combined to obtain equivalences of any of the five theories with M-theory.<ref>Becker, Becker, and Schwarz 2007, pp. 339–347</ref>]]

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 [[S-duality]]. 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. [[Type I string theory]] turns out to be equivalent by S-duality to the {{math|''SO''(32)}} heterotic string theory. Similarly, [[type IIB string theory]] is related to itself in a nontrivial way by S-duality.<ref name="Becker, Becker, and Schwarz 2007">Becker, Becker, and Schwarz 2007</ref>

Another relationship between different string theories is [[T-duality]]. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius {{math|''R''}} is equivalent to a string propagating around a circle of radius {{math|1/''R''}} in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, a string has [[momentum]] 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 [[winding number]]. If a string has momentum {{math|''p''}} and winding number {{math|''n''}} in one description, it will have momentum {{math|''n''}} and winding number {{math|''p''}} in the dual description. For example, [[type IIA string theory]] is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.<ref name="Becker, Becker, and Schwarz 2007"/>

In general, the term ''[[string duality|duality]]'' refers to a situation where two seemingly different [[physical system]]s 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 ''dual'' to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.<ref>Zwiebach 2009, p. 376</ref>

===Supersymmetry===

{{main article|Supersymmetry}}

Another important theoretical idea that plays a role in M-theory is [[supersymmetry]]. This is a mathematical relation that exists in certain physical theories between a class of particles called [[bosons]] and a class of particles called [[fermions]]. 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 [[supergravity theory]].<ref name="Duff 1998, p. 64">Duff 1998, p. 64</ref>

A theory of strings that incorporates the idea of supersymmetry is called a [[superstring theory]]. There are several different versions of superstring theory which are all subsumed within the M-theory framework. At low [[energy|energies]], superstring theories are approximated by one of the three supergravities in ten dimensions, known as [[type I supergravity|type I]], [[type IIA supergravity|type IIA]], and [[type IIB supergravity|type IIB]] supergravity. Similarly, M-theory is approximated at low energies by supergravity in eleven dimensions.<ref name="Becker, Becker, and Schwarz 2007, p. 12"/>

===Branes===

{{main article|Brane}}

In string theory and related theories such as supergravity theories, a [[brane]] 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 {{math|''p''}}, these are called {{math|''p''}}-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 {{math|''p''}}-brane sweeps out a {{math|(''p''&nbsp;+&nbsp;1)}}-dimensional volume in spacetime called its ''worldvolume''. Physicists often study [[field (physics)|fields]] analogous to the [[electromagnetic field]] which live on the worldvolume of a brane. The word brane comes from the word "membrane" which refers to a two-dimensional brane.<ref>Moore 2005</ref>

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 in M-theory attempts to better understand the properties of these branes.{{efn|For example, see the subsections on the [[6D (2,0) superconformal field theory]] and [[ABJM superconformal field theory]].}}

==History and development==

{{Main article|History of string theory}}

===Kaluza–Klein theory===

{{main article|Kaluza–Klein theory}}

In the early 20th century, physicists and mathematicians including Albert Einstein and [[Hermann Minkowski]] pioneered the use of four-dimensional geometry for describing the physical world.<ref>Yau and Nadis 2010, p. 9</ref> These efforts culminated in the formulation of Einstein's general theory of relativity, which relates gravity to the geometry of four-dimensional spacetime.<ref name="Yau and Nadis 2010, p. 10">Yau and Nadis 2010, p. 10</ref>

The success of general relativity led to efforts to apply higher dimensional geometry to explain other forces. In 1919, work by [[Theodor Kaluza]] showed that by passing to five-dimensional spacetime, one can unify gravity and [[electromagnetism]] into a single force.<ref name="Yau and Nadis 2010, p. 10"/> This idea was improved by physicist [[Oskar Klein]], who suggested that the additional dimension proposed by Kaluza could take the form of a circle with radius around {{nowrap|10<sup>−30</sup>}} cm.<ref>Yau and Nadis 2010, p. 12</ref>

The [[Kaluza–Klein theory]] and subsequent attempts by Einstein to develop [[unified field theory]] were never completely successful. In part this was because Kaluza–Klein theory predicted a particle (the [[Graviscalar|radion]]), 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 [[nuclear force]]s 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.<ref>Yau and Nadis 2010, p. 13</ref>

===Early work on supergravity===

{{main article|Supergravity}}

[[File:Edward Witten.jpg|left|thumb|upright=0.8|alt=A portrait of Edward Witten.|In the 1980s, [[Edward Witten]] contributed to the understanding of [[supergravity]] theories. In 1995, he introduced M-theory, sparking the [[second superstring revolution]].]]

New concepts and mathematical tools provided fresh insights into general relativity, giving rise to a period in the 1960s–1970s now known as the [[history of general relativity|golden age of general relativity]].<ref>Wald 1984, p. 3</ref> In the mid-1970s, physicists began studying higher-dimensional theories combining general relativity with supersymmetry, the so-called supergravity theories.<ref>van Nieuwenhuizen 1981</ref>

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 upper limit on the number of dimensions.<ref name="Duff 1998, p. 64"/> In 1978, work by [[Werner Nahm]] showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven.<ref>Nahm 1978</ref> In the same year, [[Eugène Cremmer]], [[Bernard Julia]], and [[Joël Scherk]] of the [[École Normale Supérieure]] showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.<ref>Cremmer, Julia, and Scherk 1978</ref><ref name="Duff 1998, p. 65">Duff 1998, p. 65</ref>

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 [[strong nuclear force|strong]] and [[weak nuclear force]]s, 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 [[chirality (physics)|chirality]]. [[Edward Witten]] and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.<ref name="Duff 1998, p. 65"/>

In the [[first superstring revolution]] 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.<ref name="Duff 1998, p. 65"/> Another feature of 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 [[Lagrangian (field theory)|Lagrangian]]. 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.<ref name="Duff 1998, p. 65"/>

===Relationships between string theories===

Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation.<ref name="Duff 1998, p. 65"/> However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways.<ref>Duff 1998</ref>

In the late 1970s, Claus Montonen and [[David Olive]] had conjectured a special property of certain physical theories.<ref>Montonen and Olive 1977</ref> A sharpened version of their conjecture concerns a theory called [[N = 4 supersymmetric Yang–Mills theory|{{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory]], which describes theoretical particles formally similar to the [[quark]]s and [[gluon]]s that make up [[atomic nucleus|atomic nuclei]]. The strength with which the particles of this theory interact is measured by a number called the [[coupling constant]]. The result of Montonen and Olive, now known as [[Montonen–Olive duality]], states that {{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory with coupling constant {{math|''g''}} is equivalent to the same theory with coupling constant {{math|1/''g''}}. 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<ref name="Duff 1998, p. 66">Duff 1998, p. 66</ref> by spin-moment.

In the 1990s, several theorists generalized Montonen–Olive duality to the S-duality relationship, which connects different string theories. [[Ashoke Sen]] studied S-duality in the context of heterotic strings in four dimensions.<ref>Sen 1994a</ref><ref>Sen 1994b</ref> [[Chris Hull]] and [[Paul Townsend]] showed that type IIB string theory with a large coupling constant is equivalent via S-duality to the same theory with small coupling constant.<ref>Hull and Townsend 1995</ref> Theorists also found that different string theories may be related by T-duality. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent.<ref name="Duff 1998, p. 67">Duff 1998, p. 67</ref>

===Membranes and fivebranes===

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 [[supermembranes]] or by higher-dimensional objects called branes. Such objects had been considered as early as 1962 by [[Paul Dirac]],<ref>Dirac 1962</ref> and they were reconsidered by a small but enthusiastic group of physicists in the 1980s.<ref name="Duff 1998, p. 65"/>

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.<ref>Bergshoeff, Sezgin, and Townsend 1987</ref> Intuitively, these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after this discovery, [[Michael Duff (physicist)|Michael Duff]], 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.<ref>Duff et al. 1987</ref> 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.<ref name="Duff 1998, p. 66"/>

In 1990, [[Andrew Strominger]] 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.<ref>Strominger 1990</ref> Initially, physicists were unable to prove this relationship for two important reasons. On the one hand, the Montonen–Olive 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.<ref>Duff 1998, pp. 66–67</ref> The first of these problems was solved in 1993 when [[Ashoke Sen]] established that certain physical theories require the existence of objects with both [[electric charge|electric]] and [[magnetic monopole|magnetic]] charge which were predicted by the work of Montonen and Olive.<ref>Sen 1993</ref>

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 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.<ref name="Duff 1998, p. 67"/>

===Second superstring revolution===

[[File:Limits of M-theory.svg|upright=1.6|thumb|alt=A star-shaped diagram with the various limits of M-theory labeled at its six vertices.|A schematic illustration of the relationship between M-theory, the five [[superstring theory|superstring theories]], and eleven-dimensional [[supergravity]]. 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.]]

{{main article|Second superstring revolution}}

Speaking at [[Strings (conference)|Strings]] '95 at the [[University of Southern California]] in 1995, Edward Witten of the [[Institute for Advanced Study]] 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 string theory.<ref>Witten 1995</ref> 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.<ref>Duff 1998, pp. 67–68</ref> Today this flurry of work is known as the [[second superstring revolution]].<ref>Becker, Becker, and Schwarz 2007, p. 296</ref>

One of the important developments following Witten's announcement was Witten's work in 1996 with string theorist [[Petr Hořava (theorist)|Petr Hořava]].<ref name="Hořava and Witten 1996a">Hořava and Witten 1996a</ref><ref>Hořava and Witten 1996b</ref> Witten and Hořava 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.<ref>Duff 1998, p. 68</ref>

===Origin of the term===

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, Hořava and Witten wrote

{{Blockquote|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 membranes.<ref name="Hořava and Witten 1996a"/>|sign=|source= }}

In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the ''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.<ref name="Duff 1996, sec. 1" /> Years later, he would state, "I thought my colleagues would understand that it really stood for membrane. Unfortunately, it got people confused."<ref>{{cite book |last=Gefter |first=Amanda |date=2014 |title=Trespassing on Einstein's Lawn: A Father, a Daughter, the Meaning of Nothing and the Beginning of Everything |publisher=Random House |isbn=978-0-345-531438}} at 345</ref>

==Matrix theory==

===BFSS matrix model===

{{main article|Matrix theory (physics)}}

In mathematics, a [[matrix (mathematics)|matrix]] is a rectangular array of numbers or other data. In physics, a [[matrix theory (physics)|matrix model]] 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.<ref name="Banks et al. 1997">Banks et al. 1997</ref><ref name="Connes, Douglas, and Schwarz 1998">Connes, Douglas, and Schwarz 1998</ref>

One important{{why|date=December 2016}} example of a matrix model is the [[BFSS matrix model]] proposed by [[Tom Banks (physicist)|Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], and [[Leonard Susskind]] 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 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.<ref name="Banks et al. 1997"/>{{clarify|sufficiency is established, but necessity is unclear for the reader--should be explicitly mentioned|date=December 2016}}

===Noncommutative geometry===

{{main article|Noncommutative geometry|Noncommutative quantum field theory}}

In geometry, it is often useful to introduce [[coordinate system|coordinates]]. For example, in order to study the geometry of the [[Euclidean plane]], one defines the coordinates {{math|''x''}} and {{math|''y''}} as the distances between any point in the plane and a pair of [[Cartesian coordinate system|axes]]. 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, {{math|''xy'' {{=}} ''yx''}}. This property of multiplication is known as the [[commutative law]], and this relationship between geometry and the [[commutative algebra]] of coordinates is the starting point for much of modern geometry.<ref>Connes 1994, p. 1</ref>

[[Noncommutative geometry]] 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 {{math|''xy''}} is not necessarily equal to {{math|''yx''}}). 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.<ref>Connes 1994</ref>

In a paper from 1998, [[Alain Connes]], [[Michael R. Douglas]], and [[Albert Schwarz]] showed that some aspects of matrix models and M-theory are described by a [[noncommutative quantum field theory]], a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property.<ref name="Connes, Douglas, and Schwarz 1998"/> 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.<ref>Nekrasov and Schwarz 1998</ref><ref>Seiberg and Witten 1999</ref>

==AdS/CFT correspondence==

===Overview===

{{main article|AdS/CFT correspondence}}

[[Image:Uniform tiling 433-t0 (formatted).svg|thumb|left|alt=A disk tiled by triangles and quadrilaterals which become smaller and smaller near the boundary circle.|A [[alternated octagonal tiling|tessellation]] of the [[hyperbolic plane]] by triangles and squares]]

The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as [[quantum field theory]].{{efn|A standard text is Peskin and Schroeder 1995.}} 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 [[quasiparticle]]s.{{efn|For an introduction to the applications of quantum field theory to condensed matter physics, see Zee 2010.}}

One approach to formulating M-theory and studying its properties is provided by the [[AdS/CFT correspondence|anti-de Sitter/conformal field theory (AdS/CFT) correspondence]]. Proposed by [[Juan Maldacena]] 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.<ref name="Maldacena_a"/> 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 quantum field theory in regimes where traditional calculational techniques are ineffective.<ref>Klebanov and Maldacena 2009</ref>

In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain [[vacuum solution]] of [[Einstein's equation]] called [[anti-de Sitter space]].<ref>Klebanov and Maldacena 2009, p. 28</ref> In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the [[metric tensor|metric]]) is different from the notion of distance in ordinary [[Euclidean geometry]]. It is closely related to [[hyperbolic space]], which can be viewed as a [[Poincaré disk model|disk]] as illustrated on the left.<ref name="Maldacena 2005, p. 60">Maldacena 2005, p. 60</ref> This image shows a [[tessellation]] 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.<ref name="Maldacena 2005, p. 61">Maldacena 2005, p. 61</ref>

[[File:AdS3.svg|thumb|upright=1.4|alt=A cylinder formed by stacking copies of the disk illustrated in the previous figure.|Three-dimensional [[anti-de Sitter space]] is like a stack of [[Poincaré disk model|hyperbolic disks]], each one representing the state of the universe at a given time. One can study theories of [[quantum gravity]] such as M-theory in the resulting [[spacetime]].]]

Now imagine a stack of hyperbolic disks where each disk represents the state of the [[universe]] at a given time. The resulting geometric object is three-dimensional anti-de Sitter space.<ref name="Maldacena 2005, p. 60"/> It looks like a solid [[cylinder (geometry)|cylinder]] in which any [[cross section (geometry)|cross section]] 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 [[curvature|curved]] in such a way that any point in the interior is actually infinitely far from this boundary surface.<ref name="Maldacena 2005, p. 61"/>

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 models of anti-de Sitter space.<ref name="Maldacena 2005, p. 60"/>

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 [[Minkowski space]], the model of spacetime used in nongravitational physics.<ref>Zwiebach 2009, p. 552</ref> 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.<ref>Maldacena 2005, pp. 61–62</ref>

===6D (2,0) superconformal field theory===

{{main article|6D (2,0) superconformal field theory}}

[[Image:Knot table-blank unknot.svg|left|thumb|upright=1.6|alt=A collection of knot diagrams in the plane.|The six-dimensional [[6D (2,0) superconformal field theory|(2,0)-theory]] has been used to understand results from the [[knot theory|mathematical theory of knots]].]]

One particular realization of the AdS/CFT correspondence states that M-theory on the [[product space]] {{math|''AdS''<sub>7</sub>×''S''<sup>4</sup>}} is equivalent to the so-called [[6D (2,0) superconformal field theory|(2,0)-theory]] on the six-dimensional boundary.<ref name="Maldacena_a">Maldacena 1998</ref> 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 {{math|''AdS''<sub>7</sub>}}), and there are four additional "[[compact space|compact]]" dimensions (encoded by the {{math|[[n-sphere|''S''<sup>4</sup>]]}} 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.{{efn|For a review of the (2,0)-theory, see Moore 2012.}}

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 [[effective field theory|effective quantum field theories]] 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 [[Surface (topology)|surface]], one obtains a four-dimensional quantum field theory, and there is a duality known as the [[AGT correspondence]] which relates the physics of this theory to certain physical concepts associated with the surface itself.<ref>Alday, Gaiotto, and Tachikawa 2010</ref> More recently, theorists have extended these ideas to study the theories obtained by compactifying down to three dimensions.<ref>Dimofte, Gaiotto, and Gukov 2010</ref>

In addition to its applications in quantum field theory, the (2,0)-theory has spawned important results in [[pure mathematics]]. 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 [[geometric Langlands correspondence]].<ref>Witten 2009</ref> In subsequent work, Witten showed that the (2,0)-theory could be used to understand a concept in mathematics called [[Khovanov homology]].<ref>Witten 2012</ref> Developed by [[Mikhail Khovanov]] around 2000, Khovanov homology provides a tool in [[knot theory]], the branch of mathematics that studies and classifies the different shapes of knots.<ref>Khovanov 2000</ref> Another application of the (2,0)-theory in mathematics is the work of [[Davide Gaiotto]], [[Greg Moore (physicist)|Greg Moore]], and [[Andrew Neitzke]], which used physical ideas to derive new results in [[hyperkähler manifold|hyperkähler geometry]].<ref>Gaiotto, Moore, and Neitzke 2013</ref>

===ABJM superconformal field theory===

{{main article|ABJM superconformal field theory}}

Another realization of the AdS/CFT correspondence states that M-theory on {{math|''AdS''<sub>4</sub>×''S''<sup>7</sup>}} is equivalent to a quantum field theory called the [[ABJM superconformal field theory|ABJM theory]] 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.<ref name="Aharony et al. 2008">Aharony et al. 2008</ref>

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 [[Chern–Simons theory]]. The latter theory was popularized by Witten in the late 1980s because of its applications to knot theory.<ref>Witten 1989</ref> In addition, the ABJM theory serves as a semi-realistic simplified model for solving problems that arise in condensed matter physics.<ref name="Aharony et al. 2008"/>

==Phenomenology==

===Overview===

{{main article|String phenomenology}}

[[Image:Calabi yau formatted.svg|right|thumb|alt=Visualization of a complex mathematical surface with many convolutions and self intersections.|A cross section of a [[Calabi–Yau manifold]] ]]

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 [[standard model of particle physics]]. [[Phenomenology (particle physics)|Phenomenology]] is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. [[String phenomenology]] is the part of string theory that attempts to construct realistic models of particle physics based on string and M-theory.<ref>Dine 2000</ref>

Typically, such models are based on the idea of compactification.{{efn|[[Brane world]] scenarios provide an alternative way of recovering real world physics from string theory. See Randall and Sundrum 1999.}} 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 of particle physics, together with additional undiscovered particles,<ref>Candelas et al. 1985</ref> usually [[supersymmetry|supersymmetric]] 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 [[Calabi–Yau manifold]]. This is a special kind of geometric object named after mathematicians [[Eugenio Calabi]] and [[Shing-Tung Yau]].<ref>Yau and Nadis 2010, p. ix</ref> Calabi–Yau 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.<ref>Yau and Nadis 2010, pp. 147–150</ref>

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 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.<ref>Woit 2006</ref>

===Compactification on {{math|''G''<sub>2</sub>}} manifolds===

In one approach to M-theory phenomenology, theorists assume that the seven extra dimensions of M-theory are shaped like a [[G2 manifold|{{math|''G''<sub>2</sub>}} manifold]]. This is a special kind of seven-dimensional shape constructed by mathematician [[Dominic Joyce]] of the [[University of Oxford]].<ref>Yau and Nadis 2010, p. 149</ref> These {{math|''G''<sub>2</sub>}} manifolds are still poorly understood mathematically, and this fact has made it difficult for physicists to fully develop this approach to phenomenology.<ref name="Yau and Nadis 2010, p. 150">Yau and Nadis 2010, p. 150</ref>

For example, physicists and mathematicians often assume that space has a mathematical property called [[smooth manifold|smoothness]], but this property cannot be assumed in the case of a {{math|''G''<sub>2</sub>}} manifold if one wishes to recover the physics of our four-dimensional world. Another problem is that {{math|''G''<sub>2</sub>}} manifolds are not [[complex manifold]]s, so theorists are unable to use tools from the branch of mathematics known as [[complex analysis]]. Finally, there are many open questions about the existence, uniqueness, and other mathematical properties of {{math|''G''<sub>2</sub>}} manifolds, and mathematicians lack a systematic way of searching for these manifolds.<ref name="Yau and Nadis 2010, p. 150"/>

===Heterotic M-theory===

Because of the difficulties with {{math|''G''<sub>2</sub>}} 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, Hořava, [[Burt Ovrut]], 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 Calabi–Yau manifold. If this Calabi–Yau manifold is also taken to be small, one is left with a theory in four-dimensions.<ref name="Yau and Nadis 2010, p. 150"/>

Heterotic M-theory has been used to construct models of [[brane cosmology]] 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 [[cosmic inflation]].<ref name="Yau and Nadis 2010, p. 150"/>

==References==

===Notes===

{{notelist}}

===Citations===

{{reflist|30em}}

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* {{cite arXiv |last=Witten |first=Edward |eprint=0905.2720 |title=Geometric Langlands from six dimensions |class=hep-th |date=2009 }}
* {{cite journal |last1=Witten |first1=Edward |date=2012 |title=Fivebranes and knots |journal=Quantum Topology |volume=3 |issue=1 |pages=1–137 |doi=10.4171/QT/26 |arxiv=1101.3216 |s2cid=119248828 }}
* {{cite book |last= Woit |first= Peter |title= Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law |publisher= Basic Books |date= 2006 |page= [https://archive.org/details/notevenwrongfail00woit/page/105 105] |isbn= 0-465-09275-6 |url-access= registration |url= https://archive.org/details/notevenwrongfail00woit/page/105 }}
* {{Cite book| first1 = Shing-Tung | last1 = Yau | first2 = Steve | last2 = Nadis | date = 2010 | title = The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions | publisher = Basic Books | isbn = 978-0-465-02023-2 }}
* {{cite book |last=Zee |first=Anthony |title=Quantum Field Theory in a Nutshell |edition=2nd |date=2010 |publisher=Princeton University Press |isbn=978-0-691-14034-6 |url-access=registration |url=https://archive.org/details/isbn_9780691140346 }}
* {{cite book |last1=Zwiebach |first1=Barton |title=A First Course in String Theory |date=2009 |publisher=Cambridge University Press |isbn=978-0-521-88032-9}}
{{refend}}

==Popularization==

* [http://www.bbc.co.uk/science/horizon/2001/paralleluni.shtml BBC ''Horizon'': "Parallel Universes"]&nbsp;– 2002 feature documentary by [[Horizon (British TV series)|BBC ''Horizon'']], episode [[Parallel Universes (film)|"Parallel Universes"]] focuses on the history and emergence of M-theory, and scientists involved
* [https://www.pbs.org/wgbh/nova/series/the-elegant-universe/] PBS.org-NOVA: ''The Elegant Universe'']&nbsp;– 2003 [[Emmy Award]]-winning, three-hour miniseries by [[Nova (American TV program)|''Nova'']] with [[Brian Greene]], adapted from his ''[[The Elegant Universe]]'' book (original [[PBS]] broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003)

==See also==
* [[F-theory]]
* [[Multiverse]]

==External links==
* [http://superstringtheory.com/ Superstringtheory.com]&nbsp;– The "Official String Theory Web Site", created by Patricia Schwarz. References on string theory and M-theory for the layperson and expert.
* [http://www.math.columbia.edu/~woit/wordpress/ Not Even Wrong]&nbsp;– [[Peter Woit]]'s blog on physics in general, and string theory in particular.
* [https://www.math.stonybrook.edu/Videos/IMS/video.php?f=35-Witten M-Theory – Edward Witten (1995)] – Witten's 1995 lecture introducing M-Theory.
{{String theory topics |state=collapsed}}
{{Authority control}}

[[Category:String theory]]
[[Category:1995 introductions]]