String theory

String theory

String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity.[1] It is a contender for a theory of everything (TOE), a manner of describing the known fundamental forces and matter in a mathematically complete system. The theory has yet to make novel experimental predictions at accessible energy scales, leading some scientists to claim that it cannot be considered a part of science.[2]

String theory mainly posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines ("strings"). The earliest string model, the bosonic string, incorporated only bosons, although this view developed to the superstring theory, which posits that a connection (a "supersymmetry") exists between bosons and fermions. String theories also require the existence of several extra, unobservable dimensions to the universe, in addition to the four known spacetime dimensions.

The theory has its origins in an effort to understand the strong force, the dual resonance model (1969). Subsequently five different superstring theories were developed that incorporated fermions and possessed other properties necessary for a theory of everything. Since the mid 1990s, particularly due to insights from dualities shown to relate the five theories, an eleven-dimensional theory called M-theory is believed to encompass all of the previously-distinct superstring theories.

Many theoretical physicists (e.g. Hawking, Witten, Maldacena and Susskind) believe that string theory is a step towards the correct fundamental description of nature. This is because string theory allows for the consistent combination of quantum field theory and general relativity, agrees with general insights in quantum gravity (such as the holographic principle and Black hole thermodynamics), and because it has passed many non-trivial checks of its internal consistency.[3][4][5][6][unreliable source?] According to Stephen Hawking in particular, "M-theory is the only candidate for a complete theory of the universe."[7] Nevertheless, other physicists (e.g. Feynman and Glashow) have criticized string theory for not providing any quantitative experimental predictions.[8][9]



String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but made up of 1-dimensional strings. These strings can oscillate, giving the observed particles their flavor, charge, mass and spin. String theories also include objects more general than strings, called branes. The word brane, derived from "membrane", refers to a variety of interrelated objects, such as D-branes, black p-branes and Neveu–Schwarz 5-branes. These are extended objects that are charged sources for differential form generalizations of the vector potential electromagnetic field. These objects are related to one another by a variety of dualities. Black hole-like black p-branes are identified with D-branes, which are endpoints for strings, and this identification is called Gauge-gravity duality. Research on this equivalence has led to new insights on quantum chromodynamics, the fundamental theory of the strong nuclear force.[10][11][12][13] The strings make closed loops unless they encounter D-branes, where they can open up into 1-dimensional lines. The endpoints of the string cannot break off the D-brane, but they can slide around on it.

Levels of magnification:
1. Macroscopic level – Matter
2. Molecular level
3. Atomic level – Protons, neutrons, and electrons
4. Subatomic level – Electron
5. Subatomic level – Quarks
6. String level

Since string theory is widely believed to be a consistent theory of quantum gravity, many hope that it correctly describes our universe, making it a theory of everything. There are known configurations which describe all the observed fundamental forces and matter but with a zero cosmological constant and some new fields.[14] There are other configurations with different values of the cosmological constant, which are metastable but long-lived. This leads many to believe that there is at least one metastable solution which is quantitatively identical with the standard model, with a small cosmological constant, which contains dark matter and a plausible mechanism for cosmic inflation. It is not yet known whether string theory has such a solution, nor how much freedom the theory allows to choose the details.

The full theory does not yet have a satisfactory definition in all circumstances, since the scattering of strings is most straightforwardly defined by a perturbation theory. The complete quantum mechanics of high dimensional branes is not easily defined, and the behavior of string theory in cosmological settings (time-dependent backgrounds) is not fully worked out. It is also not clear if there is any principle by which string theory selects its vacuum state, the spacetime configuration which determines the properties of our universe (see string theory landscape).

As is the case for any other quantum theory of gravity, it is widely believed that testing the theory directly would require prohibitively expensive feats of engineering. Although direct experimental testing of string theory involves grand explorations and development in engineering, there are several indirect experiments that may support string theory.

Basic properties

String theory can be formulated in terms of an action principle, either the Nambu-Goto action or the Polyakov action, which describes how strings move through space and time. In the absence of external interactions, string dynamics are governed by tension and kinetic energy, which combine to produce oscillations. The quantum mechanics of strings implies these oscillations take on discrete vibrational modes, the spectrum of the theory.

On distance scales larger than the string radius, each oscillation mode behaves as a different species of particle, with its mass, spin and charge determined by the string's dynamics. Splitting and recombination of strings correspond to particle emission and absorption, giving rise to the interactions between particles.

An analogy for strings' modes of vibration is a guitar string's production of multiple but distinct musical notes. In the analogy, different notes correspond to different particles. The only difference is the guitar is only 2-dimensional; you can strum it up, and down. In actuality the guitar strings would be every dimension, and the strings could vibrate in any direction, meaning that the particles could move through not only our dimension, but other dimensions as well.

String theory includes both open strings, which have two distinct endpoints, and closed strings making a complete loop. The two types of string behave in slightly different ways, yielding two different spectra. For example, in most string theories, one of the closed string modes is the graviton, and one of the open string modes is the photon. Because the two ends of an open string can always meet and connect, forming a closed string, there are no string theories without closed strings.

The earliest string model, the bosonic string, incorporated only bosons. This model describes, in low enough energies, a quantum gravity theory, which also includes (if open strings are incorporated as well) gauge fields such as the photon (or, more generally, any gauge theory). However, this model has problems. Most importantly, the theory has a fundamental instability, believed to result in the decay (at least partially) of spacetime itself. Additionally, as the name implies, the spectrum of particles contains only bosons, particles which, like the photon, obey particular rules of behavior. Roughly speaking, bosons are the constituents of radiation, but not of matter, which is made of fermions. Investigating how a string theory may include fermions in its spectrum led to the invention of supersymmetry, a mathematical relation between bosons and fermions. String theories which include fermionic vibrations are now known as superstring theories; several different kinds have been described, but all are now thought to be different limits of M-theory.

Some qualitative properties of quantum strings can be understood in a fairly simple fashion. For example, quantum strings have tension, much like regular strings made of twine; this tension is considered a fundamental parameter of the theory. The tension of a quantum string is closely related to its size. Consider a closed loop of string, left to move through space without external forces. Its tension will tend to contract it into a smaller and smaller loop. Classical intuition suggests that it might shrink to a single point, but this would violate Heisenberg's uncertainty principle. The characteristic size of the string loop will be a balance between the tension force, acting to make it small, and the uncertainty effect, which keeps it "stretched". Consequently, the minimum size of a string is related to the string tension.


A point-like particle's motion may be described by drawing a graph of its position (in one or two dimensions of space) against time. The resulting picture depicts the worldline of the particle (its 'history') in spacetime. By analogy, a similar graph depicting the progress of a string as time passes by can be obtained; the string (a one-dimensional object — a small line — by itself) will trace out a surface (a two-dimensional manifold), known as the worldsheet. The different string modes (representing different particles, such as photon or graviton) are surface waves on this manifold.

A closed string looks like a small loop, so its worldsheet will look like a pipe or, more generally, a Riemann surface (a two-dimensional oriented manifold) with no boundaries (i.e. no edge). An open string looks like a short line, so its worldsheet will look like a strip or, more generally, a Riemann surface with a boundary.

Interaction in the subatomic world: world lines of point-like particles in the Standard Model or a world sheet swept up by closed strings in string theory

Strings can split and connect. This is reflected by the form of their worldsheet (more accurately, by its topology). For example, if a closed string splits, its worldsheet will look like a single pipe splitting (or connected) to two pipes (often referred to as a pair of pants — see drawing at right). If a closed string splits and its two parts later reconnect, its worldsheet will look like a single pipe splitting to two and then reconnecting, which also looks like a torus connected to two pipes (one representing the ingoing string, and the other — the outgoing one). An open string doing the same thing will have its worldsheet looking like a ring connected to two strips.

Note that the process of a string splitting (or strings connecting) is a global process of the worldsheet, not a local one: locally, the worldsheet looks the same everywhere and it is not possible to determine a single point on the worldsheet where the splitting occurs. Therefore these processes are an integral part of the theory, and are described by the same dynamics that controls the string modes.

In some string theories (namely, closed strings in Type I and some versions of the bosonic string), strings can split and reconnect in an opposite orientation (as in a Möbius strip or a Klein bottle). These theories are called unoriented. Formally, the worldsheet in these theories is a non-orientable surface.


Before the 1990s, string theorists believed there were five distinct superstring theories: open type I, closed type I, closed type IIA, closed type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8).[15] The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the one whose low energy limit, with ten spacetime dimensions compactified down to four, matched the physics observed in our world today. It is now believed that this picture was incorrect and that the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory (thought to be M-theory). These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.

These dualities link quantities that were also thought to be separate. Large and small distance scales, as well as strong and weak coupling strengths, are quantities that have always marked very distinct limits of behavior of a physical system in both classical field theory and quantum particle physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related. T-duality relates the large and small distance scales between string theories, whereas S-duality relates strong and weak coupling strengths between string theories. U-duality links T-duality and S-duality.

String theories
Type Spacetime dimensions
Bosonic 26 Only bosons, no fermions, meaning only forces, no matter, with both open and closed strings; major flaw: a particle with imaginary mass, called the tachyon, representing an instability in the theory.
I 10 Supersymmetry between forces and matter, with both open and closed strings; no tachyon; group symmetry is SO(32)
IIA 10 Supersymmetry between forces and matter, with only closed strings bound to D-branes; no tachyon; massless fermions are non-chiral
IIB 10 Supersymmetry between forces and matter, with only closed strings bound to D-branes; no tachyon; massless fermions are chiral
HO 10 Supersymmetry between forces and matter, with closed strings only; no tachyon; heterotic, meaning right moving and left moving strings differ; group symmetry is SO(32)
HE 10 Supersymmetry between forces and matter, with closed strings only; no tachyon; heterotic, meaning right moving and left moving strings differ; group symmetry is E8×E8

Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional spacetime (called the bulk), while open strings have their ends attached to D-branes, which are membranes of lower dimensionality (their dimension is odd — 1, 3, 5, 7 or 9 — in type IIA and even — 0, 2, 4, 6 or 8 — in type IIB, including the time direction).

Extra dimensions

Number of dimensions

An intriguing feature of string theory is that it predicts extra dimensions. In classical string theory the number of dimensions is not fixed by any consistency criterion. However in order to make a consistent quantum theory, string theory is required to live in a spacetime of the so-called "critical dimension": we must have 26 spacetime dimensions for the bosonic string and 10 for the superstring. This is necessary to ensure the vanishing of the conformal anomaly of the worldsheet conformal field theory. Modern understanding indicates that there exist less-trivial ways of satisfying this criteria. Cosmological solutions exist in a wider variety of dimensionalities, and these different dimensions are related by dynamical transitions. The dimensions are more precisely different values of the "effective central charge", a count of degrees of freedom which reduces to dimensionality in weakly curved regimes.[16]

One such theory is the 11-dimensional M-theory, which requires spacetime to have eleven dimensions,[17] as opposed to the usual three spatial dimensions and the fourth dimension of time. The original string theories from the 1980s describe special cases of M-theory where the eleventh dimension is a very small circle or a line, and if these formulations are considered as fundamental, then string theory requires ten dimensions. But the theory also describes universes like ours, with four observable spacetime dimensions, as well as universes with up to 10 flat space dimensions, and also cases where the position in some of the dimensions is not described by a real number, but by a completely different type of mathematical quantity. So the notion of spacetime dimension is not fixed in string theory: it is best thought of as different in different circumstances.[18]

Nothing in Maxwell's theory of electromagnetism or Einstein's theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions "by both hands", and this number is fixed and independent of potential energy. String theory allows one to relate the number of dimensions to scalar potential energy. Technically, this happens because a gauge anomaly exists for every separate number of predicted dimensions, and the gauge anomaly can be counteracted by including nontrivial potential energy into equations to solve motion. Furthermore, the absence of potential energy in the "critical dimension" explains why flat spacetime solutions are possible.

This can be better understood by noting that a photon included in a consistent theory (technically, a particle carrying a force related to an unbroken gauge symmetry) must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents the photon. This energy includes a contribution from the Casimir effect, namely from quantum fluctuations in the string. The size of this contribution depends on the number of dimensions, since for a larger number of dimensions there are more possible fluctuations in the string position. Therefore, the photon in flat spacetime will be massless—and the theory consistent—only for a particular number of dimensions.[19] When the calculation is done, the critical dimensionality is not four as one may expect (three axes of space and one of time). The subset of X is equal to the relation of photon fluctuations in a linear dimension. Flat space string theories are 26-dimensional in the bosonic case, while superstring and M-theories turn out to involve 10 or 11 dimensions for flat solutions. In bosonic string theories, the 26 dimensions come from the Polyakov equation.[20] Starting from any dimension greater than four, it is necessary to consider how these are reduced to four dimensional spacetime.

Compact dimensions

Two different ways have been proposed to resolve this apparent contradiction. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra dimensions are so small as to be undetectable by present day experiments.

To retain a high degree of supersymmetry, these compactification spaces must be very special, as reflected in their holonomy. A 6-dimensional manifold must have SU(3) structure, a particular case (torsionless) of this being SU(3) holonomy, making it a Calabi–Yau space, and a 7-dimensional manifold must have G2 structure, with G2 holonomy again being a specific, simple, case. Such spaces have been studied in attempts to relate string theory to the 4-dimensional Standard Model, in part due to the computational simplicity afforded by the assumption of supersymmetry. More recently, progress has been made constructing more realistic compactifications without the degree of symmetry of Calabi–Yau or G2 manifolds.

A standard analogy for this is to consider multidimensional space as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. Indeed, think of a ball just small enough to enter the hose. Throwing such a ball inside the hose, the ball would move more or less in one dimension; in any experiment we make by throwing such balls in the hose, the only important movement will be one-dimensional, that is, along the hose. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions (and a fly flying in it would move in three dimensions). This "extra dimension" is only visible within a relatively close range to the hose, or if one "throws in" small enough objects. Similarly, the extra compact dimensions are only "visible" at extremely small distances, or by experimenting with particles with extremely small wavelengths (of the order of the compact dimension's radius), which in quantum mechanics means very high energies (see wave-particle duality).

Brane-world scenario

Another possibility is that we are "stuck" in a 3+1 dimensional (three spatial dimensions plus one time dimension) subspace of the full universe. Properly localized matter and Yang-Mills gauge fields will typically exist if the sub-space-time is an exceptional set of the larger universe.[21] These "exceptional sets" are ubiquitous in Calabi–Yau n-folds and may be described as subspaces without local deformations, akin to a crease in a sheet of paper or a crack in a crystal, the neighborhood of which is markedly different from the exceptional subspace itself. However, until the work of Randall and Sundrum,[22] it was not known that gravity too can be properly localized to a sub-spacetime. In addition, spacetime may be stratified, containing strata of various dimensions, allowing us to inhabit a 3+1-dimensional stratum -- such geometries occur naturally in Calabi–Yau compactifications.[23] Such sub-spacetimes are D-branes, hence such models are known as brane-world scenarios.

Effect of the hidden dimensions

In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza's early work demonstrated that general relativity in five dimensions actually predicts the existence of electromagnetism. However, because of the nature of Calabi–Yau manifolds, no new forces appear from the small dimensions, but their shape has a profound effect on how the forces between the strings appear in our four-dimensional universe. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.


Another key feature of string theory is the existence of D-branes. These are membranes of different dimensionality (anywhere from a zero dimensional membrane—which is in fact a point — and up, including 2-dimensional membranes, 3-dimensional volumes and so on).

D-branes are defined by the fact that worldsheet boundaries are attached to them. D-branes have mass, since they emit and absorb closed strings which describe gravitons, and — in superstring theoriescharge as well, since they couple to open strings which describe gauge interactions.

From the point of view of open strings, D-branes are objects to which the ends of open strings are attached. The open strings attached to a D-brane are said to "live" on it, and they give rise to gauge theories "living" on it (since one of the open string modes is a gauge boson such as the photon). In the case of one D-brane there will be one type of a gauge boson and we will have an Abelian gauge theory (with the gauge boson being the photon). If there are multiple parallel D-branes there will be multiple types of gauge bosons, giving rise to a non-Abelian gauge theory.

D-branes are thus gravitational sources, on which a gauge theory "lives". This gauge theory is coupled to gravity (which is said to exist in the bulk), so that normally each of these two different viewpoints is incomplete.

Gauge-gravity duality

Gauge-gravity duality is a conjectured duality between a quantum theory of gravity in certain cases and gauge theory in a lower number of dimensions. This means that each predicted phenomenon and quantity in one theory has an analogue in the other theory, with a "dictionary" translating from one theory to the other.

Description of the duality

In certain cases the gauge theory on the D-branes is decoupled from the gravity living in the bulk; thus open strings attached to the D-branes are not interacting with closed strings. Such a situation is termed a decoupling limit.

In those cases, the D-branes have two independent alternative descriptions. As discussed above, from the point of view of closed strings, the D-branes are gravitational sources, and thus we have a gravitational theory on spacetime with some background fields. From the point of view of open strings, the physics of the D-branes is described by the appropriate gauge theory. Therefore in such cases it is often conjectured that the gravitational theory on spacetime with the appropriate background fields is dual (i.e. physically equivalent) to the gauge theory on the boundary of this spacetime (since the subspace filled by the D-branes is the boundary of this spacetime). So far, this duality has not been proven in any cases, so there is also disagreement among string theorists regarding how strong the duality applies to various models.

Examples and intuition

The best known example and the first one to be studied is the duality between Type IIB superstring on AdS5 × S5 (a product space of a five-dimensional Anti de Sitter space and a five-sphere) on one hand, and N = 4 supersymmetric Yang–Mills theory on the four-dimensional boundary of the Anti de Sitter space (either a flat four-dimensional spacetime R3,1 or a three-sphere with time S3 × R). This is known as the AdS/CFT correspondence,[24][25][26][27] a name often used for Gauge / gravity duality in general.

This duality can be thought of as follows: suppose there is a spacetime with a gravitational source, for example an extremal black hole.[28] When particles are far away from this source, they are described by closed strings (i.e. a gravitational theory, or usually supergravity). As the particles approach the gravitational source, they can still be described by closed strings; alternatively, they can be described by objects similar to QCD strings,[29][30][31] which are made of gauge bosons (gluons) and other gauge theory degrees of freedom.[32] So if one is able (in a decoupling limit) to describe the gravitational system as two separate regions — one (the bulk) far away from the source, and the other close to the source — then the latter region can also be described by a gauge theory on D-branes. This latter region (close to the source) is termed the near-horizon limit, since usually there is an event horizon around (or at) the gravitational source.

In the gravitational theory, one of the directions in spacetime is the radial direction, going from the gravitational source and away (towards the bulk). The gauge theory lives only on the D-brane itself, so it does not include the radial direction: it lives in a spacetime with one less dimension compared to the gravitational theory (in fact, it lives on a spacetime identical to the boundary of the near-horizon gravitational theory). Let us understand how the two theories are still equivalent:

The physics of the near-horizon gravitational theory involves only on-shell states (as usual in string theory), while the field theory includes also off-shell correlation function. The on-shell states in the near-horizon gravitational theory can be thought of as describing only particles arriving from the bulk to the near-horizon region and interacting there between themselves. In the gauge theory these are "projected" onto the boundary, so that particles which arrive at the source from different directions will be seen in the gauge theory as (off-shell) quantum fluctuations far apart from each other, while particles arriving at the source from almost the same direction in space will be seen in the gauge theory as (off-shell) quantum fluctuations close to each other. Thus the angle between the arriving particles in the gravitational theory translates to the distance scale between quantum fluctuations in the gauge theory. The angle between arriving particles in the gravitational theory is related to the radial distance from the gravitational source at which the particles interact: the larger the angle, the closer the particles have to get to the source in order to interact with each other. On the other hand, the scale of the distance between quantum fluctuations in a quantum field theory is related (inversely) to the energy scale in this theory, so small radius in the gravitational theory translates to low energy scale in the gauge theory (i.e. the IR regime of the field theory) while large radius in the gravitational theory translates to high energy scale in the gauge theory (i.e. the UV regime of the field theory).

A simple example to this principle is that if in the gravitational theory there is a setup in which the dilaton field (which determines the strength of the coupling) is decreasing with the radius, then its dual field theory will be asymptotically free, i.e. its coupling will grow weaker in high energies.

Contact with experiment

The mathematics of string theory may lead to new insights on quantum chromodynamics, a gauge theory which is the fundamental theory of the strong nuclear force. To this end, it is hoped that a gravitational theory dual to quantum chromodynamics will be found.[33]

A mathematical technique from string theory (the AdS/CFT correspondence) has been used to describe qualitative features of quark–gluon plasma behavior in relativistic heavy-ion collisions;[10][11][12][13] the physics, however, is strictly that of standard quantum chromodynamics, which has been quantitatively modeled by lattice QCD methods with good results.[34]

Other potential forms of evidence for string theory have been proposed. One would be the discovery of large cosmic strings in space, formed when the Big Bang "stretched" some strings to astronomical proportions. Other theories, however, predict cosmic strings with a different physical basis (topological defects in various fields). Novel phenomena consistent with versions of string theory may be observed with the newly built Large Hadron Collider. One would be indications of an anomalously large strength of gravity on a microscopic scale, which would be consistent with branes interacting across extra dimensions through leakage of gravitons from one to the other. The discovery of supersymmetry could also be considered evidence since string theory was the first theory to require it, though other theories incorporate supersymmetry as well. The absence of supersymmetric particles at energies accessible to the LHC would not necessarily disprove string theory, since supersymmetry could exist but still be outside the accelerator's range.

Predictability and Testability

Unsolved problems in physics
Is there a string theory vacuum which exactly describes everything in our universe? Is it uniquely determined by low energy data?

Finding a way to confirm string theory with our current technology is a major challenge.[35] Nevertheless, all string theory models are quantum mechanical, Lorentz invariant, unitary, and contain Einstein's General Relativity as a low energy limit.[36] Therefore, to falsify string theory, it would suffice to falsify quantum mechanics, Lorentz invariance, or general relativity.[37] Hence, string theory is falsifiable and meets the definition of scientific theory according to the Popperian criterion.


String harmonics

A unique prediction of string theory is the existence of string harmonics: at sufficiently high energies—probably near the quantum gravity scale—the string-like nature of particles would become obvious. There should be heavier copies of all particles corresponding to higher vibrational states of the string. But it is not clear how high these energies are. In the most likely case, they would be 1014 times higher than those accessible in the newest particle accelerator, the LHC, making this prediction impossible to test with any particle accelerator in the foreseeable future.


String theory as currently understood makes a series of predictions for the structure of the universe at the largest scales. Many phases in string theory have very large, positive vacuum energy [38]. Regions of the universe that are in such a phase will inflate exponentially rapidly in a process known as eternal inflation. As such, the theory predicts that most of the universe is very rapidly expanding. However, these expanding phases are not stable, and can decay via the nucleation of bubbles of lower vacuum energy. Since our local region of the universe is not very rapidly expanding, string theory predicts we are inside such a bubble. The spatial curvature of the "universe" inside the bubbles that form by this process is negative, a testable prediction [39]. Moreover, other bubbles will eventually form in the parent vacuum outside the bubble and collide with it. These collisions lead to potentially observable imprints on cosmology [40] [41]. However, it is possible that neither of these will be observed if the spatial curvature is too small and the collisions are too rare.

Supersymmetry breaking

A central problem for applications is that the best understood backgrounds of string theory preserve much of the supersymmetry of the underlying theory, which results in time-invariant spacetimes: currently string theory cannot deal well with time-dependent, cosmological backgrounds. However, several models have been proposed to predict supersymmetry breaking, most notably the KKLT model,[38] which incorporates branes and fluxes to make a metastable compactification.

AdS/CFT correspondence

AdS/CFT relates string theory to gauge theory, and allows contact with low energy experiments in quantum chromodynamics. This type of string theory, which only describes the strong interactions, is much less controversial today than string theories of everything (although two decades ago, it was the other way around).[42]

Coupling constants

Grand unification natural in string theories of everything requires that the coupling constants of the four forces meet at one point under renormalization group rescaling. This is also a falsifiable statement, but it is not restricted to string theory, but is shared by grand unified theories.[43] The LHC will be used both for testing AdS/CFT, and to check if the electroweakstrong unification does happen as predicted.[44]


Some critics of string theory say that it is a failure as a theory of everything.[45][46][47][48][49][50] Notable critics include Peter Woit, Lee Smolin, Philip Warren Anderson,[51] Sheldon Glashow,[52] Lawrence Krauss,[53] and Carlo Rovelli.[54] Some common criticisms include:

  1. Very high energies needed to test quantum gravity.
  2. Lack of uniqueness of predictions due to the large number of solutions.
  3. Lack of background independence.

High Energies

It is widely believed that any theory of quantum gravity would require extremely high energies to probe directly, higher by orders of magnitude than those that current experiments such as the Large Hadron Collider[55] can attain.

However, it may not be necessary to achieve higher than as of yet possible energy needed to test quantum gravity. An alternative testing methodology is being explored called gravity resonance spectroscopy. A recent advancement in that field involves the use of neutrons to avoid problems of electro-weak forces interfering with experiments. This is a new direction and viability of the testing method is still to-be-determined.[56]

Strings themselves are the size of the Planck scale which is twenty orders of magnitude smaller than a proton. For this reason huge energies would be needed to observe them directly.

Number of solutions

String theory as it is currently understood has a huge number of solutions, called string vacua,[38] and these vacua might be sufficiently diverse to accommodate almost any phenomena we might observe at lower energies.

The vacuum structure of the theory, called the string theory landscape (or the anthropic portion of string theory vacua), is not well understood. String theory contains an infinite number of distinct meta-stable vacua, and perhaps 10520 of these or more correspond to a universe roughly similar to ours — with four dimensions, a high planck scale, gauge groups, and chiral fermions. Each of these corresponds to a different possible universe, with a different collection of particles and forces.[38] What principle, if any, can be used to select among these vacua is an open issue. While there are no continuous parameters in the theory, there is a very large set of possible universes, which may be radically different from each other. It is also suggested that the landscape is surrounded by an even more vast swampland of consistent-looking semiclassical effective field theories, which are actually inconsistent.[citation needed]

Some physicists believe this is a good thing, because it may allow a natural anthropic explanation of the observed values of physical constants, in particular the small value of the cosmological constant.[57][58] The argument is that most universes contain values for physical constants which do not lead to habitable universes (at least for humans), and so we happen to live in the most "friendly" universe. This principle is already employed to explain the existence of life on earth as the result of a life-friendly orbit around the medium-sized sun among an infinite number of possible orbits (as well as a relatively stable location in the galaxy). However, the cosmological version of the anthropic principle remains highly controversial because it would be difficult if not impossible to Popper falsify; so many do not accept it as scientific.[citation needed]

Background independence

A separate and older criticism of string theory is that it is background-dependent — string theory describes perturbative expansions about fixed spacetime backgrounds. Although the theory has some background-independence — topology change is an established process in string theory, and the exchange of gravitons is equivalent to a change in the background — mathematical calculations in the theory rely on preselecting a background as a starting point. This is because, like many quantum field theories, much of string theory is still only formulated perturbatively, as a divergent series of approximations. Although nonperturbative techniques have progressed considerably — including conjectured complete definitions in spacetimes satisfying certain asymptotics — a full non-perturbative definition of the theory is still lacking. Some see background independence as a fundamental requirement of a theory of quantum gravity, particularly since general relativity is already background independent. Some hope that M-theory, or a non-perturbative treatment of string theory (such as "background independent open string field theory") will have a background-independent formulation.


Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein. The first person to add a fifth dimension to general relativity was German mathematician Theodor Kaluza in 1919, who noted that gravity in five dimensions describes both gravity and electromagnetism in four. In 1926, the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension--- it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions.

String theory was originally developed during the late 1960s and early 1970s as a never completely successful theory of hadrons, the subatomic particles like the proton and neutron which feel the strong interaction. In the 1960s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories which did not presume that they were composed of any fundamental particles, but would construct their interactions from self-consistency conditions on the S-matrix. The S-matrix approach was started by Werner Heisenberg in the 1940s as a way of constructing a theory which did not rely on the local notions of space and time, which Heisenberg believed break down at the nuclear scale. While the scale was off by many orders of magnitude, the approach he advocated was ideally suited for a theory of quantum gravity.

Working with experimental data, R. Dolen, D. Horn and C. Schmid[59] developed some sum rules for hadron exchange. When a particle and antiparticle scatter, virtual particles can be exchanged in two qualitatively different ways. In the s-channel, the two particles annihilate to make temporary intermediate states which fall apart into the final state particles. In the t-channel, the particles exchange intermediate states by emission and absorption. In field theory, the two contributions add together, one giving a continuous background contribution, the other giving peaks at certain energies. In the data, it was clear that the peaks were stealing from the background--- the authors interpreted this as saying that the t-channel contribution was dual to the s-channel one, meaning both described the whole amplitude and included the other.

The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude which had the property of Dolen-Horn-Schmid duality, later renamed world-sheet duality. The amplitude needed poles where the particles appear, on straight line trajectories, and there is a special mathematical function whose poles are evenly spaced on half the real line--- the Gamma function--- which was widely used in Regge theory. By manipulating combinations of Gamma functions, Veneziano was able to find a consistent scattering amplitude with poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at high energy. The amplitude could fit near-beam scattering data as well as other Regge type fits, and had a suggestive integral representation which could be used for generalization.

Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle which appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon. Miguel Virasoro and Joel Shapiro found a different amplitude now understood to be that of closed strings, while Ziro Koba and Holger Nielsen generalized Veneziano's integral representation to multiparticle scattering. Veneziano and Sergio Fubini introduced an operator formalism for computing the scattering amplitudes which was a forerunner of world-sheet conformal theory, while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on the states. Claud Lovelace calculated a loop amplitude, and noted that there is an inconsistency unless the dimension of the theory is 26. Charles Thorn, Peter Goddard and Richard Brower went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to 26.

In 1969 Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind recognized that the theory could be given a description in space and time in terms of strings. The scattering amplitudes were derived systematically from the action principle by Peter Goddard, Jeffrey Goldstone, Claudio Rebbi and Charles Thorn, giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the Virasoro conditions.

In 1970, Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. John Schwarz and André Neveu added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10. Stanley Mandelstam formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave a different formulation of the bosonic string, as a string field theory, with infinitely many particle types and with fields taking values not on points, but on loops and curves.

In 1974, Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle which obeyed the correct Ward identities to be a graviton. John Schwarz and Joel Scherk came to the same conclusion and made the bold leap to suggest that string theory was a theory of gravity, not a theory of hadrons. They reintroduced Kaluza–Klein theory as a way of making sense of the extra dimensions. At the same time, quantum chromodynamics was recognized as the correct theory of hadrons, shifting the attention of physicists and apparently leaving the bootstrap program in the dustbin of history.

String theory eventually made it out of the dustbin, but for the following decade all work on the theory was completely ignored. Still, the theory continued to develop at a steady pace thanks to the work of a handful of devotees. Ferdinando Gliozzi, Joel Scherk, and David Olive realized in 1976 that the original Ramond and Neveu Schwarz-strings were separately inconsistent and needed to be combined. The resulting theory did not have a tachyon, and was proven to have space-time supersymmetry by John Schwarz and Michael Green in 1981. The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively. In 1979, Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein equations of General Relativity, emerge from the Renormalization group equations for the two-dimensional field theory. Schwarz and Green discovered T-duality, and constructed two different superstring theories--- IIA and IIB related by T-duality, and type I theories with open strings. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices.

In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino. This led him, in collaboration with Luis Alvarez-Gaumé to study violations of the conservation laws in gravity theories with anomalies, concluding that type I string theories were inconsistent. Green and Schwarz discovered a contribution to the anomaly that Witten and Alvarez-Gaumé had missed, which restricted the gauge group of the type I string theory to be SO(32). In coming to understand this calculation, Edward Witten became convinced that string theory was truly a consistent theory of gravity, and he became a high-profile advocate. Following Witten's lead, between 1984 and 1986, hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution.

During this period, David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm discovered heterotic strings. The gauge group of these closed strings was two copies of E8, and either copy could easily and naturally include the standard model. Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten found that the Calabi-Yau manifolds are the compactifications which preserve a realistic amount of supersymmetry, while Lance Dixon and others worked out the physical properties of orbifolds, distinctive geometrical singularities allowed in string theory. Cumrun Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of mirror symmetry. David Gross and Vipul Periwal discovered that string perturbation theory was divergent in a way that suggested that new non-perturbative objects were missing.

In the 1990s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and identified these with the black-hole solutions of supergravity. These were understood to be the new objects suggested by the perturbative divergences, and they opened up a new field with rich mathematical structure. It quickly became clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the physical interpretation of the strings and branes was revealed--- they are a type of black hole. Leonard Susskind had incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly-excited string states with ordinary thermal black hole states. As suggested by 't Hooft, the fluctuations of the black hole horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all nearby objects too.

In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten gave a speech on string theory that essentially united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called M-theory. M-theory was also foreshadowed in the work of Paul Townsend at approximately the same time. The flurry of activity which began at this time is sometimes called the second superstring revolution.

During this period, Tom Banks, Willy Fischler Stephen Shenker and Leonard Susskind formulated a full holographic description of M-theory on IIA D0 branes,[60] the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the holographic principle. Andrew Strominger and Cumrun Vafa calculated the entropy of certain configurations of D-branes and found agreement with the semi-classical answer for extreme charged black holes. Petr Horava and Edward Witten found the eleven-dimensional formulation of the heterotic string theories, showing that orbifolds solve the chirality problem. Witten noted that the effective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered in terms of the location of the branes.

In 1997 Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to the horizon, which for extreme charged black holes looks like an anti de Sitter space. He noted that in this limit the gauge theory describes the string excitations near the branes. So he hypothesized that string theory on a near-horizon extreme-charged black-hole geometry, an anti-deSitter space times a sphere with flux, is equally well described by the low-energy limiting gauge theory, the N=4 supersymmetric Yang-Mills theory. This hypothesis, which is called the AdS/CFT correspondence, was further developed by Steven Gubser, Igor Klebanov and Alexander Polyakov, and by Edward Witten, and it is now well-accepted. It is a concrete realization of the holographic principle, which has far-reaching implications for black holes, locality and information in physics, as well as the nature of the gravitational interaction. Through this relationship, string theory has been shown to be related to gauge theories like quantum chromodynamics and this has led to more quantitative understanding of the behavior of hadrons, bringing string theory back to its roots.

See also


  1. ^ Sunil Mukhi(1999)"The Theory of Strings: A Detailed Introduction"
  2. ^ Woit, Peter, Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law, Basic Books, 2007, ISBN 978-0465092765
  3. ^ Joseph Polchinski, "All Strung Out?", American Scientist, January-February 2007 Volume 95, Number 1
  4. ^ "On the right track. Interview with Professor Edward Witten. ", Frontline, Volume 18 - Issue 03, Feb. 03 - 16, 2001
  5. ^ Leonard Susskind, "Hold fire! This epic vessel has only just set sail...", Times Higher Education, 25 August 2006
  6. ^ Geoff Brumfiel, "Our Universe: Outrageous fortune", Nature, Nature 439, 10-12 (5 January 2006) | doi:10.1038/439010a; Published online 4 January 2006
  7. ^ Hawking, Stephen (2010). The Grand Design. Bantam Books. 
  8. ^ "NOVA - The elegant Universe"
  9. ^ Jim Holt, "Unstrung", The New Yorker, October 2, 2006
  10. ^ a b H. Nastase, More on the RHIC fireball and dual black holes, BROWN-HET-1466, arXiv:hep-th/0603176, March 2006,
  11. ^ a b H. Liu, K. Rajagopal, U. A. Wiedemann, An AdS/CFT Calculation of Screening in a Hot Wind, MIT-CTP-3757, arXiv:hep-ph/0607062 July 2006,
  12. ^ a b H. Liu, K. Rajagopal, U. A. Wiedemann, Calculating the Jet Quenching Parameter from AdS/CFT, Phys.Rev.Lett.97:182301,2006 arXiv:hep-ph/0605178
  13. ^ a b H. Nastase, The RHIC fireball as a dual black hole, BROWN-HET-1439, arXiv:hep-th/0501068, January 2005,
  14. ^ Burt A. Ovrut (2006). "A Heterotic Standard Model". Fortschritte der Physik 54-(2-3): 160–164. Bibcode 2006ForPh..54..160O. doi:10.1002/prop.200510264. 
  15. ^ S. James Gates, Jr., Ph.D., Superstring Theory: The DNA of Reality "Lecture 23 – Can I Have That Extra Dimension in the Window?", 0:04:54, 0:21:00.
  16. ^ Simeon Hellerman and Ian Swanson(2006): "Dimension-changing exact solutions of string theory".; Ofer Aharony and Eva Silverstein(2006):"Supercritical stability, transitions and (pseudo)tachyons".
  17. ^ M. J. Duff, James T. Liu and R. Minasian Eleven Dimensional Origin of String/String Duality: A One Loop Test Center for Theoretical Physics, Department of Physics, Texas A&M University
  18. ^ Polchinski, Joseph (1998). String Theory, Cambridge University Press.
  19. ^ The calculation of the number of dimensions can be circumvented by adding a degree of freedom which compensates for the "missing" quantum fluctuations. However, this degree of freedom behaves similar to spacetime dimensions only in some aspects, and the produced theory is not Lorentz invariant, and has other characteristics which don't appear in nature. This is known as the linear dilaton or non-critical string.
  20. ^ "Quantum Geometry of Bosonic Strings – Revisited"
  21. ^ See, for example, T. Hübsch, "A Hitchhiker’s Guide to Superstring Jump Gates and Other Worlds", in Proc. SUSY 96 Conference, R. Mohapatra and A. Rasin (eds.), Nucl. Phys. (Proc. Supl.) 52A (1997) 347–351
  22. ^ L. Randall and R. Sundrum, "An Alternative to compactification" Phys. Rev. Lett. 83 (1999) 4690–4693
  23. ^ P. Aspinwall, D. Morrison and B. Greene, "Calabi–Yau moduli space, mirror manifolds and space-time topology change in string theory", Nucl. Phys. B416 (1994) 414–480
  24. ^ J. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200
  25. ^ S. S. Gubser, I. R. Klebanov and A. M. Polyakov (1998). "Gauge theory correlators from non-critical string theory". Physics Letters B428: 105–114. arXiv:hep-th/9802109. Bibcode 1998PhLB..428..105G. doi:10.1016/S0370-2693(98)00377-3. 
  26. ^ Edward Witten (1998). "Anti-de Sitter space and holography". Advances in Theoretical and Mathematical Physics 2: 253–291. arXiv:hep-th/9802150. Bibcode 
  27. ^ Aharony, O.; S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz (2000). "Large N Field Theories, String Theory and Gravity". Phys. Rept. 323 (3-4): 183–386. arXiv:hep-th/9905111. doi:10.1016/S0370-1573(99)00083-6. .
  28. ^ Robbert Dijkgraaf,Erik Verlinde and Herman Verlinde(1997)"5D Black Holes and Matrix Strings"
  29. ^ Minoru Eto,Koji Hashimoto and Seiji Terashima(2007)"QCD String as Vortex String in Seiberg-Dual Theory"
  30. ^ Harvey B.Meyer(2005)"Vortices on the worldsheet of the QCD string"
  31. ^ Koji Hashimoto(2007)"Cosmic Strings, QCD Strings and D-branes"
  32. ^ Piljin Yi(2007)"Story of baryons in a gravity dual of QCD"
  33. ^ For example: T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog.Theor.Phys.113:843–882,2005, arXiv:hep-th/0412141, December 2004
  34. ^ See for example Recent Results of the MILC research program, taken from the MILC Collaboration homepage
  35. ^ David Gross, Perspectives, String Theory: Achievements and Perspectives - A conference
  36. ^ J. Polchinski, String Theory, Cambridge University Press, Cambridge, UK (1998)
  37. ^ N. Comins, W. Kaufmann, Discovering the Universe: From the Stars to the Planets, W.H. Freeman & Co., p. 357 (2008)
  38. ^ a b c d S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, de Sitter Vacua in String Theory, Phys.Rev.D68:046005,2003, arXiv:hep-th/0301240
  39. ^ B. Freivogel, M. Kleban, M. Rodriguez Martinez, and L. Susskind, Observational consequences of a landscape, JHEP 0603 (2006) 039, arXiv:hep-th/0505232
  40. ^ M. Kleban, T. Levi, and K. Sigurdson, Observing the landscape with cosmic wakes, arXiv:1109.3473
  41. ^ S. Nadis, "How we could see another universe," Astronomy, June 2009
  42. ^ S. James Gates, Jr., Ph.D., Superstring Theory: The DNA of Reality "Lecture 21 - Can 4D Forces (without Gravity) Love Strings?", 0:26:06-0:26:21, cf. 0:24:05-0:26-24.
  43. ^ Idem, "Lecture 19 - Do-See-Do and Swing your Superpartner Part II" 0:16:05-0:24:29.
  44. ^ Idem, Lecture 21, 0:20:10-0:21:20.
  45. ^ Peter Woit's Not Even Wrong weblog
  46. ^ Lee Smolin's The Trouble With Physics webpage
  47. ^ The n-Category Cafe
  48. ^ John Baez weblog
  49. ^ P. Woit (Columbia University), String theory: An Evaluation,February 2001, arXiv:physics/0102051
  50. ^ P. Woit, Is String Theory Testable? INFN Rome March 2007
  51. ^ "string theory is the first science in hundreds of years to be pursued in pre-Baconian fashion, without any adequate experimental guidance", New York Times, 4 January 2005
  52. ^ "there ain't no experiment that could be done nor is there any observation that could be made that would say, `You guys are wrong.' The theory is safe, permanently safe" NOVA interview)
  53. ^ "String theory [is] yet to have any real successes in explaining or predicting anything measurable" New York Times, 8 November 2005)
  54. ^ see his Dialog on Quantum Gravity, arXiv:hep-th/0310077)
  55. ^ Elias Kiritsis(2007)"String Theory in a Nutshell"
  56. ^
  57. ^ N. Arkani-Hamed, S. Dimopoulos and S. Kachru, Predictive Landscapes and New Physics at a TeV, arXiv:hep-th/0501082, SLAC-PUB-10928, HUTP-05-A0001, SU-ITP-04-44, January 2005
  58. ^ L. Susskind The Anthropic Landscape of String Theory, arXiv:hep-th/0302219, February 2003
  59. ^ Dolen, Horn, Schmid "Finite-Energy Sum Rules and Their Application to πN Charge Exchange"
  60. ^ Banks, Fischler, Shenker and Susskind "M Theory As A Matrix Model: A Conjecture"

Further reading

Popular books and articles

Two nontechnical books that are critical of string theory:

  • Smolin, Lee (2006). The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. New York: Houghton Mifflin Co.. p. 392. ISBN 0-618-55105-0. 
  • Woit, Peter (2006). Not Even Wrong - The Failure of String Theory And the Search for Unity in Physical Law. London: Jonathan Cape &: New York: Basic Books. p. 290. ISBN 978-0-465-09275-8. 


  • Becker, Katrin, Becker, Melanie, and John H. Schwarz (2007) String Theory and M-Theory: A Modern Introduction . Cambridge University Press. ISBN 0-521-86069-5
  • Binétruy, Pierre (2007) Supersymmetry: Theory, Experiment, and Cosmology. Oxford University Press. ISBN 978-0-19-850954-7.
  • Dine, Michael (2007) Supersymmetry and String Theory: Beyond the Standard Model. Cambridge University Press. ISBN 0-521-85841-0.
  • Paul H. Frampton (1974). Dual Resonance Models. Frontiers in Physics. ISBN 0-805-32581-6. 
  • Gasperini, Maurizio (2007) Elements of String Cosmology. Cambridge University Press. ISBN 978-0-521-86875-4.
  • Michael Green, John H. Schwarz and Edward Witten (1987) Superstring theory. Cambridge University Press. The original textbook.
  • Kiritsis, Elias (2007) String Theory in a Nutshell. Princeton University Press. ISBN 978-0-691-12230-4.
  • Johnson, Clifford (2003). D-branes. Cambridge: Cambridge University Press. ISBN 0-521-80912-6. 
  • Polchinski, Joseph (1998) String Theory. Cambridge University Press.
  • Szabo, Richard J. (Reprinted 2007) An Introduction to String Theory and D-brane Dynamics. Imperial College Press. ISBN 978-1-86094-427-7.
  • Zwiebach, Barton (2004) A First Course in String Theory. Cambridge University Press. ISBN 0-521-83143-1. Contact author for errata.

Technical and critical:

  • Penrose, Roger (February 22 2005). The Road to Reality: A Complete Guide to the Laws of the Universe. Knopf. p. 1136. ISBN 0-679-45443-8. 

Online material

External links

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