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{{shortShort description|Theory of subatomic structure}}

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{{Other uses}}

{{seeintroSeeintro|Introduction to M-theory}}

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{{String theory}}

In [[physics]], '''string theory''' is a [[Mathematical theory|theoretical framework]] in which the [[Point particle|point-like particles]] of [[particle physics]] are replaced by [[Dimension (mathematics and physics)|one-dimensional]] objects called [[String (physics)|strings]]. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks justacts like an ordinarya particle, with its [[mass]], [[charge (physics)|charge]], and other properties determined by the [[vibration]]al state of the string. In string theory, one of the many vibrational states of the string corresponds to the [[graviton]], a [[quantum mechanics|quantum mechanical]] particle that carries the [[gravity|gravitational force]]. Thus, string theory is a theory of [[quantum gravity]].

String theory is a broad and varied subject that attempts to address a number of deep questions of [[fundamental physics]]. String theory has contributed a number of advances to [[mathematical physics]], which have been applied to a variety of problems in [[black hole]] physics, early universe [[Physical cosmology|cosmology]], [[nuclear physics]], and [[condensed matter physics]], and it has stimulated a number of major developments in [[pure mathematics]]. Because string theory potentially provides a unified description of gravity and particle physics, it is a candidate for a [[theory of everything]], a self-contained [[mathematical model]] that describes all [[Fundamental interaction|fundamental force]]s and forms of [[matter]]. Despite much work on these problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of its details.

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String theory was first studied in the late 1960s as a theory of the [[strong nuclear force]], before being abandoned in favor of [[quantum chromodynamics]]. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, [[bosonic string theory]], incorporated only the class of [[particle]]s known as [[boson]]s. It later developed into [[superstring theory]], which posits a connection called [[supersymmetry]] between bosons and the class of particles called [[fermion]]s. Five consistent versions of superstring theory were developed before it was conjectured in the mid-1990s that they were all different limiting cases of a single theory in eleven dimensions known as [[M-theory]]. In late 1997, theorists discovered an important relationship called the [[anti-de Sitter/conformal field theory correspondence]] (AdS/CFT correspondence), which relates string theory to another type of physical theory called a [[quantum field theory]].

One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous [[stringString theory landscape|landscape]] of possible universes[[universe]]s, which has complicated efforts to develop theories of particle physics based on string theory. These issues have led some in the community to criticize these approaches to physics, and to question the value of continued research on string theory unification.

== Fundamentals ==

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In the 20th century, two theoretical frameworks emerged for formulating the laws of physics. The first is [[Albert Einstein]]'s [[general theory of relativity]], a theory that explains the force of [[gravity]] and the structure of [[spacetime]] at the macro-level. The other is [[quantum mechanics]], a completely different formulation, which uses known [[probability]] principles to describe physical phenomena at the micro-level. By the late 1970s, these two frameworks had proven to be sufficient to explain most of the observed features of the [[universe]], from [[elementary particle]]s to [[atom]]s to the evolution of stars and the universe as a whole.<ref name="Becker, Becker 2007, p. 1">[[#Becker|Becker, Becker and Schwarz]], p. 1</ref>

In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of [[quantum gravity]].<ref name="Becker, Becker 2007, p. 1"/> The general theory of relativity is formulated within the framework of [[classical physics]], whereas the other [[fundamental interaction|fundamental forces]] are described within the framework of quantum mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.<ref>[[#Zwiebach|Zwiebach]], p. 6</ref> In addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of [[atomic nucleus|atomic nuclei]], [[black hole]]s, and the early universe.{{efn|For example, physicists are still working to understand the phenomenon of [[quark confinement]], the paradoxes of [[black holes]], and the origin of [[dark energy]].}}

String theory is a [[mathematical theory|theoretical framework]] that attempts to address these questions.

String theory is a [[mathematical theory|theoretical framework]] that attempts to address these questions and many others. The starting point for string theory is the idea that the [[point particle|point-like particles]] of [[particle physics]] can also be modeled as one-dimensional objects called [[string (physics)|strings]]. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can [[vibration|vibrate]] in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle consistent with non-string models of elementary particles, with its [[mass]], [[charge (physics)|charge]], and other properties determined by the vibrational state of the string. String theory's application as a form of quantum gravity proposes a vibrational state responsible for the [[graviton]], a yet unproven quantum particle that is theorized to carry gravitational force.<ref name="Becker, Becker 2007, pp. 2">[[#Becker|Becker, Becker and Schwarz]], pp. 2–3</ref>

String theory is a [[mathematical theory|theoretical framework]] that attempts to address these questions and many others. The starting point for string theory is the idea that the [[point particle|point-like particles]] of [[particle physics]] can also be modeled as one-dimensional objects called [[string (physics)|strings]]. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can [[vibration|vibrate]] in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle consistent with non-string models of elementary particles, with its [[mass]], [[charge (physics)|charge]], and other properties determined by the vibrational state of the string. String theory's application as a form of quantum gravity proposes a vibrational state responsible for the [[graviton]], a yet unproven quantum particle that is theorized to carry gravitational force.<ref name="Becker, Becker 2007, pp. 2">[[#Becker|Becker, Becker and Schwarz]], pp. 2–3</ref>

One of the main developments of the past several decades in string theory was the discovery of certain 'dualities', mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered a number of these dualities between different versions of string theory, and this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as [[M-theory]].<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 9–12</ref>

One of the main developments of the past several decades in string theory was the discovery of certain 'dualities'—mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered a number of these dualities between different versions of string theory, and this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as [[M-theory]].<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 9–12</ref>

Studies of string theory have also yielded a number of results on the nature of black holes and the gravitational interaction. There are certain paradoxes that arise when one attempts to understand the quantum aspects of black holes, and work on string theory has attempted to clarify these issues. In late 1997 this line of work culminated in the discovery of the [[anti-de Sitter/conformal field theory correspondence]] or (AdS/CFT for short).<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 14–15</ref> This is a theoretical result that relates string theory to other physical theories which are better understood theoretically. The AdS/CFT correspondence has implications for the study of black holes and quantum gravity, and it has been applied to other subjects, including [[nuclear physics|nuclear]]<ref name="Klebanov and Maldacena 2009"/> and [[condensed matter physics]].<ref name="Merali 2011"/><ref name=Sachdev/>

Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that it will eventually be developed to the point where it fully describes our universe, making it a [[theory of everything]]. One of the goals of current research in string theory is to find a solution of the theory that reproduces the observed spectrum of elementary particles, with a small [[cosmological constant]], containing [[dark matter]] and a plausible mechanism for [[cosmic inflation]]. While there has been progress toward these goals, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of details.<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 3, 15–16</ref>

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{{main|String (physics)}}

[[Image:World lines and world sheet.svg|left|thumb|250pxupright=1.2|Interaction in the quantum world: [[worldline]]s of point-like [[particles]] or a [[worldsheet]] swept up by closed [[string (physics)|strings]] in string theory]]

The application of quantum mechanics to physical objects such as the [[electromagnetic field]], which are extended in space and time, is known as [[quantum field theory]]. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields.<ref name="Zee 2010"/>

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=== Dualities ===

[[File:Dualities in String Theory.svg|right|thumb|alt=A diagram indicating the relationships between M-theory and the five superstring theories.|450pxupright=2|A diagram of string theory dualities. Blue edges indicate [[S-duality]]. Red edges indicate [[T-duality]].]]

{{main|S-duality|T-duality}}

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== M-theory ==

{{main|M-theory}}

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=== Unification of superstring theories ===

[[File:Limits of M-theory.svg|right|350pxupright=1.6|thumb|alt=A star-shaped diagram with the various limits of M-theory labeled at its six vertices.|A schematic illustration of the relationship between [[M-theory]], the five [[superstring theory|superstring theories]], and eleven-dimensional [[supergravity]]. The shaded region represents a family of different physical scenarios that are possible in M-theory. In certain limiting cases corresponding to the cusps, it is natural to describe the physics using one of the six theories labeled there.]]

In the 1970s, many physicists became interested in [[supergravity]] theories, which combine general relativity with supersymmetry. Whereas general relativity makes sense in any number of dimensions, supergravity places an upper limit on the number of dimensions.<ref>[[#Duff1998|Duff]], p. 64</ref> In 1978, work by [[Werner Nahm]] showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven.<ref name=Nahm/> In the same year, [[Eugene Cremmer]], [[Bernard Julia]], and [[Joël Scherk]] of the [[École Normale Supérieure]] showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.<ref name=Cremmer/><ref name="Duff 1998, p. 65">[[#Duff1998|Duff]], p. 65</ref>

Initially, many physicists hoped that by compactifying [[eleven-dimensional supergravity]], it might be possible to construct realistic models of our four-dimensional world. The hope was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the [[strong nuclear force|strong]] and [[weak nuclear force]]s, and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as [[chirality (physics)|chirality]]. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.<ref name="Duff 1998, p. 65"/>

In the [[first superstring revolution]] in 1984, many physicists turned to string theory as a unified theory of particle physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects.<ref name="Duff 1998, p. 65"/> Another feature of string theory that many physicists were drawn to in the 1980s and 1990s was its high degree of uniqueness. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary [[Lagrangian (field theory)|Lagrangian]]. In string theory, the possibilities are much more constrained: by the 1990s, physicists had argued that there were only five consistent supersymmetric versions of the theory.<ref name="Duff 1998, p. 65"/>

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In the branch of physics called [[statistical mechanics]], [[entropy]] is a measure of the randomness or disorder of a physical system. This concept was studied in the 1870s by the Austrian physicist [[Ludwig Boltzmann]], who showed that the [[thermodynamics|thermodynamic]] properties of a [[gas]] could be derived from the combined properties of its many constituent [[molecule]]s. Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give a precise definition of entropy as the [[natural logarithm]] of the number of different states of the molecules (also called ''microstates'') that give rise to the same macroscopic features.<ref>[[#Yau|Yau and Nadis]], pp. 187–188</ref>

In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases, the entropy scales with the volume. In the 1970s, 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 aremay lost toescape its gravitational attraction.<ref name=Bekenstein/> When combined with ideas of the physicist [[Stephen Hawking]],<ref name=Hawking1975/> Bekenstein's work yielded a precise formula for the entropy of a black hole. The [[Bekenstein–Hawking formula]] expresses the entropy {{math|''S''}} as

: <math>S= \frac{c^3kA}{4\hbar G}</math>

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=== Derivation within string theory ===

In a paper from 1996, [[Andrew Strominger]] and [[Cumrun Vafa]] showed how to derive the Beckenstein–HawkingBekenstein–Hawking formula for certain black holes in string theory.<ref name="Strominger and Vafa 1996"/> Their calculation was based on the observation that D-branes—which look like fluctuating membranes when they are weakly interacting—become dense, massive objects with event horizons when the interactions are strong. 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. Their calculation reproduced the Bekenstein–Hawking formula exactly, including the factor of {{math|1/4}}.<ref>[[#Yau|Yau and Nadis]], pp. 190–192</ref> Subsequent work by Strominger, Vafa, and others refined the original calculations and gave the precise values of the "quantum corrections" needed to describe very small black holes.<ref name=MSW/><ref name=OST/>

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 hole]]s in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge.<ref>[[#Yau|Yau and Nadis]], pp. 192–193</ref> Strominger and Vafa also restricted attention to black holes in five-dimensional spacetime with unphysical supersymmetry.<ref>[[#Yau|Yau and Nadis]], pp. 194–195</ref>

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{{main|AdS/CFT correspondence}}

One approach to formulating string theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. This is a theoretical result whichthat implies that string theory is in some cases equivalent to a quantum field theory. In addition to providing insights into the mathematical structure of string theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.<ref name="Klebanov and Maldacena 2009"/> The AdS/CFT correspondence was first proposed by [[Juan Maldacena]] in late 1997.<ref name=Maldacena1998/> Important aspects of the correspondence were elaborated in articles by [[Steven Gubser]], [[Igor Klebanov]], and [[Alexander Markovich Polyakov]],<ref name=Gubser/> and by Edward Witten.<ref name=Witten1998/> By 2010, Maldacena's article had over 7000 citations, becoming the most highly cited article in the field of [[high energy physics]].{{efn|{{cite web |url=http://www.slac.stanford.edu/spires/topcites/2010/eprints/to_hep-th_annual.shtml |title=Top Cited Articles during 2010 in hep-th |author=<!--Staff writer(s); no by-line.--> |access-date=25 July 2013}}}}

=== Overview of the correspondence ===

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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.<ref name="Maldacena 2005, p. 60"/> It looks like a solid [[cylinder (geometry)|cylinder]] in which any [[cross section (geometry)|cross section]] is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is [[curvature|curved]] in such a way that any point in the interior is actually infinitely far from this boundary surface.<ref name="Maldacena 2005, p. 61"/>

[[File:AdS3.svg|thumb|right|alt=A cylinder formed by stacking copies of the disk illustrated in the previous figure.|350pxupright=1.6|Three-dimensional [[anti-de Sitter space]] is like a stack of [[Poincaré disk model|hyperbolic disks]], each one representing the state of the universe at a given time. The resulting [[spacetime]] looks like a solid [[cylinder (geometry)|cylinder]].]]

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.<ref name="Maldacena 2005, p. 60"/>

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 nongravitationalnon-gravitational physics.<ref>[[#Zwiebach|Zwiebach]], p. 552</ref> One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory. 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.<ref>[[#Maldacena2005|Maldacena 2005]], pp. 61–62</ref>

=== Applications to quantum gravity ===

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The AdS/CFT correspondence has also been used to study aspects of condensed matter physics. Over the decades, [[experimental physics|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. Some condensed matter theorists including [[Subir Sachdev]] hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string theory and learn more about their behavior.<ref name="Merali 2011"/>

So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an [[insulator (electricity)|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 the [[Planck's constant]], the fundamental parameter of quantum mechanics, which does not enter into the description of the other [[phase (matter)|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.<ref name=Sachdev/>

== Phenomenology ==

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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: electromagnetism and the strong and weak nuclear forces. 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.<ref name=Candelas1985/> One popular way of deriving realistic physics from string theory is to start with the heterotic theory in ten dimensions and assume that the six extra dimensions of spacetime are shaped like a six-dimensional Calabi–Yau manifold. Such compactifications offer many ways of extracting realistic physics from string theory.<ref>{{cite arXiv|last1=Cvetic|first1=M|authorlink1=Mirjam Cvetič|last2=Halverson|first2=J.|authorlink2=|last3=Shiu|first3=G.|authorlink3=Gary Shiu|last4=Taylor|first4=W.|authorlink4=|date=2022|title=Snowmass White Paper: String Theory and Particle Physics|pages=|class=hep-th|eprint=2204.01742}}</ref> Other similar methods can be used to construct realistic or semi-realistic models of our four-dimensional world based on M-theory.<ref>[[#Yau|Yau and Nadis]], pp. 147–150</ref>

=== Cosmology ===

{{main|String cosmology}}

[[File:IlcWMAP 9yr moll40962012.png|thumb|right|300pxupright=1.4|A map of the [[cosmic microwave background]] produced by the [[Wilkinson Microwave Anisotropy Probe]]]]

The Big Bang theory is the prevailing [[physical cosmology|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 [[redshift]]s, 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 the 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.<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 530–531</ref>

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{{main|Monstrous moonshine}}

[[File:Labeled Triangle Reflections.svg|left|thumb|200pxupright=1|alt=An equilateral triangle with a line joining each vertex to the midpoint of the opposite side|An equilateral triangle can be rotated through 120°, 240°, or 360°, or reflected in any of the three lines pictured without changing its shape.]]

[[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 {{math|''S''₀}}, {{math|''S''₁}}, or {{math|''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 (mathematics)|group]]. In this particular example, the group is known as the [[dihedral group]] of [[order (group theory)|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]].<ref name=Dummit/>

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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 {{math|10⁵³}} elements, more than a thousand times the number of atoms in the Earth.<ref name="Klarreich 2015"/>

[[Image:KleinInvariantJ.jpg|right|thumb|300pxupright=1.4|A graph of the [[j-invariant|{{math|''j''}}-function]] in the complex plane]]

A seemingly unrelated construction is the [[j-invariant|{{math|''j''}}-function]] of [[number theory]]. This object belongs to a special class of functions called [[modular function]]s, whose graphs form a certain kind of repeating pattern.<ref>[[#Gannon|Gannon]], p. 2</ref> Although this function appears in a branch of mathematics that 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 (mathematician)|John McKay]] and [[John G. Thompson|John Thompson]] noticed that certain numbers arising in the analysis of the monster group (namely, the dimensions of its [[irreducible representation]]s) are related to numbers that appear in a formula for the {{math|''j''}}-function (namely, the coefficients of its [[Fourier series]]).<ref>[[#Gannon|Gannon]], p. 4</ref> This relationship was further developed by [[John Horton Conway]] and [[Simon P. Norton|Simon Norton]]<ref name=Conway/> who called it [[monstrous moonshine]] because it seemed so far fetched.<ref>[[#Gannon|Gannon]], p. 5</ref>

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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.<ref>[[#Gannon|Gannon]], p. 8</ref><ref name=Borcherds/> 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|date=February 2016}} version of string theory.<ref name=FLM/> In 1998, Borcherds was awarded the [[Fields medal]] for his work.<ref>[[#Gannon|Gannon]], p. 11</ref>

Since the 1990s, the connection between string theory and moonshine has led to further results in mathematics and physics.<ref name="Klarreich 2015"/> In 2010, physicists [[Tohru Eguchi]], [[Hirosi Ooguri]], and [[Yuji Tachikawa (physicist)|Yuji Tachikawa]] discovered connections between a different sporadic group, the [[Mathieu group M24|Mathieu group {{math|''M''₂₄}}]], and a certain version{{which|date=November 2016}} of string theory.<ref name=EOT/> [[Miranda Cheng]], John Duncan, and [[Jeffrey A. Harvey]] proposed a generalization of this moonshine phenomenon called [[umbral moonshine]],<ref name=CDH/> and their conjecture was proved mathematically by Duncan, Michael Griffin, and [[Ken Ono]].<ref name=DGO/> 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.<ref name=Witten2007/>

== History ==

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[[File:GabrieleVeneziano.jpg|right|thumb|upright|[[Gabriele Veneziano]]]]

The result was widely advertised by [[Murray Gell-Mann]], leading [[Gabriele Veneziano]] to construct a [[Veneziano scattering amplitude|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—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 model|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 Ángel Virasoro (physicist)|Miguel Virasoro]] and Joel Shapiro found a different amplitude now understood to be that of closed strings, while [[Ziro Koba]] and [[Holger Bech Nielsen|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 (physicist)|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 1969–701969–1970, [[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 (physicist)|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 algebra|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 Henry Schwarz|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.

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=== Compatibility with dark energy ===

It remains unknown whether string theory is compatible with a metastable, positive [[cosmological constant]].

Some putative examples of such solutions do exist, such as the model described by Kachru ''et al''. in 2003.<ref>{{cite journal |last1=Kachru |first1=Shamit |last2=Kallosh |first2=Renata |last3=Linde |first3=Andrei |last4=Trivedi |first4=Sandip P. |title=de Sitter Vacua in String Theory |journal=[[Phys. Rev. D]] |date=2003 |volume=68 |issue=4 |page=046005 |doi=10.1103/PhysRevD.68.046005 |arxiv=hep-th/0301240 |bibcode=2003PhRvD..68d6005K |s2cid=119482182 | issn=0556-2821 }}</ref> In 2018, a group of four physicists advanced a controversial conjecture which would imply that [[Swampland (physics)|no such universe exists]]. This is contrary to some popular models of [[dark energy]] such as [[Lambda-CDM model|Λ-CDM]], which requires a positive vacuum energy. However, string theory is likely compatible with certain types of [[quintessence (physics)|quintessence]], where dark energy is caused by a new field with exotic properties.<ref>{{cite web |last1=Wolchover |first1=Natalie |title=Dark Energy May Be Incompatible With String Theory |url=https://www.quantamagazine.org/dark-energy-may-be-incompatible-with-string-theory-20180809/ |website=[[Quanta Magazine]] |publisher=Simons Foundation |access-date=2 April 2020 |date=9 August 2018 |archive-date=15 November 2020 |archive-url=https://web.archive.org/web/20201115210807/https://www.quantamagazine.org/dark-energy-may-be-incompatible-with-string-theory-20180809/ |url-status=live }}</ref>

=== Background independence ===

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== Notes ==

{{Notelist}}

{{notelist}}

{{reflist|group=note}}

== References ==

{{reflist|30em|refs=

<ref name="Aspinwall et al. 2009">{{cite book |editor1-first=Paul |editor1-last=Aspinwall |editor2-first=Tom |editor2-last=Bridgeland |editor3-first=Alastair |editor3-last=Craw |editor4-first=Michael |editor4-last=Douglas |editor5-first=Mark |editor5-last=Gross |editor6-first=Anton |editor6-last=Kapustin |editor7-first=Gregory |editor7-last=Moore |editor8-first=Graeme |editor8-last=Segal |editor9-first=Balázs |editor9-last=Szendröi |editor10-first=P.M.H. |editor10-last=Wilson |title=Dirichlet Branes and Mirror Symmetry |year=2009 |publisher=American Mathematical Society | series = [[Clay Mathematics Monographs]] | volume = 4 | isbn=978-0-8218-3848-8|page=13}}</ref>

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<ref name=Lian1997>{{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=1997 |title=Mirror principle, I |journal=Asian Journal of Mathematics |volume=1 |issue=4 |pages=729–763|bibcode=1997alg.geom.12011L |arxiv=alg-geom/9712011 |doi=10.4310/ajm.1997.v1.n4.a5|s2cid=8035522 }}</ref>

<ref name=Lian1999>{{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=1999a |title=Mirror principle, II |journal=Asian Journal of Mathematics |volume=3 |pages=109–146|bibcode=1999math......5006L |arxiv=math/9905006 |doi=10.4310/ajm.1999.v3.n1.a6|s2cid=17837291 }}<{{br>}}

{{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=1999b |title=Mirror principle, III |journal=Asian Journal of Mathematics |volume=3 |issue=4 |pages=771–800|bibcode=1999math.....12038L |arxiv=math/9912038 |doi=10.4310/ajm.1999.v3.n4.a4}}</ref>

<ref name=Lian2000>{{cite journal |last1=Lian |first1=Bong |last2=Liu |first2=Kefeng |last3=Yau |first3=Shing-Tung |year=2000 |title=Mirror principle, IV |journal=Surveys in Differential Geometry |pages=475–496|bibcode=2000math......7104L |arxiv=math/0007104 |doi=10.4310/sdg.2002.v7.n1.a15 |volume=7|s2cid=1099024 }}</ref>

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<ref name=Nekrasov>{{cite journal |last1=Nekrasov |first1=Nikita |last2=Schwarz |first2=Albert |date=1998 |title=Instantons on noncommutative {{math|'''R'''⁴}} and (2,0) superconformal six dimensional theory |journal=Communications in Mathematical Physics |volume=198 |issue=3 |pages=689–703 |doi=10.1007/s002200050490|bibcode=1998CMaPh.198..689N |arxiv = hep-th/9802068 |s2cid=14125789 }}</ref>

<ref name="Polchinski 2007">{{cite journal |last1=Polchinski |first1=Joseph |title=All Strung Out? |journal=American Scientist |volume=95 |pages=72 |date=2007 |url=http://www.americanscientist.org/bookshelf/pub/all-strung-out |access-date=29 December 2016|doi=10.1511/2007.63.72 |url-access=subscription }}</ref>

<ref name=Randall>{{cite journal |last1=Randall |first1=Lisa |last2=Sundrum |first2=Raman |date=1999 |title=An alternative to compactification |journal=Physical Review Letters |volume=83 |issue=23 |pages=4690–4693 |doi=10.1103/PhysRevLett.83.4690 |arxiv=hep-th/9906064 |bibcode = 1999PhRvL..83.4690R |s2cid=18530420 }}</ref>

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<ref name=Sen1994a>{{cite journal |last1=Sen |first1=Ashoke |date=1994 |title=Strong-weak coupling duality in four-dimensional string theory |journal=International Journal of Modern Physics A |volume=9 |issue=21 |pages=3707–3750 |bibcode=1994IJMPA...9.3707S |doi=10.1142/S0217751X94001497 |arxiv = hep-th/9402002 |s2cid=16706816 }}</ref>

<ref name=Sen1994b>{{cite journal |last1=Sen |first1=Ashoke |date=1994 |title=Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and {{math|''SL''(2,'''Z''')}} invariance in string theory |journal=Physics Letters B |volume=329 |issue=2 |pages=217–221 |doi=10.1016/0370-2693(94)90763-3|arxiv = hep-th/9402032 |bibcode = 1994PhLB..329..217S |s2cid=17534677 }}</ref>

<ref name=Strominger1998>{{cite journal |last1=Strominger |first1=Andrew |date=1998 |title=Black hole entropy from near-horizon microstates |journal=Journal of High Energy Physics |volume=1998 |issue=2 |doi=10.1088/1126-6708/1998/02/009|arxiv = hep-th/9712251 |bibcode = 1998JHEP...02..009S |page=009|s2cid=2044281 }}</ref>

<ref name="Strominger and Vafa 1996">{{cite journal |last1=Strominger |first1=Andrew |last2=Vafa |first2=Cumrun |date=1996 |title=Microscopic origin of the Bekenstein–Hawking entropy |journal=Physics Letters B |volume=379 |issue=1 |pages=99–104 |arxiv = hep-th/9601029 |bibcode = 1996PhLB..379...99S |doi = 10.1016/0370-2693(96)00345-0 |s2cid=1041890 }}</ref>

<ref name=SYZ>{{cite journal |last1=Strominger |first1=Andrew |last2=Yau |first2=Shing-Tung |last3=Zaslow |first3=Eric |year=1996 |title=Mirror symmetry is T-duality |journal=Nuclear Physics B |volume=479 |issue=1 |pages=243–259|arxiv = hep-th/9606040 |bibcode = 1996NuPhB.479..243S |doi = 10.1016/0550-3213(96)00434-8 |s2cid=14586676 }}</ref>

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<ref name=Witten1998>{{cite journal | last1=Witten| first1=Edward | title=Anti-de Sitter space and holography | journal=Advances in Theoretical and Mathematical Physics | volume=2 | issue=2 | year=1998 | pages=253–291 | arxiv=hep-th/9802150|bibcode = 1998AdTMP...2..253W | doi=10.4310/ATMP.1998.v2.n2.a2 | s2cid=10882387 }}</ref>

<ref name=Witten2007>{{cite arXiv | last1=Witten| first1=Edward | title=Three-dimensional gravity revisited | year=2007 |eprint=0706.3359 |class=hep-th }}</ref>

<ref name="Zee 2010">{{cite book |last=Zee |first=Anthony |title=Quantum Field Theory in a Nutshell |edition=2nd |date=2010 |publisher=Princeton University Press |isbn=978-0-691-14034-6 |url-access=registration |url=https://archive.org/details/isbn_9780691140346 |chapter=Parts V and VI}}</ref>

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===Bibliography===

{{refbeginRefbegin|30em}}

{{refbegin|30em}}

* {{cite book|ref=Becker |last1=Becker |first1=Katrin |last2=Becker |first2=Melanie |author-link2=Melanie Becker|last3=Schwarz |first3=John |title=String theory and M-theory: A modern introduction |date=2007 |publisher=Cambridge University Press |isbn=978-0-521-86069-7}}

* {{cite journal |ref=Duff1998|last1=Duff |first1=Michael |date=1998 |title=The theory formerly known as strings |journal=Scientific American |volume=278 |issue=2 |pages=64–9 |doi=10.1038/scientificamerican0298-64|bibcode = 1998SciAm.278b..64D }}

* {{cite book |ref=Gannon|last1=Gannon |first1=Terry |title=Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms, and Physics |date=2023 |publisher=Cambridge University Press|bibcode=2023mbmb.book.....G }}

* {{cite book |ref=Hori|editor1-first=Kentaro |editor1-last=Hori |editor2-first=Sheldon |editor2-last=Katz |editor3-first=Albrecht |editor3-last=Klemm |editor4-first=Rahul |editor4-last=Pandharipande |editor5-first=Richard |editor5-last=Thomas |editor6-first=Cumrun |editor6-last=Vafa |editor7-first=Ravi |editor7-last=Vakil |editor8-first=Eric |editor8-last=Zaslow |title=Mirror Symmetry |year=2003 |series=[[Clay Mathematics Monographs]]|volume=1|publisher=American Mathematical Society |url=http://math.stanford.edu/~vakil/files/mirrorfinal.pdf |isbn=978-0-8218-2955-4 |url-status=dead |archive-url=https://web.archive.org/web/20060919020706/http://math.stanford.edu/~vakil/files/mirrorfinal.pdf |archive-date=2006-09-19 }}

* {{cite journal|ref=Maldacena2005|title=The Illusion of Gravity |last=Maldacena |first=Juan |date=2005 |journal=Scientific American |url=http://www.sns.ias.edu/~malda/sciam-maldacena-3a.pdf |access-date=29 December 2016 |bibcode=2005SciAm.293e..56M |volume=293 |pages=56–63 |doi=10.1038/scientificamerican1105-56 |issue=5 |pmid=16318027 |url-status=dead |archive-url=https://web.archive.org/web/20141101181409/http://www.sns.ias.edu/~malda/sciam-maldacena-3a.pdf |archive-date=November 1, 2014 }}

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* {{Cite book|ref=Yau| first1 = Shing-Tung | last1 = Yau | first2 = Steve | last2 = Nadis | year = 2010 | title = The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions | publisher = Basic Books | isbn = 978-0-465-02023-2 }}

* {{cite book |ref=Zwiebach|last1=Zwiebach |first1=Barton |title=A First Course in String Theory |date=2009 |publisher=Cambridge University Press |isbn=978-0-521-88032-9}}

{{refendRefend}}

== Further reading ==

=== Popular science ===

* {{Cite book| first = Brian | last = Greene | year = 2003 | title = The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory | publisher = W.W. Norton & Company | ___location = New York | isbn = 978-0-393-05858-1 | title-link = The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory }}

* {{Cite book| first = Brian | last = Greene | year = 2004 | title = The Fabric of the Cosmos: Space, Time, and the Texture of Reality | publisher = Alfred A. Knopf | ___location = New York | isbn = 978-0-375-41288-2 | title-link = The Fabric of the Cosmos: Space, Time, and the Texture of Reality | bibcode = 2004fcst.book.....G }}

* {{cite book | last=Gubser | first=Steven Scott | title=The Little Book of String Theory | publisher=Princeton University Press | publication-place=Princeton, N. J. | date=2010 | isbn=978-0-691-14289-0}}

* {{Cite book| first = Roger | last = Penrose | year = 2005 | title = The Road to Reality: A Complete Guide to the Laws of the Universe | publisher = Knopf | isbn = 978-0-679-45443-4 | title-link = The Road to Reality: A Complete Guide to the Laws of the Universe }}

* {{Cite book| first = Lee | last = Smolin | year = 2006 | title = The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next | publisher = Houghton Mifflin Co. | ___location = New York | isbn = 978-0-618-55105-7 | title-link = The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next }}

* {{Citecite book| first = Peter | last = Woit | year first= 2006Peter | title = Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law | publisher = New York: Basic Books | ___location publication-place=New London:York Jonathan| Cape &date=2006-09-04 | isbn = 978-0-465-09275-8 <!--| bothoclc=67840232 are| correcturl=https://www.worldcat.org/title/67840232 | access-date=2025->05-31}} - UK edition published by Jonathan Cape, London, 2006

=== Textbooks ===

* {{cite book |last1=Becker |first1=K. |last2=Becker |first2=M. |last3=Schwarz |first3=J. H. |title=String Theory and M-Theory: A Modern Introduction |publisher=Cambridge University Press |date=2006 |isbn=978-0521860697 }}

* {{cite book |last1=Blumenhagen |first1=R. |last2=Lüst |first2=D. |last3=Theisen |first3=S. |title=Basic Concepts of String Theory |publisher=Springer |date=2012 |isbn=978-3642294969 }}

* {{cite book |last1=Green |first1=Michael |last2=Schwarz |first2=John |last3=Witten |first3=Edward |title=Superstring theory. Vol.Theory |volume=1: ''Introduction'' |publisher=Cambridge University Press |date=2012 |isbn=978-1107029118 }}

* {{cite book |last1=Green |first1=Michael |last2=Schwarz |first2=John |last3=Witten |first3=Edward |title=Superstring theory. Vol.Theory |volume=2: ''Loop amplitudes, anomalies and phenomenology'' |publisher=Cambridge University Press |date=2012 |isbn=978-1107029132 }}

* {{cite book |last1=Ibáñez |first1=L.E. |last2=Uranga |first2=A.M. |title=String Theory and Particle Physics: An Introduction to String Phenomenology |publisher=Cambridge University Press |date=2012 |isbn=978-0521517522 }}

* {{cite book |last1=Kiritsis |first1=E. |title=String Theory in a Nutshell |publisher=Princeton University Press |date=2019 |isbn=978-0691155791 }}

* {{cite book |last=Ortín |first=T. |title=Gravity and Strings |publisher=Cambridge University Press |date=2015 |isbn=978-0521768139 }}

* {{cite book |last1=Polchinski |first1=Joseph |year=1998 |title=String Theory Vol. |volume=1: ''An Introduction to the Bosonic String'' |publisher=Cambridge University Press |isbn=978-0-521-63303-1}}

* {{cite book |last=Polchinski |first=Joseph |year=1998 |title=String Theory Vol. |volume=2: ''Superstring Theory and Beyond'' |publisher=Cambridge University Press |isbn=978-0-521-63304-8}}

* {{cite book |last1=West|first1=P. |title=Introduction to Strings and Branes |publisher=Cambridge University Press |date=2012 |isbn=978-0521817479 }}

* {{cite book |last1=Zwiebach |first1=Barton |title=A First Course in String Theory |date=2009 |publisher=Cambridge University Press |isbn=978-0-521-88032-9}}

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{{Commons category}}

{{Wikiquote}}

* [https://www.bbc.co.uk/science/horizon/2001/paralleluni.shtml bbc-horizon:BBC's parallel-uni]''Parallel —Universes''], 2002 feature documentary by ''[[Horizon (British TV series)|BBC Horizon]]'', episode "[[Parallel Universes (film)|Parallel Universes]]", focusfocuses on history and emergence of M-theory, and scientists involved.

'''Websites'''

* [https://www.pbs.org/wgbh/nova/physicsseries/the-elegant-universe.html/ pbs.org-nova:Nova's elegant-uni]''The —Elegant Universe''], 2003 [[Emmy Award]]-winning–winning, three-hour miniseries by ''[[Nova (American TV program)|Nova]]'' with [[Brian Greene]], adapted from his ''[[The Elegant Universe]]'' (original [[PBS]] broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003).

* [http://www.math.columbia.edu/~woit/blog/ Not Even Wrong]—A blog critical of string theory

* [https://whystringtheory.com/ Why String Theory]—An introduction to string theory.

'''Video'''

* [https://www.bbc.co.uk/science/horizon/2001/paralleluni.shtml bbc-horizon: parallel-uni] — 2002 feature documentary by [[Horizon (British TV series)|BBC Horizon]], episode [[Parallel Universes (film)|Parallel Universes]] focus on history and emergence of M-theory, and scientists involved.

* [https://www.pbs.org/wgbh/nova/physics/elegant-universe.html pbs.org-nova: elegant-uni] — 2003 [[Emmy Award]]-winning, three-hour miniseries by [[Nova (American TV program)|Nova]] with [[Brian Greene]], adapted from his [[The Elegant Universe]] (original [[PBS]] broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003).

{{String theory topics |state=collapsed}}

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[[Category:String theory| ]]

[[Category:Mathematical physics]]

[[Category:Concepts in physics]]

[[Category:Dimension]]

[[Category:Mathematical physics]]

[[Category:Multi-dimensional geometry]]

[[Category:Physical cosmology]]

[[Category:Physics beyond the Standard Model]]