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The appearance of various types of branes is discussed. The connection to BPS vacua of supergravity is presented afterwards. Some issues of warped compactifications are also addressed.

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These comprise corrections to gravitational interactions as well as the cosmological constant problem. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username.

Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. 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.

Branes and applications in string theory and M-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: electromagnetism, the strong and weak nuclear forces , 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.

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.

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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: by the s, physicists had argued that there were only five consistent supersymmetric versions of the theory.

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.


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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.

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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.


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  8. 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.

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    Lecture 10 - String Theory and M-Theory

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