Introduction to Strings and Branes

D16 , arXiv: A19 , arXiv: B39 , arXiv: Progress and Problems , John H. What can we hope to learn? November 24 - 29, C33 S67, arXiv: Carnegie Institution Centennial Symposium in November A19S1 , arXiv: Dirac Centennial Symposium, Tallahassee, Dec.

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Sackler Colloquium on Challenges to the Standard Paradigm. Beyond the Desert , Oulu Finland , June D73 , arXiv: Alan Kostelecky, Stuart Samuel , Phys. 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.

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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.

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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.

Strings and Branes

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.

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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.

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. Typically, such models are based on the idea of compactification. By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles, [64] usually 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.

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. 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. In one approach to M-theory phenomenology, theorists assume that the seven extra dimensions of M-theory are shaped like a G 2 manifold. This is a special kind of seven-dimensional shape constructed by mathematician Dominic Joyce of the University of Oxford. For example, physicists and mathematicians often assume that space has a mathematical property called smoothness , but this property cannot be assumed in the case of a G 2 manifold if one wishes to recover the physics of our four-dimensional world.

But it is still deeply satisfying to be assured that electromagnetism is the complete theory that in principle underlies and unifies all chemical phenomena. We will have more to say about the idea of unified theories in what follows. Like electromagnetism, every other fundamental interaction must have its own mediating particle. Precisely three other kinds of fundamental interactions are known. One of these is the familiar force of gravitation, while the other two are nuclear forces that were only discovered in this century: The former is, in particular, responsible for the binding of protons and neutrons to make up the nucleus of an atom, while the latter is a totally distinct force and gives rise to phenomena such as radioactive decay.

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The weak force is the only one which violates left-right symmetry or "parity". Gravitation, like electromagnetism, is a long-range force, which is why these two have been known for a long time. The two nuclear forces are short-ranged, and are therefore not commonly observed at everyday distance scales. So, we may ask what is the elementary particle associated to each of these interactions. To gravitation, we associate the "graviton", a particle that has not been directly observed but is strongly believed to exist.

To the strong nuclear force we associate a set of particles called "gluons" because of their glue-like binding properties, and to the weak nuclear force we associate another set of particles called "W and Z bosons". There is compelling evidence for the existence of gluons, while W and Z bosons produced at accelerators have been tracked directly.