In

theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

,

**M-theory** is an extension of

string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories (and supersedes them). Though a full description of the theory is not known, the low-entropy dynamics are known to be supergravity interacting with 2- and 5-dimensional

membraneIn theoretical physics, a membrane, brane, or p-brane is a spatially extended mathematical concept that appears in string theory and related theories...

s.

This idea is the unique

supersymmetricIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

theory in eleven dimensions, with its low-entropy matter content and interactions fully determined, and can be obtained as the strong coupling limit of type IIA string theory because a new dimension of space emerges as the

coupling constantIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...

increases.

Drawing on the work of a number of string theorists (including

Ashoke SenAshoke Sen , FRS, is an Indian theoretical physicist. He has made a number of major original contributions to the subject of string theory, including his landmark paper on strong-weak coupling duality or S-duality, which was influential in changing the course of research in the field...

,

Chris HullChris Hull is a professor of Theoretical Physics at Imperial College London. Hull is known for his work on string theory, M-theory, and generalized complex structures...

,

Paul TownsendPaul Kingsley Townsend FRS is a British physicist, currently a Professor of Theoretical Physics in Cambridge University's Department of Applied Mathematics and Theoretical Physics. He is notable for his work on string theory....

, Michael Duff and John Schwarz),

Edward WittenEdward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....

of the

Institute for Advanced StudyThe Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

suggested its existence at a conference at

USCThe University of Southern California is a private, not-for-profit, nonsectarian, research university located in Los Angeles, California, United States. USC was founded in 1880, making it California's oldest private research university...

in 1995, and used M-theory to explain a number of previously observed

dualitiesString duality is a class of symmetries in physics that link different string theories, theories which assume that the fundamental building blocks of the universe are strings instead of point particles....

, initiating a flurry of new research in string theory called the second superstring revolution.

In the early 1990s it was shown that the various superstring theories were related by dualities which allow the description of an object in one super string theory to be related to the description of a different object in another super string theory. These relationships imply that each of the super string theories is a different aspect of a single underlying theory, proposed by Witten, and named "M-theory".

Originally the letter M in M-theory was taken from

*membrane*, a construct designed to generalize the strings of string theory. However, as Witten was more skeptical about membranes than his colleagues, he opted for "M-theory" rather than "Membrane theory". Witten has since stated that the interpretation of the M can be a matter of taste for the user of the name.

M-theory (and string theory) has been criticized for lacking predictive power or being untestable. Further work continues to find mathematical constructs that join various surrounding theories. However, the tangible success of M-theory can be questioned, given its current incompleteness and limited predictive power.

### Prior to May 1994

Before 1994 there were five known consistent superstring theories (henceforth referred to as string theories), which were given the names

Type I string theoryIn theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...

, Type IIA string theory, Type IIB string theory,

heteroticIn physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...

SO(32) (the HO string) theory, and heterotic

*E*_{8}×*E*_{8}In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

(the HE string) theory. The five theories all share essential features that relate them to the name of string theory. Each theory is fundamentally based on vibrating, one-dimensional strings at approximately the length of the

Planck length. Calculations have also shown that each theory requires more than the normal four

spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

dimensions (although all extra dimensions are in fact spatial). When the theories are analyzed in detail, significant differences appear.

### Type I string theory and supplements

The Type I string theory has vibrating strings like the rest of the string theories. These strings vibrate both in closed loops, so that the strings have no ends, and as open strings with two loose ends. The open loose strings are what separates the Type I string theory from the other four string theories. This was a feature that the other string theories did not contain.

### String vibrational patterns

The calculations of the String Vibrational Patterns show that the list of string vibrational patterns and the way each pattern interacts and influences others vary from one theory to another. These and other differences hindered the development of the string theory as being the theory that united

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

and

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

successfully. Attempts by the physics community to eliminate four of the theories, leaving only one string theory, have not been successful.

### M-theory

M-theory attempts to unify the five string theories by examining certain identifications and dualities. Thus each of the five string theories become special cases of M-theory.

As the names suggest, some of these string theories were thought to be related to each other. In the early 1990s, string theorists discovered that some relations were so strong that they could be thought of as an identification.

### Type IIA and Type IIB

The Type IIA string theory and the Type IIB string theory were known to be connected by

T-dualityT-duality is a symmetry of quantum field theories with differing classical descriptions, of which the relationship between small and large distances in various string theories is a special case. Discussion of the subject originated in a paper by T. S. Buscher and was further developed by Martin...

; this essentially meant that the IIA string theory description of a circle of radius R is exactly the same as the IIB description of a circle of radius 1/R, where distances are measured in units of the Planck length.

This was a profound result. First, this was an intrinsically quantum mechanical result; the identification did not hold in the realm of

classical physicsWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

. Second, because it is possible to build up any space by gluing circles together in various ways, it would seem that any space described by the IIA string theory can also be seen as a different space described by the IIB theory. This implies that the IIA string theory can identify with the IIB string theory: any object which can be described with the IIA theory has an equivalent, although seemingly different, description in terms of the IIB theory. This suggests that the IIA string theory and the IIB string theory are really aspects of the same underlying theory.

### Other dualities

There are other dualities between the other string theories. The

heteroticIn physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...

SO(32) and the heterotic

*E*_{8}×*E*_{8}In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

theories are also related by T-duality; the heterotic SO(32) description of a circle of radius R is exactly the same as the heterotic

*E*_{8}×*E*_{8}In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

description of a circle of radius 1/R. This implies that there are really only three superstring theories, which might be called (for discussion) the Type I theory, the Type II theory, and the heterotic theory.

There are still more dualities, however. The Type I string theory is related to the heterotic SO(32) theory by

S-dualityIn theoretical physics, S-duality is an equivalence of two quantum field theories or string theories. An S-duality transformation maps states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory...

; this means that the Type I description of

weakly interactingWeak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...

particles can also be seen as the heterotic SO(32) description of very

strongly interactingIn particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

particles. This identification is somewhat more subtle, in that it identifies only extreme limits of the respective theories. String theorists have found strong evidence that the two theories are really the same, even away from the extremely strong and extremely weak limits, but they do not yet have a proof strong enough to satisfy mathematicians. However, it has become clear that the two theories are related in some fashion; they appear as different limits of a single underlying theory.

### Only two string theories

Given the above commonalities there appear to be only two string theories: the heterotic string theory (which is also the type I string theory) and the type II theory. There are relations between these two theories as well, and these relations are in fact strong enough to allow them to be identified.

### Last step

This last step is best explained first in a certain limit. In order to describe our world, strings must be extremely tiny objects. So when one studies string theory at low energies, it becomes difficult to see that strings are extended objects — they become effectively zero-dimensional (pointlike). Consequently, the quantum theory describing the low energy limit is a theory that describes the dynamics of these points moving in spacetime, rather than strings. Such theories are called quantum field theories. However, since string theory also describes gravitational interactions, one expects the low-energy theory to describe particles moving in gravitational backgrounds. Finally, since superstring string theories are supersymmetric, one expects to see supersymmetry appearing in the low-energy approximation. These three facts imply that the low-energy approximation to a superstring theory is a supergravity theory.

### Supergravity theories

The possible supergravity theories were classified by Werner Nahm in the 1970s. In 10 dimensions, there are only two supergravity theories, which are denoted Type IIA and Type IIB. This similar denomination is not a coincidence; the Type IIA string theory has the Type IIA supergravity theory as its low-energy limit and the Type IIB string theory gives rise to Type IIB supergravity. The heterotic SO(32) and heterotic

*E*_{8}×*E*_{8}In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

string theories

*also* reduce to Type IIA and Type IIB supergravity in the low-energy limit. This suggests that there may indeed be a relation between the heterotic/Type I theories and the Type II theories.

In 1994, Edward Witten outlined the following relationship: The Type IIA supergravity (corresponding to the heterotic SO(32) and Type IIA string theories) can be obtained by dimensional reduction from the single unique eleven-dimensional supergravity theory. This means that if one studied supergravity on an eleven-dimensional spacetime that looks like the product of a ten-dimensional spacetime with another very small one-dimensional manifold, one gets the Type IIA supergravity theory. (And the Type IIB supergravity theory can be obtained by using T-duality.) However, eleven-dimensional supergravity is not consistent on its own — it does not make sense at extremely high energy, and likely requires some form of completion. It seems plausible, then, that there is some quantum theory — which Witten dubbed M-theory — in eleven-dimensions which gives rise at low energies to eleven-dimensional supergravity, and is related to ten-dimensional string theory by dimensional reduction. Dimensional reduction to a circle yields the Type IIA string theory, and dimensional reduction to a line segment yields the heterotic SO(32) string theory.

### Same underlying theory

M-theory would implement the notion that all of the different string theories are different special cases.

### Recent developments

In late 2007, Bagger and Lambert set off renewed interest in M-theory with the discovery of a candidate

LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

description of coincident M2-branes, based on a

non-associativeIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...

generalization of

Lie AlgebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

, Nambu 3-algebra or Filippov 3-algebra. Practitioners hope the Bagger–Lambert–Gustavsson action will provide the long-sought microscopic description of M-theory.

## Nomenclature

When Edward Witten named M-theory, he did not specify what the

*M* stood for—perhaps because the nascent theory wasn't fully defined. Some, including Sheldon Glashow, speculate that Witten chose the letter because it resembles an inverted

*W*. According to Witten, "

*M* can stand variously for 'magic', 'mystery', or 'matrix', according to one's taste."

Faced with this ambiguous initial, countless scientists and commentators have offered their own expansions of the

*M*—some sincere, others facetious.

*M* should stand for

membraneIn theoretical physics, a membrane, brane, or p-brane is a spatially extended mathematical concept that appears in string theory and related theories...

, say some. Meanwhile,

Michio Kakuis an American theoretical physicist, the Henry Semat Professor of Theoretical Physics in the City College of New York of City University of New York, the co-founder of string field theory, and a "communicator" and "popularizer" of science...

, Michael Duff,

Neil TurokNeil Geoffrey Turok is the Director of Perimeter Institute for Theoretical Physics. He is the son of Mary and Ben Turok, activists in the anti-apartheid movement and the African National Congress.-Career:...

, and others suggest

*mother* or

*master* (i.e., the "mother of all theories" or the "master theory").

Although Witten coined the term

*M-theory* to refer to his model of an eleven-dimensional universe, other scientists have generalized the moniker for application to any of various meta-theories involving string theory and

brane cosmologyBrane cosmology refers to several theories in particle physics and cosmology motivated by, but not exclusively derived from, superstring theory and M-theory.-Brane and bulk:...

. (

Ashoke SenAshoke Sen , FRS, is an Indian theoretical physicist. He has made a number of major original contributions to the subject of string theory, including his landmark paper on strong-weak coupling duality or S-duality, which was influential in changing the course of research in the field...

proposed

*u-theory* (ur, '

überÜber comes from the German language. It has one umlaut. It is a cognate of both Latin super and Greek ὑπέρ...

', 'ultimate', 'underlying', or perhaps 'unified') as a more distinctive appellation.) When unqualified,

*M-theory* now usually denotes this more general definition, rather than the one Witten originally advanced.

## M-theory and membranes

In the standard string theories, strings are assumed to be the single fundamental constituent of the universe. M-theory adds another fundamental constituent - membranes. Like the tenth spatial dimension, the approximate equations in the original five superstring models proved too weak to reveal membranes.

### P-branes

A membrane, or brane, is a multidimensional object, usually called a P-brane, with P referring to the number of dimensions in which it exists. The value of 'P' can range from zero to nine, thus giving branes dimensions from zero (0-brane ≡ point particle) to nine - five more than the world we are accustomed to inhabiting. The inclusion of p-branes does not render previous work in string theory wrong on account of not taking note of these P-branes. P-branes are much more massive ("heavier") than strings, and when all higher-dimensional P-branes are much more massive than strings, they can be ignored, as researchers had done unknowingly in the 1970s.

### Strings with "loose ends"

Shortly after Witten's breakthrough in 1995,

Joseph PolchinskiJoseph Polchinski is a physicist working on string theory. He graduated from Canyon del Oro High School in Tucson, Arizona in 1971, obtained his B.S. degree from Caltech in 1975, and his Ph.D. from the University of California, Berkeley in 1980 under the supervision of Stanley Mandelstam...

of the

University of California, Santa BarbaraThe University of California, Santa Barbara, commonly known as UCSB or UC Santa Barbara, is a public research university and one of the 10 general campuses of the University of California system. The main campus is located on a site in Goleta, California, from Santa Barbara and northwest of Los...

discovered a fairly obscure feature of string theory. He found that in certain situations the endpoints of strings (strings with "loose ends") would not be able to move with complete freedom as they were attached, or stuck within certain regions of space. Polchinski then reasoned that if the endpoints of open strings are restricted to move within some p-dimensional region of space, then that region of space must be occupied by a p-brane. These type of "sticky" branes are called

Dirichlet-P-branesIn string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Hořava in 1989...

, or D-P-branes. His calculations showed that the newly discovered D-P-branes had exactly the right properties to be the objects that exert a tight grip on the open string endpoints, thus holding down these strings within the p-dimensional region of space they fill.

### Strings with closed loops

Not all strings are confined to p-branes. Strings with closed loops, like the

gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

, are completely free to move from membrane to membrane. Of the four force carrier particles, the

gravitonIn physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

is unique in this way. Researchers speculate that this is the reason why investigation through the weak force, the strong force, and the electromagnetic force have not hinted at the possibility of extra dimensions. These force carrier particles are strings with endpoints that confine them to their p-branes. Further testing is needed in order to show that extra spatial dimensions indeed exist through experimentation with gravity.

## Membrane interactions

One of the reasons M-theory is so difficult to formulate is that the numbers of different types of membranes in the various dimensions increases exponentially. For example once one gets to 3 dimensional surfaces, one has to deal with solid objects with

knotA knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object—the "load"...

-shaped holes, and then one needs the whole of

knot theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

just to classify them.

Since M-theory is thought to operate in 11 dimensions this problem then becomes very difficult. But just like

string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

, in order for the theory to satisfy causality, the theory must be local, and so the

topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

changing must occur at a single point. The basic orientable 2-brane interactions are easy to show. Orientable 2-branes are

toriIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

with multiple holes cut out of them.

## Matrix theory

The original formulation of M-theory was in terms of a (relatively) low-energy effective field theory, called 11-dimensional

SupergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...

. Though this formulation provided a key link to the low-energy limits of string theories, it was recognized that a full high-energy formulation (or "UV-completion") of M-theory was needed.

### Analogy with water

For an analogy, the supergravity description is like treating water as a continuous, incompressible fluid. This is effective for describing long-distance effects such as waves and currents, but inadequate to understand short-distance/high-energy phenomena such as evaporation, for which a description of the underlying molecules is needed. What, then, are the underlying degrees of freedom of M-theory?

### BFSS model

BanksTom Banks is a theoretical physicist at University of California, Santa Cruz and a professor at Rutgers University. His work centers around string theory and its applications to high energy particle physics and cosmology. He received his Ph.D...

,

FischlerWilly Fischler born in 1949 in Antwerpen, Belgium is a theoretical physicist and string theorist. He is currently the Jane and Roland Blumberg Centennial Professor of Physics at the University of Texas at Austin, where he is affiliated with the Weinberg theory group.Fischler is, among other things,...

,

ShenkerStephen Hart Shenker is an American theoretical physicist who works on string theory. He is a professor at Stanford University and former director of the Stanford Institute for Theoretical Physics. His brother Scott Shenker is a computer scientist...

and

SusskindLeonard Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...

(BFSS) conjectured that Matrix theory could provide the answer. They demonstrated that a theory of 9 very large matrices, evolving in time, could reproduce the supergravity description at low energy, but take over for it as it breaks down at high energy. While the supergravity description assumes a continuous space-time, Matrix theory predicts that, at short distances, non-commutative geometry takes over, somewhat similar to the way the continuum of water breaks down at short distances in favor of the graininess of molecules.

### IKKT model

Another matrix string theory equivalent to Type IIB string theory was constructed in 1996 by Ishibashi, Kawai, Kitazawa, and Tsuchiya.

## Mysterious duality

A conjecture developed by

Cumrun VafaCumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...

,

Amer IqbalAmer Iqbal, Ph.D., is a Pakistani theoretical physicist. He is primarily known for his work in string theory and mathematical physics. He is the associate professor of Physics at the School of Science and Engineering at the Lahore University of Management Sciences...

, and Andrew Neitzke in 2001, called "mysterious duality", concerns similarities between M-theory and

del Pezzo surfaceIn mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class...

s.

It concerns a set of mathematical similarities between objects and laws describing M-theory on

*k*-dimensional

toriIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

(i.e. type II superstring theory on

**T**^{k − 1} for

*k* > 0) on one side, and

geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

of del Pezzo surfaces (for example, the

cubic surfaceA cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...

s) on the other side. The main observation is that the

large diffeomorphismIn mathematics and theoretical physics, a large diffeomorphism is a diffeomorphism that cannot be continuously connected to the identity diffeomorphism ....

s of del Pezzo surfaces match the

Weyl groupIn mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

of the U-duality group of the corresponding compactification of M-theory. The elements of the second homology of the del Pezzo surfaces are mapped to various BPS objects of different dimensions in M-theory.

The

complex projective plane **P**^{2}**C** is related to M-theory in 11 dimensions. When

*k* points are blown-up, the del Pezzo surface describes M-theory on a

*k*-torus, and the exceptional del Pezzo surface, namely

**P**^{1}**C** ×

**P**^{1}**C**, is connected with type IIB string theory in 10 dimensions.