ten-dimensional superstring theory but rather eleven-dimensional M-theory. . First, many of the most important questions in string theory, in particular how to. String theory is not, in contrast to general relativity and quantum field the- ory, a theory in the strict sense. There is, e.g., no axiomatic formulation and there is no. algebraic approaches to problems in M-theory, Robert Berman on appli- cations of . String theory has been said to be“21th century physics cast into the 20th.

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abstract and seemingly remote from the real world, M theory already has String theory is the most promising approach to quantum gravity [4]. This is a series of lectures on superstring/M-theory for cosmologists. It is definitely not a technical introduction to M-theory and almost all technical details will be. Abstract. String theory, ot its modern incarnation M-theory, gives a huge attempts to understand string theory quantum mechanics has been the foundation.

Type I string theory turns out to be equivalent by S-duality to the SO 32 heterotic string theory. String theory may provide a concrete model for the realization of inflation — a period of dramatic exponential expansion that the universe is likely to have experienced at the earliest times…. Additional sources 1. D1-brane , D3-brane , D5-brane. 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.

One of the vibrational states of a string gives rise to the graviton , a quantum mechanical particle that carries gravitational force.

There are several versions of string theory: The different theories allow different types of strings, and the particles that arise at low energies exhibit different 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. This theory, like its string theory predecessors, is an example of a quantum theory of gravity.

It describes a force just like the familiar gravitational force subject to the rules of quantum mechanics. In everyday life, there are three familiar dimensions of space: 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. 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.

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 , 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.

Compactification is one way of modifying the number of dimensions in a physical theory. 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. 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 SO 32 heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality.

Another relationship between different string theories is T-duality. Here one considers strings propagating around a circular extra dimension.

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 p and winding number n in one description, it will have momentum n and winding number 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.

In general, the term duality refers to a situation where two seemingly different physical systems 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. 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. 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 energies , the superstring theories are approximated by supergravity in ten spacetime dimensions. Similarly, M-theory is approximated at low energies by supergravity in eleven dimensions.

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 p , these are called 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.

Physicists often study 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. 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. 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. The success of general relativity led to efforts to apply higher dimensional geometry to explain other forces.

In , work by Theodor Kaluza showed that by passing to five-dimensional spacetime, one can unify gravity and electromagnetism into a single force. 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 [ which? 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 forces 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.

New concepts and mathematical tools provided fresh insights into general relativity, giving rise to a period in the s—70s now known as the golden age of general relativity. 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. 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: 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. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.

In the first superstring revolution in , 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. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary Lagrangian.

In string theory, the possibilities are much more constrained: Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation.

In the late s, Claus Montonen and David Olive had conjectured a special property of certain physical theories. The strength with which the particles of this theory interact is measured by a number called the coupling constant. 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 [25] by spin-moment. In the s, 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. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent.

String theory extends ordinary particle physics by replacing zero-dimensional point particles by one-dimensional objects called strings. In the late s, 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 by Paul Dirac , [30] and they were reconsidered by a small but enthusiastic group of physicists in the s.

Supersymmetry severely restricts the possible number of dimensions of a brane. In , Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes. Shortly after this discovery, 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. 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. In , 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. 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. 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 , 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. Speaking at the string theory conference at the University of Southern California in , 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. Their work shed light on the mathematical structure of M-theory and suggested possible ways of connecting M-theory to real world physics. 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.

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.

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.

In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a 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. One important [ why? 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.

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.

In geometry, it is often useful to introduce coordinates. For example, in order to study the geometry of the Euclidean plane , one defines the coordinates x and y as the distances between any point in the plane and a pair of 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.

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.

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 xy is not necessarily equal to 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.

In a paper from , 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. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.

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. Quantum field theories are also used throughout condensed matter physics to model particle-like objects called quasiparticles. It is closely related to hyperbolic space , which can be viewed as a disk as illustrated on the left.

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. 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.

Time runs along the vertical direction in this picture. As with the hyperbolic plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface. 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. 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.

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. In this example, the spacetime of the gravitational theory is effectively seven-dimensional hence the notation AdS 7 , and there are four additional " compact " dimensions encoded by the S 4 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. 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 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 , 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.

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.

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.

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 s because of its applications to knot theory. 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. Johnson More generally, this torus may be taken to be an elliptic curve and this may vary over the 9d base space as an elliptic fibration.

Applying T-duality to one of the compact direction yields a dimensional theory which may now be thought of as encoded by a dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity vacuum is the input data that F-theory is concerned with. By following through the above diagram, one finds how this determines the coupling constant in the type II string theory:.

Johnson 97 , Blumenhagen Seet also at cubical structure in M-theory. M-theory on G2-manifolds. M-theory on Calabi-Yau manifolds. The AdS-CFT duality for the black M5-brane may be turned around to deduce from the 6d 2,0 -superconformal QFT on the M5-brane scattering amplitudes in the dimensional bulk -spacetime, hence in putative M-theory. This approach to computing putative M-theory scattering amplitudes is due to ChesterPerlmutter M2-brane , M5-brane.

Horava-Witten theory , M9-brane. M-theory on 8-manifolds.

Diaconescu-Moore-Witten anomaly. First indications for M-theory came from the supermembrane Green-Schwarz sigma-model now called the M2-brane. A comprehensive collection of early articles is in. For some time though the success of string theory in dimensions caused resistence to the idea of a theory of membranes in dimensions, an account is in Duff 99 and in brevity on the first pages of. A public talk announcing the conjecture that the strong-coupling limit of type IIA string theory is dimensional supergravity KK-compactified on a circle is at 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.

The argument that the regularized M2-brane worldvolume theory is the BFSS matrix model is discussed in. A11 arXiv: Despite the magic and mystery, the relation to the original abbreviation for membrane-theory was highlighted again for instance in.

Early articles clarifying the relation to type II string theory now known as F-theory include.

B arXiv: Discussion of the cohomological charge quantization in type II RR-fields as cocycles in KR-theory in relation to the M-theory supergravity C-field is in.

For more on this perspective as 10d type II as a self-dual higher gauge theory in the boudnary of a kind of d Chern-Simons theory is in.

More complete discussion of the decomposition of the supergravity C-field as one passes from 11d to 10d is in. Discussion of M-theory as arising from type II string theory via the effect of D0-branes is in. Discussion of phenomena of M-theory in higher geometry and generalized cohomology is in. See also the references at exceptional generalized geometry. A-type nodal curve cycle degenertion locus of elliptic fibration Sen 97, section 2.

SU - gauge enhancement. D6-brane with O6-planes. D7-branes with O7-planes. D-type nodal curve cycle degenertion locus of elliptic fibration Sen 97, section 3.

SO - gauge enhancement. E6 -, E7 -, E8 - gauge enhancement.