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String duality is a class of symmetries in physics that link different string theories, [1] theories which assume that the fundamental building blocks of the universe are strings instead of point particles.
Before the so-called "duality revolution" there were believed to be five distinct versions of string theory, plus the (unstable) bosonic and gluonic theories.
| String theories | ||
|---|---|---|
| Type | Spacetime dimensions | Details |
| Bosonic | 26 | Only bosons, no fermions means 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 closed strings and open strings, no tachyon, group symmetry is SO(32) |
| IIA | 10 | Supersymmetry between forces and matter, with closed strings and open strings bound to D-branes, no tachyon, massless fermions spin both ways (nonchiral) |
| IIB | 10 | Supersymmetry between forces and matter, with closed strings and open strings bound to D-branes, no tachyon, massless fermions only spin one way (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 space-time (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).
Before the 1990s, string theorists believed there were five distinct superstring theories: type I, types IIA and IIB, and the two heterotic string theories (SO(32) and E8×E8). The thinking was that out of these five candidate theories, only one was the actual theory of everything, and that theory was the theory whose low energy limit, with ten dimensions spacetime compactified down to four, matched the physics observed in our world today. It is now known that the five superstring theories are not fundamental, [2] but are instead different limits of a more fundamental theory, dubbed M-theory. These theories are related by transformations called dualities. If two theories are related by a duality transformation, each observable of the first theory can be mapped in some way to the second theory to yield equivalent predictions. The two theories are then said to be dual to one another under that transformation. Put differently, the two theories are two mathematically different descriptions of the same phenomena. A simple example of a duality is the equivalence of particle physics upon replacing matter with antimatter; describing our universe in terms of anti-particles would yield identical predictions for any possible experiment.
String dualities often link quantities that appear to be separate: Large and small distance scales, strong and weak coupling strengths. These quantities 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.
Suppose we are in ten spacetime dimensions, [3] which means we have nine space dimensions and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized momentum around the circle, because its momentum is linked to its wavelength (see wave–particle duality), and 2πR must be a multiple of that. In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in standard units, for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy). A string, in addition to traveling around the circle, may also wrap around it. The number of times the string winds around the circle is called the winding number, and that is also quantized (as it must be an integer). Winding around the circle requires energy, because the string must be stretched against its tension, so it contributes an amount of energy of the form , where is a constant called the string length and w is the winding number (an integer). Now (for a given maximum energy) there will be many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R. In fact, a theory with large R and a theory with small R are equivalent, where the role of momentum in the first is played by the winding in the second, and vice versa. Mathematically, taking R to and switching n and w will yield the same equations. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.
This type of duality is called T-duality. T-duality relates type IIA superstring theory to type IIB superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle (one with a large radius and the other with a small radius) then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories. T-duality also relates type I superstring theory to both type IIA and type IIB superstring theories with certain boundary conditions (termed orientifold).
Formally, the location of the string on the circle is described by two fields living on it, one which is left-moving and another which is right-moving. The movement of the string center (and hence its momentum) is related to the sum of the fields, while the string stretch (and hence its winding number) is related to their difference. T-duality can be formally described by taking the left-moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding.
Every force has a coupling constant, which is a measure of its strength, and determines the chances of one particle to emit or absorb another particle. For electromagnetism, the coupling constant is proportional to the square of the electric charge. When physicists study the quantum behavior of electromagnetism, they can't solve the whole theory exactly, because every particle may emit and absorb many other particles, which may also do the same, endlessly. So events of emission and absorption are considered as perturbations and are dealt with by a series of approximations, first assuming there is only one such event, then correcting the result for allowing two such events, etc. (this method is called Perturbation theory). This is a reasonable approximation only if the coupling constant is small, which is the case for electromagnetism. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.
This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the oscillation modes of the string, called the dilaton. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called S-duality. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of perturbation theory, but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.
Superstring theories related by S-duality are: type I superstring theory with heterotic SO(32) superstring theory, and type IIB theory with itself.
Furthermore, type IIA theory in strong coupling behaves like an 11-dimensional theory, with the dilaton field playing the role of an eleventh dimension. This 11-dimensional theory is known as M-theory.
Unlike the T-duality, however, S-duality has not been proven to even a physics level of rigor for any of the aforementioned cases. It remains, strictly speaking, a conjecture, although most string theorists believe in its validity.
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity.
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity.
In theoretical physics, T-duality is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories describes strings propagating in a spacetime shaped like a circle of some radius , while the other theory describes strings propagating on a spacetime shaped like a circle of radius proportional to . The idea of T-duality was first noted by Bala Sathiapalan in an obscure paper in 1987. The two T-dual theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, momentum in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description.
In theoretical physics, S-duality is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theoretical physics because it relates a theory in which calculations are difficult to a theory in which they are easier.
In theoretical physics, supergravity is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra a superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way.
In particle physics, the hypothetical dilaton particle is a particle of a scalar field that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compactifications of extra dimensions. In Brans–Dicke theory of gravity, Newton's constant is not presumed to be constant but instead 1/G is replaced by a scalar field and the associated particle is the dilaton.
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
In theoretical physics, F-theory is a branch of string theory developed by Iranian-American physicist Cumrun Vafa. The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in the form of F-theory compactified on elliptically fibered Calabi–Yau four-folds. The letter "F" supposedly stands for "Father".
In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string. There are two kinds of heterotic superstring theories, the heterotic SO(32) and the heterotic E8 × E8, abbreviated to HO and HE. Apart from that there exist seven more heterotic string theories which are not supersymmetric and hence are only of secondary importance in most applications. Heterotic string theory was first developed in 1985 by David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm (the so-called "Princeton string quartet"), in one of the key papers that fueled the first superstring revolution.
In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented and the only one which contains not only closed strings, but also open strings.
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories have extended supersymmetry which is maximal amount of supersymmetry — namely 32 supercharges — in ten dimensions. Both theories are based on oriented closed strings. On the worldsheet, they differ only in the choice of GSO projection.
In theoretical physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic.
Many first principles in quantum field theory are explained, or get further insight, in string theory.
The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum gravity, particle and condensed matter physics, cosmology, and pure mathematics.
The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invariant background. However, there are recent developments which make possible Lorentz invariant quantization of string theory in 4-dimensional Minkowski space-time.
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
Higher-dimensional supergravity is the supersymmetric generalization of general relativity in higher dimensions. Supergravity can be formulated in any number of dimensions up to eleven. This article focuses upon supergravity (SUGRA) in greater than four dimensions.
This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, and high energy physics.
In string theory, the Ramond–Neveu–Schwarz (RNS) formalism is an approach to formulating superstrings in which the worldsheet has explicit superconformal invariance but spacetime supersymmetry is hidden, in contrast to the Green–Schwarz formalism where the latter is explicit. It was originally developed by Pierre Ramond, André Neveu and John Schwarz in the RNS model in 1971, which gives rise to type II string theories and can also give type I string theory. Heterotic string theories can also be acquired through this formalism by using a different worldsheet action. There are various ways to quantize the string within this framework including light-cone quantization, old canonical quantization, and BRST quantization. A consistent string theory is only acquired if the spectrum of states is restricted through a procedure known as a GSO projection, with this projection being automatically incorporated in the Green–Schwarz formalism.
