Group theory is the branch of mathematics that studies the concept of symmetry. 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. One can rotate it through 120°, 240°, or 360°, or one can reflect in any of the lines labeled S_{0}, S_{1}, or S_{2} in the picture. 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.^{ [104] }
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.^{ [lower-alpha 6] } One of the major achievements of contemporary group theory is the classification of finite simple groups, a mathematical theorem which provides a list of all possible finite simple groups.^{ [104] }
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.^{ [105] }
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.^{ [106] } Although this function appears in a branch of mathematics which seems very different from the theory of finite groups, the two subjects turn out to be intimately related. In the late 1970s, 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).^{ [107] } This relationship was further developed by John Horton Conway and Simon Norton ^{ [108] } who called it monstrous moonshine because it seemed so far fetched.^{ [109] }
In 1992, 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.^{ [110] }^{ [111] } Borcherds' work used ideas from string theory in an essential way, extending earlier results of Igor Frenkel, James Lepowsky, and Arne Meurman, who had realized the monster group as the symmetries of a particular^{[ which? ]} version of string theory.^{ [112] } In 1998, Borcherds was awarded the Fields medal for his work.^{ [113] }
Since the 1990s, the connection between string theory and moonshine has led to further results in mathematics and physics.^{ [105] } In 2010, physicists Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa discovered connections between a different sporadic group, the Mathieu group M_{24}, and a certain version^{[ which? ]} of string theory.^{ [114] } Miranda Cheng, John Duncan, and Jeffrey A. Harvey proposed a generalization of this moonshine phenomenon called umbral moonshine,^{ [115] } and their conjecture was proved mathematically by Duncan, Michael Griffin, and Ken Ono.^{ [116] } Witten has also speculated that the version of string theory appearing in monstrous moonshine might be related to a certain simplified model of gravity in three spacetime dimensions.^{ [117] }
Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein. The first person to add a fifth dimension to a theory of gravity was Gunnar Nordström in 1914, who noted that gravity in five dimensions describes both gravity and electromagnetism in four. Nordström attempted to unify electromagnetism with his theory of gravitation, which was however superseded by Einstein's general relativity in 1919. Thereafter, German mathematician Theodor Kaluza combined the fifth dimension with general relativity, and only Kaluza is usually credited with the idea. In 1926, the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension—it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions.
String theory was originally developed during the late 1960s and early 1970s as a never completely successful theory of hadrons, the subatomic particles like the proton and neutron that feel the strong interaction. In the 1960s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories 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 1940s 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 26. Charles Thorn, Peter Goddard and Richard Brower went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to 26.
In 1969–70, 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 1971, Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. John Schwarz and André Neveu added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10. Stanley Mandelstam formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave a different formulation of the bosonic string, as a string field theory, with infinitely many particle types and with fields taking values not on points, but on loops and curves.
In 1974, Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle 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 1977 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 1984. The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively. In 1979, Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein equations of general relativity, emerge from the renormalization group equations for the two-dimensional field theory. Schwarz and Green discovered T-duality, and constructed two superstring theories—IIA and IIB related by T-duality, and type I theories with open strings. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices.
In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino. This led him, in collaboration with Luis Álvarez-Gaumé, to study violations of the conservation laws in gravity theories with anomalies, concluding that type I string theories were inconsistent. Green and Schwarz discovered a contribution to the anomaly that Witten and Alvarez-Gaumé had missed, which restricted the gauge group of the type I string theory to be SO(32). In coming to understand this calculation, Edward Witten became convinced that string theory was truly a consistent theory of gravity, and he became a high-profile advocate. Following Witten's lead, between 1984 and 1986, hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution.^{[ citation needed ]}
During this period, David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm discovered heterotic strings. The gauge group of these closed strings was two copies of E8, and either copy could easily and naturally include the standard model. Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten found that the Calabi–Yau manifolds are the compactifications 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.^{[ citation needed ]}
In the 1990s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and identified these with the black-hole solutions of supergravity. These were understood to be the new objects suggested by the perturbative divergences, and they opened up a new field with rich mathematical structure. It quickly became clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the physical interpretation of the strings and branes was revealed—they are a type of black hole. Leonard Susskind had incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly excited string states with ordinary thermal black hole states. As suggested by 't Hooft, the fluctuations of the black hole horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all nearby objects too.
In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten gave a speech on string theory that in essence united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called M-theory. M-theory was also foreshadowed in the work of Paul Townsend at approximately the same time. The flurry of activity that began at this time is sometimes called the second superstring revolution.^{ [31] }
During this period, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated matrix theory, a full holographic description of M-theory using IIA D0 branes.^{ [48] } This was the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the holographic principle. It is an example of a gauge-gravity duality and is now understood to be a special case of the AdS/CFT correspondence. 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.^{ [59] } Petr Hořava and Witten found the eleven-dimensional formulation of the heterotic string theories, showing that orbifolds solve the chirality problem. Witten noted that the effective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered in terms of the location of the branes.
In 1997, Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to the horizon, which for extreme charged black holes looks like an anti-de Sitter space.^{ [68] } He noted that in this limit the gauge theory describes the string excitations near the branes. So he hypothesized that string theory on a near-horizon extreme-charged black-hole geometry, an anti-de Sitter space times a sphere with flux, is equally well described by the low-energy limiting gauge theory, the N = 4 supersymmetric Yang–Mills theory. This hypothesis, which is called the AdS/CFT correspondence, was further developed by Steven Gubser, Igor Klebanov and Alexander Polyakov,^{ [69] } and by Edward Witten,^{ [70] } and it is now well-accepted. It is a concrete realization of the holographic principle, which has far-reaching implications for black holes, locality and information in physics, as well as the nature of the gravitational interaction.^{ [53] } Through this relationship, string theory has been shown to be related to gauge theories like quantum chromodynamics and this has led to more quantitative understanding of the behavior of hadrons, bringing string theory back to its roots.^{[ citation needed ]}
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^{500}, and these might be sufficiently diverse to accommodate almost any phenomenon that might be observed at low energies.^{ [118] }
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^{500} 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.^{ [119] }
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.^{ [119] } The anthropic principle is the idea that some of the numbers appearing in the laws of physics are not fixed by any fundamental principle but must be compatible with the evolution of intelligent life. In 1987, 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.^{ [120] } Weinberg suggested that there might be a huge number of possible consistent universes, each with a different value of the cosmological constant, and observations indicate a small value of the cosmological constant only because humans happen to live in a universe that has allowed intelligent life, and hence observers, to exist.^{ [121] }
String theorist Leonard Susskind has argued that string theory provides a natural anthropic explanation of the small value of the cosmological constant.^{ [122] } According to Susskind, the different vacuum states of string theory might be realized as different universes within a larger multiverse. 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.^{ [123] } Many prominent theorists and critics have disagreed with Susskind's conclusions.^{ [124] } According to Woit, "in this case [anthropic reasoning] is nothing more than an excuse for failure. 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."^{ [125] }
No vacuum in the string theory landscape is known to support a metastable, positive cosmological constant, except possibly one unconfirmed model described by Kachru et al. in 2003.^{ [126] } In 2018, a group of four physicists advanced a controversial conjecture which would imply that no such universe exists. This is contrary to some popular models of dark energy such as Λ-CDM, which requires a positive vacuum energy. However, string theory is likely compatible with certain types of quintessence, where dark energy is caused by a new field with exotic properties.^{ [127] }
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.^{ [128] }
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.^{ [129] }
Others have disagreed with Smolin's characterization of string theory. In a review of Smolin's book, string theorist Joseph Polchinski writes
[Smolin] is mistaking an aspect of the mathematical language being used for one of the physics being described. 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. Indeed, as Smolin belatedly notes, [AdS/CFT] provides a solution to this problem, one that is unexpected and powerful.^{ [130] }
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.^{ [130] } Smolin has responded by saying that the AdS/CFT correspondence, as it is currently understood, may not be strong enough to resolve all concerns about background independence.^{ [131] }
Since the superstring revolutions of the 1980s and 1990s, string theory has become the dominant paradigm of high energy theoretical physics.^{ [132] } Some string theorists have expressed the view that there does not exist an equally successful alternative theory addressing the deep questions of fundamental physics. In an interview from 1987, 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.^{ [133] }
Several other high-profile theorists and commentators have expressed similar views, suggesting that there are no viable alternatives to string theory.^{ [134] }
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.^{ [135] } In his book The Road to Reality , mathematical physicist Roger Penrose expresses similar views, stating "The often frantic competitiveness that this ease of communication engenders leads to bandwagon effects, where researchers fear to be left behind if they do not join in."^{ [136] } Penrose also claims that the technical difficulty of modern physics forces young scientists to rely on the preferences of established researchers, rather than forging new paths of their own.^{ [137] } Lee Smolin expresses a slightly different position in his critique, claiming that string theory grew out of a tradition of particle physics which discourages speculation about the foundations of physics, while his preferred approach, loop quantum gravity, encourages more radical thinking. 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.^{ [138] }
Smolin goes on to offer a number of prescriptions for how scientists might encourage a greater diversity of approaches to quantum gravity research.^{ [139] }
In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravity. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with renormalization in general relativity. In string theory, believed to be a consistent theory of quantum gravity, the graviton is a massless state of a fundamental string.
The holographic principle is a tenet of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region—such as a light-like boundary like a gravitational horizon. First proposed by Gerard 't Hooft, it was given a precise string-theory interpretation by Leonard Susskind, who combined his ideas with previous ones of 't Hooft and Charles Thorn. As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. The prime example of holography is the AdS/CFT correspondence.
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 the spring of 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution.
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored, such as in the vicinity of black holes or similar compact astrophysical objects where the effects of gravity are strong, such as neutron stars.
A theory of everything, final theory, ultimate theory, or master theory is a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe. Finding a TOE is one of the major unsolved problems in physics. String theory and M-theory have been proposed as theories of everything. Over the past few centuries, two theoretical frameworks have been developed that, together, most closely resemble a TOE. These two theories upon which all modern physics rests are general relativity and quantum mechanics. General relativity is a theoretical framework that only focuses on gravity for understanding the universe in regions of both large scale and high mass: stars, galaxies, clusters of galaxies, etc. On the other hand, quantum mechanics is a theoretical framework that only focuses on three non-gravitational forces for understanding the universe in regions of both small scale and low mass: sub-atomic particles, atoms, molecules, etc. Quantum mechanics successfully implemented the Standard Model that describes the three non-gravitational forces – strong nuclear, weak nuclear, and electromagnetic force – as well as all observed elementary particles.
Edward Witten is an American mathematical and theoretical physicist. He is currently the Charles Simonyi Professor in the School of Natural Sciences at the Institute for Advanced Study. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity. He is considered to be the practical founder of M-theory.
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 an imaginary 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, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles.
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.
Andrew Eben Strominger is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his work on Calabi–Yau compactification and topology change in string theory, and on the stringy origin of black hole entropy. He is a senior fellow at the Society of Fellows, and is the Gwill E. York Professor of Physics.
Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite", can in fact be viewed as "elementary" quantized particles with electrons playing the reverse role of "composite" topological solitons; the viewpoints are equivalent and the situation dependent on the duality. It was later proven to hold true when dealing with a N = 4 supersymmetric Yang–Mills theory. It is named after Finnish physicist Claus Montonen and British physicist David Olive after they proposed the idea in their academic paper Magnetic monopoles as gauge particles? where they state:
There should be two "dual equivalent" field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles.
In theoretical physics, topological string theory is a version of string theory. Topological string theory was first thought of and is studied by physicists such as Edward Witten and Cumrun Vafa.
In theoretical physics, the anti-de Sitter/quantum chromodynamics correspondence is a program to describe quantum chromodynamics (QCD) in terms of a dual gravitational theory, following the principles of the AdS/CFT correspondence in a setup where the quantum field theory is not a conformal field 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.
In non-technical terms, M-theory presents an idea about the basic substance of the universe. As of 2020, science has produced no experimental evidence to support the concept that M-theory is a description of the real world. Although a complete mathematical formulation of M-theory is not known, the general approach is the leading contender for a universal "Theory of Everything" that unifies gravity with other forces such as electromagnetism. M-theory aims to unify quantum mechanics with general relativity's gravitational force in a mathematically consistent way. In comparison, other theories such as loop quantum gravity are considered by physicists and researchers/students as less elegant because they posit gravity to be completely different from forces such as the electromagnetic force.
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.
In two spatial and one time dimensions, general relativity turns out to have no propagating gravitational degrees of freedom. In fact, it can be shown that in a vacuum, spacetime will always be locally flat. This makes (2+1)-dimensional topological gravity a topological theory with no gravitational local degrees of freedom.
This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, and high energy physics.
Gary T. Horowitz is an American theoretical physicist who works on string theory and quantum gravity.
Atish Dabholkar is an Indian theoretical physicist. He is currently the Director of the Abdus Salam International Centre for Theoretical Physics (ICTP) with the rank of Assistant Director-General, UNESCO. Prior to that, he was head of ICTP's High Energy, Cosmology and Astroparticle Physics section, and also Directeur de Recherche at the Centre National de la Recherche Scientifique (CNRS) at Sorbonne University in the "Laboratoire de Physique Théorique et Hautes Énergies" (LPTHE).
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