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Rotational motion. The frontiers of physics: CD-ROM drives. Torque and angular momentum. SUPERSTRINGS AND OTHER THINGS viii. PDF | On Feb 10, , Carlos I. Calle and others published Superstrings and Other Things, Second Edition. Request PDF on ResearchGate | Superstrings and Other Things: A Guide to Physics, Second Edition | Superstrings and Other Things: A Guide to Physics.


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superstrings and other things pdf. Superstrings and Other Things: A Guide to Physics takes readers on a fascinating journey through physics. Written in an. superstrings and other things a guide to physics second edition superstrings and other things pdf. This site is temporarily unavailable. Hosted by Network. Superstrings and other things Supergravity and Superstrings: A Geometric Perspective, Volume 3: Asimov, Isaac - Of Time and Space and Other Things.

Evolution of cosmic superstring networks: Jon Urrestilla. The network consists of gauge U 1 strings of two different kinds and their bound states, arising due to an attractive interaction potential.

Bound strings constitute only a small fraction of the total string length in the network. Both types of string are naturally formed in the course of brane-antibrane annihilation at the end of brane inflation [2, 3, 4, 5, 6]. Fundamental F and D-strings produced in the aftermath of this annihilation can form p, q bound states combining p F -strings and q D-strings. As a result the strings are expected to form an interconnected F D-network [5, 6], with different types of string joined in 3-way Y -type junctions.

Similar string networks can also be formed in field theory; a simple example has been recently given by Saffin [7].

His model includes two Abelian Higgs models, with an additional coupling between the Higgses. An even simpler example is the usual Abelian Higgs model. With a suitable choice of the Higgs and gauge couplings, corresponding to the type-I regime, this model allows stable strings with arbitrary windings, which can be joined in 3-way junctions [8]. In the latter case, when two non-commuting strings cross, a third string starts stretching between them, resulting in two Y -junctions.

The evolution of cosmic string networks has been a subject of much recent discussion and debate. A similar model was later used for ZN networks having N strings joined at each vertex [16]. In Z3 models, the vertices can carry an unconfined magnetic charge.

The energy of the network is then efficiently dissipated by gauge field radiation from these magnetic monopoles. Another energy loss mechanism is the formation of closed loops and of small nets disconnected from the main network.

In the absence of magnetic charges, and if loop and net formation turn out to be inefficient, the remaining energy loss channel is the gravitational radiation. However, most recent work points to scaling evolution. These models allow for several types of string with different tensions and use the velocity-dependent one-scale model of string evolution [17, 18] see also [19]. The models make somewhat different assumptions about the physics of F - and D-string interaction.

If this picture is correct, it provides an important additional mechanism of energy loss by the network. Avgoustidis and Shellard [18] assume, on the other hand, that the energy released in the zipping process goes to increase the kinetic energy of strings, and thus remains in the network, and allow different length scales for different string types.

Network evolution has also been studied in field-theory numerical simulations. Hindmarsh and Saffin [21] performed a full field theory simulation of global Z3 strings. The models used in [16, 20, 21] have some important differences from superstring networks. First, all types of string in these models have the same tension, while in an F D-network the tensions of all p, q strings are generally different.

Second, the global symmetry breaking models used in [16, 20, 21] allowed for an additional energy loss mechanism — the radiation of massless Goldstone bosons — which is known to be rather efficient. On the other hand, superstring networks are expected to have only gravitational-strength couplings to massless or light bosons.

A field theory simulation of an interconnected string network has been recently developed by Rajantie, Sakellariadou and Stoica [23].

The dynamic range of their simulations was not sufficient to reach any conclusions about the statistical properties of the network and its scaling behaviour or lack thereof. The main focus of Rajantie et.

They find that this interaction disrupts the string bound states in the network. This result is probably of little relevance for superstring F D-networks, since, as we already noted, superstring interactions are expected to have gravitational strength and thus have little effect on network dynamics.

The details of the model and of the simulation are given in the next section. The results are presented in Section 3. Our conclusions are summarized and discussed in Section 4.

Table I gives the corresponding string tensions, as well as the binding energies per unit length of string , which are relatively large. Numerical setup Our aim was to perform real-time lattice simulations of model 3 , for as long a time as our facilities allowed us. There is a well-known problem in such simulations: As a result the string width quickly drops below the resolution threshold of the simulation.

Following [31], the equations of motion 12 can be written as: As earlier work has shown [30, 31], there is little difference in string dynamics for different values of s. We discretized the modified equations of motion 14 on a lattice using the standard lattice link variable approach [32] and performed the simulations on the UK National Cosmology Supercomputer [33]. The simulation box consisted of lattice points, with periodic boundary conditions.

Scaling evolution regime is expected to be an attractor, and indeed earlier work has shown [31, 34, 35, 36] that this regime is approached from a wide range of initial configurations. Nonetheless, constructing initial conditions for this kind of simulation is a nontrivial task.

The challenge is to find some initial configuration that leads to scaling as fast as possible, in order to maximize the dynamical range. For the results presented here, we used the following procedure: This configuration was then smoothed out by averaging over nearest neighbours, and this procedure repeated 20 times, in order to get rid of some excess initial gradient energy.

Note that this initial configuration satisfies the lattice Gauss law, and due to the lattice-link variable procedure, the Gauss law is guaranteed to be satisfied throughout the simulation.

Superstrings and other things

In order to automatically detect the strings in the simulation and compute their length, we calculated the net winding of the phases around plaquettes. One can then trace the string following the winding and estimate the lengths. An AB string can be traced by the sites where both A and B phases wind. Unfortunately, there are two drawbacks of this procedure: On the other hand, there are places where inside a clear segment of AB string, A and B phases do not wind in exactly the same plaquettes, but there is a slight displacement we will show an example in the following Section.

The total length of AB string is not considerably affected by this process; the main difference corresponds to a more realistic count of the number N of Y -junctions formed, that is, it helps in not overcounting AB-segments and AB Y -junctions. With this choice, the corrected value of N is roughly a factor of 4 smaller than one would get from the raw data.

As we already mentioned, our choice of parameters was largely motivated by the effort to increase the dynamic range of the simulation. For example, we allocated only a few lattice points per string thickness. As a result, our discretized representation of the field theory string solutions is not particularly accurate. For a rapidly moving string, this may result in spurious damping, with the kinetic energy of the string being dissipated into particles [37].

Moore et. In the present paper, our focus is not so much on the dynamics of oscillating loops, as it is on the overall charac- teristics of the large network.

Picture of a typical simulation of a p,q network. The green and blue colours correspond to A and B strings respectively, whereas the red colour shows the AB segments. It can be clearly seen how Y junctions are formed all over the simulation. In all cases an interconnected network was formed with A-, B- and 1, 1 AB-strings. No higher- p, q strings were observed.

As mentioned in the previous section, within a single AB bound state, A and B strings can be displaced by a single lattice point, making the code decide that it is in fact two separate segments.

Figure 3 shows a fragment of the simulation box, with a somewhat long AB string depicted. Those accidental displacements should not be taken into account, and with the help of the parameter dAB max introduced earlier , the displacements are reassessed, and the segment is counted as one. Fragment of the simulation box showed in Fig. Green and blue correspond to A and B string, whereas the red colour corresponds to an AB segment.

A long AB segment can be seen in the picture, but at some points the A and B string miss each other by just a lattice point. With the help of the parameter dAB max those accidental displacements are accounted for, and long segments such as the one in the figure are counted as one. Throughout this paper we shall use comoving length scales. Remarkably, the graphs for the flat, radiation and matter regimes are almost identical. Toward the end of the simulation, the two terms in Eq.

Note that all three cases exhibit an approximately linear behaviour, nearly independent of the regime. The values of the linear growth coefficients of different lengths, as defined in the body of the paper, obtained by fitting the simulation data. We see that fAB remains nearly constant, at the value 0. This is in conflict with analytic models [17, 18] predicting that the energy of the network should be more or less equally divided between A, B and AB-strings.

Visual inspection of the simulation movies suggests, on the contrary, that formation of bound segments by intersecting A and B strings occurs rather infrequently, probably when the relative velocity of the colliding strings is sufficiently small [38, 39]. Even though the fraction of bound string is small, the interaction of A and B-strings has a significant effect on the network evolution. To quantify this effect, we ran some simulations with the same initial conditions for A-string fields as before, but with B-string fields starting in their ground state, so that no B strings are formed.

This value is in agreement with earlier U 1 simulations by Vincent et. Fraction of total string length in bound strings, for flat, radiation and matter regimes.

The percentage is fairly low; between 0. The shorter bound segments in flat spacetime are probably due to larger string velocities. The average length of bound segments lAB , as defined in Eq. The evolution is approximately linear, with slope given in Table II. Non-scaling of A and B segments All results presented so far are consistent with scaling evolution, with all characteristic length scales of the network growing proportionally to the horizon.

In fact, Fig. The average length of A-segments lA is shown for flat, radiation and matter regimes. In all three cases, lA is nearly constant at late times. The behaviour of lB is essentially the same. Values of the parameters in simulation with a second damping period, introduced to achieve scaling of the segment lengths lA and lB.

Towards a true scaling regime We attempted to shorten this transient regime, or avoid it altogether, by increasing the duration of the initial damped period. Getting to the end of the transient regime by the end of the simulation proved to be a difficult task, for which we had to push the limits of the stability of the code by using a rather large time step, a rather large damping coefficient, and evolving the system beyond the half-light-crossing time of the simulation box.

The difference from the rest of our simulations is in the second period of damping imposed after the first period which is common to all simulations and is used to relax the system from the highly excited initial state and allow string formation. All three lines seem to approach a linear behaviour.

The fairly linear behaviour of lA is the main change. However, because of our extreme choice of parameters and large error bars in lA , and because the change in lA during the linear regime is relatively small, this conclusion should be regarded as tentative. Loops and small nets We observed the formation of small independent nets a few times in the course of the simulation, but these occasions were rather rare, so AB-strings belonged predominantly to the infinite network, with at most one small net in addition.

The fraction of the total string length in disconnected A- or B-loops is shown in Fig. Percentage of string length for both A and B strings in loops that do not belong to the main network. The length distribution of independent loops is plotted in Fig. The figure uses logarithmic binning and shows the distribution of loops at four different times. Such loops should arguably be regarded as part of the infinite network.

These values are in agreement with previous field theoretical simulations where a loops were found to account for a few percent of the total string length [40]. Simulation movies show that loops that decouple from the network do not oscillate as they would in Nambu-Goto simulations, but rather shrink and disintegrate.

Left Distribution of independent loops for the radiation era, averaged over 20 simulations, for four different time steps. The figure shows that there are rather large independent loops formed during the simulation, and that the peak of the distribution tends to move towards larger sizes with time.

Right Loop distribution averaged over 20 simulations in radiation era. The distribution appears to follow Eqn.

Additional damping, through particle emission from loops, may be due to the presence of short-wavelength string excitations in the initial conditions, as indicated by the numerical results of Refs.

The short lifetimes of the loops in our simulations explain, at least in part, the relatively small amount of string in loops, as compared to the Nambu-Goto simulations. In any case, we checked the loop distribution in our simulations for scaling behavior. In a situation where the loop distribution scales, all the lines should line on top of one another.

We see that the graph in Fig. The model has two types of Abelian gauge strings, A and B, with an attractive interaction which gives rise to bound AB-strings. Starting with a randomized high-energy field distribution, we found that an interconnected string network is indeed formed, consisting of A and B strings, as well as their 1, 1 bound states, joined together at Y -type junctions.

No higher p, q -strings were observed in the simulations.

Other characteristic length scales, such as the length of AB segments, the typical correlation length of A and B strings or the typical distance between Y junctions also scale.

Also, the average length of AB-segments is much shorter than the length of A- or B-string segments. This is in contrast with analytic models [17, 18] predicting all lengths to be fairly equal. From movies of the simulations one can see that AB-segments do not always form when an A-string meets a B-string even for relatively high bounding energies ; on the contrary, the formation of bound segments is rather infrequent.

Even though, with our general initial condition configuration, most of the typical distances in the network showed a scaling behaviour, the average comoving lengths of A and B segments did not. These lengths remained much larger than the typical correlation length throughout the simulation.

Calle C.I. Superstrings and Other Things: A Guide to Physics

We found that the system does seem to approach a regime where all characteristic lengths scale linearly with time. If this were the true scaling regime, it would provide us, among other things, with the means to calculate CMB power spectra predictions from the field theoretical model as in [31, 47, 48, 49]. But due to a relatively short duration of the linear evolution and to likely presence of spurious damping, our simulations cannot be relied upon for a quantitatively accurate description of scaling.

Some of these loops are very large and will very likely reconnect into the main network again. We examined the length distribution of loops in the network and found that, even though out parameter choice is not expected to resolve accurately string dynamics, these distributions seem to scale. The network properties in our simulations are closer to those of superstring networks than they were in earlier simulations that used Z3 -strings or non-linear sigma-models.

However, there is still an important differences. This feature can be accounted for in Nambu-Goto and analytic models. The efficiency of various energy loss mechanisms by the string network remains a topic for future research.

Energy loss to loop production appears to be substantial, considering that the length in loops at any time is a few percent of the total and that the loops do not stay around for long and rapidly decay. Another important energy loss mechanism in field theory simulations is direct particle emission from strings [40, 52]. In fact, the analysis in [18] shows that emission of particles and of tiny loops which immediately decay into particles is the dominant energy loss mechanism for a single U 1 string network, so it probably dominates in our simulations as well.

This issue is not completely settled, since some of the recent Nambu-Goto simulations [43] and analytic treatments [53] indicate continuous production of microscopic loops throughout the network evolution. In summary, what have we learnt from our simulations? We have demonstrated that an interconnected network of strings can indeed form at a symmetry breaking phase transition.

This network shows no tendency to freeze to a static configuration. On the contrary, it appears to approach scaling, with all characteristic lengths growing linearly with time.

The latter property leads to relatively frequent self-intersections and allows the network to lose a substantial fraction of its energy in the form of closed loops. Our simulations also indicate that the true scaling evolution may be preceded by a transient regime in which the comoving lengths of A and B-segments remain nearly constant in time.

Some of the shortcomings of our approach can be overcome in Nambu-Goto-type simulations e. In the s, the physicist Jacob Bekenstein suggested that the entropy of a black hole is instead proportional to the surface area of its event horizon , the boundary beyond which matter and radiation is lost to its gravitational attraction. The Bekenstein—Hawking formula expresses the entropy S as. Like any physical system, a black hole has an entropy defined in terms of the number of different microstates that lead to the same macroscopic features.

The Bekenstein—Hawking entropy formula gives the expected value of the entropy of a black hole, but by the s, physicists still lacked a derivation of this formula by counting microstates in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the viability of any theory of quantum gravity such as string theory. In a paper from , Andrew Strominger and Cumrun Vafa showed how to derive the Beckenstein—Hawking formula for certain black holes in string theory.

In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of different ways of placing D-branes in spacetime so that their combined mass and charge is equal to a given mass and charge for the resulting black hole.

The black holes that Strominger and Vafa considered in their original work were quite different from real astrophysical black holes. One difference was that Strominger and Vafa considered only extremal black holes in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge.

Although it was originally developed in this very particular and physically unrealistic context in string theory, the entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity.

Indeed, in , Strominger argued that the original result could be generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry. This is a theoretical result which implies that string theory is in some cases equivalent to a quantum field theory.

It is closely related to hyperbolic space , which can be viewed as a disk as illustrated on the left. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior. One can imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time.

The resulting geometric object is three-dimensional anti-de Sitter space. Time runs along the vertical direction in this picture. As with the hyperbolic plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface.

This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.

An important feature of anti-de Sitter space is its boundary which looks like a cylinder in the case of three-dimensional anti-de Sitter space. One property of this boundary is that, within a small region on the surface around any given point, it looks just like Minkowski space , the model of spacetime used in nongravitational physics.

The claim is that this quantum field theory is equivalent to a gravitational theory, such as string theory, in the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory.

In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding. One reason for this is that the correspondence provides a formulation of string theory in terms of quantum field theory, which is well understood by comparison.

Another reason is that it provides a general framework in which physicists can study and attempt to resolve the paradoxes of black holes. In , Stephen Hawking published a calculation which suggested that black holes are not completely black but emit a dim radiation due to quantum effects near the event horizon.

This property is usually referred to as unitarity of time evolution. The apparent contradiction between Hawking's calculation and the unitarity postulate of quantum mechanics came to be known as the black hole information paradox. This state of matter arises for brief instants when heavy ions such as gold or lead nuclei are collided at high energies.

The physics of the quark—gluon plasma is governed by a theory called quantum chromodynamics , but this theory is mathematically intractable in problems involving the quark—gluon plasma. The calculation showed that the ratio of two quantities associated with the quark—gluon plasma, the shear viscosity and volume density of entropy, should be approximately equal to a certain universal constant. In , the predicted value of this ratio for the quark—gluon plasma was confirmed at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory.

Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques.

So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an insulator. A superfluid is a system of electrically neutral atoms that flows without any friction. Such systems are often produced in the laboratory using liquid helium , but recently experimentalists have developed new ways of producing artificial superfluids by pouring trillions of cold atoms into a lattice of criss-crossing lasers.

These atoms initially behave as a superfluid, but as experimentalists increase the intensity of the lasers, they become less mobile and then suddenly transition to an insulating state.

During the transition, the atoms behave in an unusual way. For example, the atoms slow to a halt at a rate that depends on the temperature and on Planck's constant , the fundamental parameter of quantum mechanics, which does not enter into the description of the other phases.

This behavior has recently been understood by considering a dual description where properties of the fluid are described in terms of a higher dimensional black hole. In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing models of real world physics that combine general relativity and particle physics.

Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. String phenomenology is the part of string theory that attempts to construct realistic or semi-realistic models based on string theory.

Partly because of theoretical and mathematical difficulties and partly because of the extremely high energies needed to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to any of these models being a correct fundamental description of nature.

This has led some in the community to criticize these approaches to unification and question the value of continued research on these problems.

The currently accepted theory describing elementary particles and their interactions is known as the standard model of particle physics. This theory provides a unified description of three of the fundamental forces of nature: Despite its remarkable success in explaining a wide range of physical phenomena, the standard model cannot be a complete description of reality.

This is because the standard model fails to incorporate the force of gravity and because of problems such as the hierarchy problem and the inability to explain the structure of fermion masses or dark matter. String theory has been used to construct a variety of models of particle physics going beyond the standard model. Typically, such models are based on the idea of compactification. Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions.

By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles.

Such compactifications offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct realistic or semi-realistic models of our four-dimensional world based on M-theory. The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. Despite its success in explaining many observed features of the universe including galactic redshifts , the relative abundance of light elements such as hydrogen and helium , and the existence of a cosmic microwave background , there are several questions that remain unanswered.

For example, the standard Big Bang model does not explain why the universe appears to be same in all directions, why it appears flat on very large distance scales, or why certain hypothesized particles such as magnetic monopoles are not observed in experiments. Currently, the leading candidate for a theory going beyond the Big Bang is the theory of cosmic inflation. Developed by Alan Guth and others in the s, inflation postulates a period of extremely rapid accelerated expansion of the universe prior to the expansion described by the standard Big Bang theory.

The theory of cosmic inflation preserves the successes of the Big Bang while providing a natural explanation for some of the mysterious features of the universe. In the theory of inflation, the rapid initial expansion of the universe is caused by a hypothetical particle called the inflaton.

The exact properties of this particle are not fixed by the theory but should ultimately be derived from a more fundamental theory such as string theory. While these approaches might eventually find support in observational data such as measurements of the cosmic microwave background, the application of string theory to cosmology is still in its early stages.

In addition to influencing research in theoretical physics , string theory has stimulated a number of major developments in pure mathematics. Like many developing ideas in theoretical physics, string theory does not at present have a mathematically rigorous formulation in which all of its concepts can be defined precisely. As a result, physicists who study string theory are often guided by physical intuition to conjecture relationships between the seemingly different mathematical structures that are used to formalize different parts of the theory.

These conjectures are later proved by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics. After Calabi—Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi—Yau manifold.

In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry. Regardless of whether Calabi—Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between different string theories has significant mathematical consequences.

The Calabi—Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry , a branch of mathematics concerned with counting the numbers of solutions to geometric questions. Enumerative geometry studies a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials.

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For example, the Clebsch cubic illustrated on the right is an algebraic variety defined using a certain polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface.

Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi—Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five. This problem was solved by the nineteenth-century German mathematician Hermann Schubert , who found that there are exactly 2, such lines.

In , geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is , By the year , most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish. Originally, these results of Candelas were justified on physical grounds. However, mathematicians generally prefer rigorous proofs that do not require an appeal to physical intuition.

Inspired by physicists' work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry.

Group theory is the branch of mathematics that studies the concept of symmetry.

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For example, one can consider a geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle without changing its shape. Each of these operations is called a symmetry , and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a group. In this particular example, the group is known as the dihedral group of order 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a finite group.

Mathematicians often strive for a classification or list of all mathematical objects of a given type. It is generally believed that finite groups are too diverse to admit a useful classification. A more modest but still challenging problem is to classify all finite simple groups.

These are finite groups which may be used as building blocks for constructing arbitrary finite groups in the same way that prime numbers can be used to construct arbitrary whole numbers by taking products. This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances.

The largest sporadic group, the so-called monster group , has over 10 53 elements, more than a thousand times the number of atoms in the Earth. A seemingly unrelated construction is the j -function of number theory.

This object belongs to a special class of functions called modular functions , whose graphs form a certain kind of repeating pattern. In the late s, mathematicians John McKay and John Thompson noticed that certain numbers arising in the analysis of the monster group namely, the dimensions of its irreducible representations are related to numbers that appear in a formula for the j -function namely, the coefficients of its Fourier series.

In , Richard Borcherds constructed a bridge between the theory of modular functions and finite groups and, in the process, explained the observations of McKay and Thompson. Since the s, the connection between string theory and moonshine has led to further results in mathematics and physics.

Harvey proposed a generalization of this moonshine phenomenon called umbral moonshine , [] and their conjecture was proved mathematically by Duncan, Michael Griffin, and Ken Ono. 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.

Thereafter, German mathematician Theodor Kaluza combined the fifth dimension with general relativity , and only Kaluza is usually credited with the idea.

In , 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 s and early s as a never completely successful theory of hadrons , the subatomic particles like the proton and neutron that feel the strong interaction. In the s, 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 that 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 s as a way of constructing a theory that 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 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 that 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 that 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 that 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 that 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 that 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 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 In —70, 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 , Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. In the fermion theories, the critical dimension was 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 , Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle that 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 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 The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively.

In , 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. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices. In the early s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino.

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 and , hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution.

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 that 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.

Daniel Friedan , Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring using conformal field theory techniques. David Gross and Vipul Periwal discovered that string perturbation theory was divergent. Stephen Shenker showed it diverged much faster than in field theory suggesting that new non-perturbative objects were missing.

Superstrings and other things

In the s, 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 , at the annual conference of string theorists at the University of Southern California USC , Edward Witten gave a speech on string theory that in essence united the five string theories that existed at the time, and giving birth to a new 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 that began at this time is sometimes called the second superstring revolution.

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.

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 , 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. 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.

To construct models of particle physics based on string theory, physicists typically begin by specifying a shape for the extra dimensions of spacetime.

Each of these different shapes corresponds to a different possible universe, or "vacuum state", with a different collection of particles and forces. String theory as it is currently understood has an enormous number of vacuum states, typically estimated to be around 10 , and these might be sufficiently diverse to accommodate almost any phenomena that might be observed at low energies. Many critics of string theory have expressed concerns about the large number of possible universes described by string theory.

In his book Not Even Wrong , Peter Woit , a lecturer in the mathematics department at Columbia University , has argued that the large number of different physical scenarios renders string theory vacuous as a framework for constructing models of particle physics.

According to Woit,. The possible existence of, say, 10 consistent different vacuum states for superstring theory probably destroys the hope of using the theory to predict anything. If one picks among this large set just those states whose properties agree with present experimental observations, it is likely there still will be such a large number of these that one can get just about whatever value one wants for the results of any new observation.

Some physicists believe this large number of solutions is actually a virtue because it may allow a natural anthropic explanation of the observed values of physical constants , in particular the small value of the cosmological constant. In , Steven Weinberg published an article in which he argued that the cosmological constant could not have been too large, or else galaxies and intelligent life would not have been able to develop. String theorist Leonard Susskind has argued that string theory provides a natural anthropic explanation of the small value of the cosmological constant.

The fact that the observed universe has a small cosmological constant is just a tautological consequence of the fact that a small value is required for life to exist. Speculative scientific ideas fail not just when they make incorrect predictions, but also when they turn out to be vacuous and incapable of predicting anything. One of the fundamental properties of Einstein's general theory of relativity is that it is background independent , meaning that the formulation of the theory does not in any way privilege a particular spacetime geometry.

One of the main criticisms of string theory from early on is that it is not manifestly background independent. In string theory, one must typically specify a fixed reference geometry for spacetime, and all other possible geometries are described as perturbations of this fixed one. In his book The Trouble With Physics , physicist Lee Smolin of the Perimeter Institute for Theoretical Physics claims that this is the principal weakness of string theory as a theory of quantum gravity, saying that string theory has failed to incorporate this important insight from general relativity.

Others have disagreed with Smolin's characterization of string theory. In a review of Smolin's book, string theorist Joseph Polchinski writes.

New physical theories are often discovered using a mathematical language that is not the most suitable for them… In string theory it has always been clear that the physics is background-independent even if the language being used is not, and the search for more suitable language continues. Polchinski notes that an important open problem in quantum gravity is to develop holographic descriptions of gravity which do not require the gravitational field to be asymptotically anti-de Sitter.

Since the superstring revolutions of the s and s, string theory has become the dominant paradigm of high energy theoretical physics. In an interview from , Nobel laureate David Gross made the following controversial comments about the reasons for the popularity of string theory:.

The most important [reason] is that there are no other good ideas around. That's what gets most people into it. When people started to get interested in string theory they didn't know anything about it.

In fact, the first reaction of most people is that the theory is extremely ugly and unpleasant, at least that was the case a few years ago when the understanding of string theory was much less developed.

It was difficult for people to learn about it and to be turned on. So I think the real reason why people have got attracted by it is because there is no other game in town. All other approaches of constructing grand unified theories, which were more conservative to begin with, and only gradually became more and more radical, have failed, and this game hasn't failed yet. Several other high-profile theorists and commentators have expressed similar views, suggesting that there are no viable alternatives to string theory.

Many critics of string theory have commented on this state of affairs. In his book criticizing string theory, Peter Woit views the status of string theory research as unhealthy and detrimental to the future of fundamental physics. He argues that the extreme popularity of string theory among theoretical physicists is partly a consequence of the financial structure of academia and the fierce competition for scarce resources.

According to Smolin,. String theory is a powerful, well-motivated idea and deserves much of the work that has been devoted to it. If it has so far failed, the principal reason is that its intrinsic flaws are closely tied to its strengths—and, of course, the story is unfinished, since string theory may well turn out to be part of the truth.

The real question is not why we have expended so much energy on string theory but why we haven't expended nearly enough on alternative approaches. Smolin goes on to offer a number of prescriptions for how scientists might encourage a greater diversity of approaches to quantum gravity research. From Wikipedia, the free encyclopedia. This article is about physics. For string algorithms, see String computer science. For other uses, see String disambiguation. For a more accessible and less technical introduction to this topic, see Introduction to M-theory.

Related concepts. Verlinde H. Verlinde Witten Yau Zaslow. Main article: String physics. Main articles: S-duality and T-duality. Matrix theory physics. String phenomenology. String cosmology. Mirror symmetry string theory. Monstrous moonshine. History of string theory. String theory landscape.

Background independence. Retrieved 25 July Archived from the original on November 5, Retrieved December 31, CS1 maint: Dirichlet Branes and Mirror Symmetry. Clay Mathematics Monographs. American Mathematical Society. A conjecture". Physical Review D. String theory and M-theory: A modern introduction.

Cambridge University Press. Bekenstein, Jacob Physics Letters B. Borcherds, Richard Inventiones Mathematicae. Nuclear Physics B. Communications in Number Theory and Physics. Connes, Alain Noncommutative Geometry.

Academic Press. Journal of High Energy Physics. Conway, John; Norton, Simon London Math. Foundations of Physics.Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle that appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon.

New physical theories are often discovered using a mathematical language that is not the most suitable for them… In string theory it has always been clear that the physics is background-independent even if the language being used is not, and the search for more suitable language continues. Help Center Find new research papers in: The exact properties of this particle are not fixed by the theory but should ultimately be derived from a more fundamental theory such as string theory.

The results are presented in Section 3. Experimental Mathematics. The fraction of the total string length in disconnected A- or B-loops is shown in Fig. Although it was originally developed in this very particular and physically unrealistic context in string theory, the entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity.