3.4 String Theories

String Theory: Already the old Greek philosophers believed that matter was made of atoms. Starting in the 20th century, particle accelerators have broken the atoms first in electrons and nuclei and, with the help of more powerful machines, nuclei were broken down in hundred of subatomic particles.
String Theory, as well as the much improved M-Theory, are based on the concept that subatomic particles are in fact strings. Each subparticle is a string vibrating at a different frequency. In other words there are like musical notes that one can play, let us say, on a violin strings or on drum membrane. These are not ordinary strings, or membranes since they only exist in 10 or 11dimensional hyperspace.
As an example, physicists thought for a long time that electrons were very small point particles. According to Strings and M-Theories, electrons are not point particles, they are tiny vibrating strings. This is also true for the subatomic particles. These strings can vibrate at different frequencies and a vibrating string is seen as different subatomic particles depending on its vibration frequency. In other words, subatomic particles are all the same string but with different vibrating frequencies or musical notes.
String, and especially M- Theories, could be the link unifying relativity and quantum theories into a "Theory of Everything". However, this unification can only happen in 10 or 11 dimensional hyperspace. The unification these two theories are difficult because:
- General Relativity is the theory of the very large; black holes, big bang, quasars, the expanding universe,...
- Quantum Theory deals with the world of the very small: atom, protons, neutrons, quarks,... It is based on the fact that energy comes in discreet packets called quanta. It only gives the probability that a given event will occur.

String Theory was first proposed in 1968 by Gabriele Veneziano and, independently, by Makito Suzuki, both from CERN. Reading an old mathematical book, they learned about the Euler Beta Functions discovered by Euler in the 18th century. They noticed that they seemed to describe the subatomic world.
Elementary particle physics aims to predict the mathematical structure of the S-Matrix for the strong interactions. Physicists tried to construct this matrix from experimental data obtained with the big accelerators. Veneziano and Suzuki found the S- Matrix using Euler functions described in his old book. The result was not perfect at first, but it was however a first good approximation.
According to the String Theory each particle is only vibrations or note on the string. A latest version included an infinite series of terms, the Veneziano and Suzuki' being the first, and mot important term. Another improvement is called "Field Theory of Strings" but soon some problems appeared in the String Theory. It was shown that the Veneziano/Suzuki model worked only in 26 dimensions and the Super String Theory of Neven, Schwarz and Ramond existed only in 10 dimensions. Most classic theories, including those of Newton and Einstein, work in any dimensions.
John Schwarz and Joel Scherk's model required the presence of an unknown particle of mass zero and spin 2. They suggested that this particle was the graviton (foreseen by Einstein but not found yet). This means that their theory does not only deal with the strong nuclear force but also with gravity. This seemed the base o a "Theory of Everything".
In 1984 John Schwarz and Mike Green showed that the String Theory had none of the inconsistencies that killed all the other theories (no mathematical divergences, no anomalies). In short it was a good candidate to become the "Theory of Everything".
According to the String Theory, if one had a very powerful microscope able to see an electron, one would not see a point, but a vibrating string of Planck length (10E-33 cm).The electrons and other subatomic particles are thought to look like a point only because they are so small.
Plucking an electron string could transform it in a neutrino; plucking it gain it could now appear to be a quark. And these are only examples. In fact by plucking the electron could be transformed in any subatomic particle, all of them being no more than different musical notes that one can play on a string or superstring.
Strings can interact by splitting or rejoining, creating for instance the splitting of a neutron or creating the interaction we know between protons and electrons. Through String Theory one can reproduce all the laws of atomic and nuclear physics. Moreover, Einstein's theories of relativity are a by-product of the String theory. The graviton, the lowest vibration of the string, is in fact the road to find Einstein's theory of gravity in quantum form.
The fact that String Theory works only in higher dimensions is not a problem as it was found that these higher dimensions (above the 3 dimensions of space plus time) are very small (atom size) so w cannot observe them.
According to String Theory, the universe had initially 10 dimensions and all the forces were unified. However this universe was unstable and 6 of the 10 dimensions curled in a tiny invisible ball leaving the known remaining four dimensions to expand following the big bang.
String Theory succeeded where the other failed or two reasons:
i- The string being an extended object avoids the divergence found in point particles. In particular the gravity force becomes infinite when one approaches a point particle (according to Newton's Theory, gravity to proportional to the inverse of the distance squared that is it is proportional to 1/r2. For r = 0, the force is infinite).On the opposite, it remains finite in String theory.
ii- The strings have larger symmetries that cancels the other divergences and anomalities of the theory. It allows interchanging all the particles found in nature. This is called "Supersymmetry".

In supersymmetry all subatomic particles have symmetrical partners called "Sparticles" but we have never seen them. Some physicists believe that this is due tithe fact that they are too massive to be created in accelerators.
Examples: Electron, Selectron
Lepton, Slepton
Quark, Squark
String Theory can derive many of the qualitative features of the standard model.

M-Theory: String equations have billions of solutions and this makes the theory unattractive, even if many looked like our universe and close to the Standard Model. Finding precisely the Standard Model was and is near impossible. Moreover there are 5 self-consistent string theories, and it is difficult that there could be five distinct unified theories.
The Veneziano String Theory is based on both open strings (2 ends type I) as well as closed strings (circular loop strings, type II). Fully self-consistent can only be derived for close strings (loops, type II).
In 1994, scientists found that ten-dimensional String Theory was an approximation to an eleven-dimensional theory that's now known as M-Theory. It was then shown that all five String Theories were the same, only different approximations to the M-Theory. M-Theory also explained Super Gravity Theory. Super gravity has only 2 particles with zero mass (the graviton and the gravitino) and it is also an eleven-dimensional theory. On the opposite, M-Theory has an infinity of particles of different masses. It can be shown that Super Gravity is only a part of M-Theory (the part with particles of zero mass). Moreover, if we curl one of the dimensions of the M-Theory, the membrane becomes a string, and the M-Theory becomes the type II String Theory. In this way we have unified all ten and eleven-dimensional theories in a single theory, the M- Theory.
M-Theory does not only include String Theory, it introduces many membranes of different dimensions:
- Point particles are called "Zero Branes" (they have no dimension).
- A string is a "One Brane" (because it is a one-dimensional object).
- A membrane is a "Two-Branes" like a surface of a ball.
- Our universe is some kind of a "Three-Branes", a three-dimensional object.
The higher dimensions of the M-Theory are not small; they may be large enough to be seen in laboratories. In String Theory, 6 of the 10 dimensions are reduced to a small ball (a Calabri-Yan manifold) so they cannot be seen with today's instruments and one cannot enter in them. In M-Theory, one can see our universe as a .membrane floating in a larger universe. As a result not all higher dimensions have to be reduced to a small ball. Some can be big, infinite even.
According to the M-Theory, there are three types of universes:
i- Type I universe explains why we live in a four space/time universe when M-Theory works in 11 dimensions. This universe is believed to have started perfectly symmetrical. It did not expand because strings in all dimensions tightened it in a small radius ball. In these small dimensions the universe is bound by strings and anti-strings. If a string and an anti-string collide, they annihilate each other. In higher dimensions there is so much room that strings and anti-strings rarely collide and never unravel. In three or fewer dimensions, stings and anti-strings will collide preventing the universe expanding following their big bang.
ii- Type II universes show that if they can give birth to other universe, they can also collide creating other universes. In these cases, the big bangs are the result of the collision of two, parallel branes - universes. These are called "Ekpirotic" universes. According to this theory, at the beginning we had two flat homogeneous and parallel three branes. At that point this universe was cold and empty, and gravity pulled the two branes together. When they collided, the huge amount of energy liberated was converted in matter and radiation. The force of the collision pushed the two membranes apart; following this they cooled rapidly creating our universe. The cooling and expansion went on for trillions of years until the temperature and density were close to zero. The universe was then empty and inert but gravity pulled again the two membranes together and the same cycle was repeated.
iii- Type III assumes that initially the universe was infinitely old, cold and nearly empty. Gravity created clumps of matter which condensed so much that a black hole was formed. When the matter within the black hole was compressed by gravity to the size of the Planck length, string theory took over and it exploded in a big bang followed by the well known expansion leading to our universe.
NB: The M- Theory also makes strange predictions such:
- The universe is a hologram.
- The universe is a computer program.
These theories are still in an early stage now and are better left out.
The M-Theory is not yet in its final form. It could very well be that it will become the Theory of Everything but, as of today, it is not yet there.

- Bosonic string theory: It is the original version of string theory, developed in the late 1960s. Although it has many attractive features, it has a pair of features that render it unattractive as a physical model. Firstly it predicts only the existence of bosons whereas we know many physical particles are fermions. Secondly, it predicts the existence of a particle whose mass is imaginary implying that it travels faster than light. The existence of such a particle, commonly known as a tachyon, would conflict with much of what we know about physics, and such particles have never been observed. Another feature of bosonic string theory is that in general the theory displays inconsistencies due to the conformal anomaly. In a space-time of 26 dimensions, however, with 25 dimensions of space and one of time, the inconsistencies cancel. Another way to look at this is that in general bosonic string theory predicts unphysical particle states called 'ghosts'. So bosonic string theory predicts a 26 dimensional space-time. This high dimensionality isn't a problem for bosonic string theory because it can be formulated in such a way that along the 22 excess dimensions, space-time is folded up to form a small torus. This would leave only the familiar four dimensions of space-time visible.

- Brane cosmology: It refers to several theories in particle physics and cosmology motivated by, but not rigorously derived from, superstring theory and M-theory. The central idea is that the visible, four-dimensional universe is restricted to a brane inside a higher-dimensional space, called the "bulk". The additional dimensions may be taken to be compact, in which case the observed universe contains the extra dimensions, and then no reference to the bulk is appropriate in this context. In the bulk model, other branes may be moving through this bulk. Interactions with the bulk, and possibly with other branes, can influence our brane and thus introduce effects not seen in more standard cosmological models.

- Cosmic string: It is a hypothetical 1-dimensional topological defect in the fabric of spacetime. Cosmic strings are hypothesized to form when different regions of spacetime undergo phase changes, resulting in domain boundaries between the two regions when they meet. This is somewhat analogous to the boundaries that form between crystal grains in solidifying liquids, or the cracks that form when water freezes into ice. In the case of our universe, such phase changes may have occurred in the early days as the universe formed. Cosmic strings, if they exist, would be extremely thin with diameters on the same order as a proton. They would have immense density, however, and so would represent significant gravitational sources. A cosmic string 1.6 kilometres in length would exert more gravity than the Earth. Cosmic strings would form a network of loops in the early universe, and their gravity could have been responsible for the original clumping of matter into galactic superclusters. A cosmic string's vibrations, which are thought to oscillate near the speed of light, can cause part of the string to pinch off into an isolated loop. These loops have a finite lifespan due to decay via gravitational radiation.

- Dual resonance model: It is a term used in theoretical physics which refers to the early investigation on strong interactions of string theory. It was based upon the observation than the amplitudes for the s-channel scatterings matched exactly with the amplitudes for the t-channel scatterings among mesons and also the Regge trajectory. The study of dual resonance models was very popular from 1968 to 1974. It fell rapidly out of favour around 1974 mainly because it was superseded by quantum chromodynamics as the accepted theory of strong interactions.

- Eleven dimensional supergravity theory: This maximal supergravity is the classical limit of M-theory. There is, classically, only one 11-dimensional supergravity theory. Like all maximal supergravities, it contains a single supermultiplet, the supergravity supermultiplet. This supermultiplet contains the graviton, a Majorana gravitino and a 3-form gauge field often called the C-field. It contains two p-brane solutions, a 2-brane and a 5-brane, which are electrically and magnetically charged, respectively, with respect to the C-field. The supergravity 2-brane and 5-brane are the long-wavelength limits of the M2-brane and M5-brane in M-theory.

- Heterotic string: It is a peculiar mixture (or hybrid) of the bosonic string and the. In string theory, the left-moving and the right-moving excitations almost do not talk to each other, and it is possible to construct a string theory whose left-moving (counter-clockwise) excitations "think" that they live on a bosonic string propagating in D = 26 dimensions, while the right-moving (clock-wise) excitations "think" that they belong to a superstring in D = 10 dimensions. The mismatched 16 dimensions must be compacted on an even, self-dual lattice (a discrete subgroup of a linear space). There are two possible even self-dual lattices in 16 dimensions, and it leads to two types of the heterotic string. They differ by the gauge group in 10 dimensions. One gauge group is SO(32) (the HO string) while the other is E8×E8 (the HE string). These two gauge groups also turned out to be the only two anomaly-free gauge groups that can be coupled to the N = 1 supergravity in 10 dimensions other than U(1)496 and E8 × U(1)248, which is suspected to lie in the swampland.

- M-theory: It is a proposed "master theory" that unifies the five superstring theories. The various superstring theories are related by dualities, which allow physicists to relate the description of an object in one super string theory to the description of a different object in another super string theory. These relationships imply that each of the super string theories is a different aspect of a single underlying theory. M-theory is not yet complete; however it can be applied in many situations (usually by exploiting string theoretic dualities).

- String duality: It is a class of symmetries in physics that link different string theories, 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 field theory: It is a proposal to define string theory in such a way that the background independence is respected. String field theory can be understood as a quantum field theory with infinitely many fields which are unified into one master "string field". In perturbative string theory, scattering amplitudes are found by summing a genus expansion of Feynman diagrams, in analogy with the loop expansion in quantum field theory. However, this procedure does not follow from first principles, but rather from symmetry arguments and intuition. When quantized, the action governing the string field would, in principle, reproduce all the Feynman diagrams of splitting and joining perturbative strings, but also encode non-perturbative effects.

- String theory: It is an incomplete mathematical approach to theoretical physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point particles that form the basis for the standard model of particle physics. By replacing the point-like particles with strings, an apparently consistent quantum theory of gravity emerges, which has not been achievable under quantum field theory. Usually, the term string theory includes a group of related superstring theories and a few related frameworks such as M-theory, which seeks to unite them all.

- String theory landscape or anthropic landscape: It refers to the large number of possible false vacua in string theory. The "landscape" includes so many possible configurations that it is thought by some physicists that the known laws of physics, the Standard Model and General relativity with a positive cosmological constant, occurs in at least one of them. The anthropic landscape refers to the collection of those portions of the landscape that are suitable for supporting human life, an application of the anthropic principle that selects a subset of the theoretically possible configurations. In string theory the number of false vacua is commonly quoted as 10500. The large number of possibilities arises from different choices of Calabi-Yau manifolds and different values of generalized magnetic fluxes over different homology cycles.

- Super menbrane theory: This is also a supersymmetric theory that exists in 11 dimensions. Supermembrane can have two or more dimensions as it represents a surface. Two-brane and five-brane membranes are self consistent in 11 dimensions.
However, super membranes have problems: They are difficult to work with, and their quantum theory diverges. These membranes are unstable, and decay in point particles.

- Superstring theory: It is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings. It is considered one of the most promising candidate theories of quantum gravity. Superstring theory is shorthand for supersymmetric string theory because unlike bosonic string theory, it is the version of string theory that incorporates fermions and supersymmetry.

Number of superstring theories
Theoretical physicists were troubled by the existence of five separate string theories. This has been solved by the second superstring revolution in the 1990s during which the five string theories were discovered to be different limits of a single underlying theory: M-theory.

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
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 closed strings only, no tachyon, massless fermions spin both ways (nonchiral)
IIB 10 Supersymmetry between forces and matter, with closed strings only, 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

The five consistent superstring theories are:
. The type I string has one supersymmetry in the ten-dimensional sense (16 supercharges). This theory is special in the sense that it is based on unoriented open and closed strings, while the rest are based on oriented closed strings.
. The type II string theories have two supersymmetries in the ten-dimensional sense (32 supercharges). There are actually two kinds of type II strings called type IIA and type IIB. They differ mainly in the fact that the IIA theory is non-chiral (parity conserving) while the IIB theory is chiral (parity violating).
. The heterotic string theories are based on a peculiar hybrid of a type I superstring and a bosonic string. There are two kinds of heterotic strings differing in their ten-dimensional gauge groups: the heterotic E8×E8 string and the heterotic SO(32) string. (The name heterotic SO(32) is slightly inaccurate since among the SO(32) Lie groups, string theory singles out a quotient Spin(32)/Z2 that is not equivalent to SO(32).)

- T-duality: It is a symmetry between small and large distances that is not present in Kaluza-Klein compactification of a particle theory. It relates different string theories that everyone thought were completely unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution. T-duality is a symmetry of string theory, relating type IIA and type IIB string theory, and the two heterotic string theories. T-duality transformations act on spaces in which at least one direction has the topology of a circle. Under the transformation, the radius R of that direction will be changed to 1/R, and "wrapped" string states will be exchanged with high-momentum string states in the dual theory. For example, one might begin with an IIA string wrapped once around the direction in question. Under T-duality, it will be mapped to an IIB string which has momentum in that direction. An IIA string with a winding number of two (wrapped twice) will be mapped to an IIB string with two units of momentum, and so on. The total squared mass of a closed string

is invariant under the exchange , and the interactions and all other physical phenomena can be proved invariant under this operation, too. T-duality acting on D-branes changes their dimension by +1 or -1.