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For a more accessible and less technical introduction to this topic, see Introduction to M-theory.
M-theory is a theory in physics that unifies all consistent versions of superstring theory. The existence of such a theory was first conjectured by Edward Witten at the string theory conference at the University of Southern California in the summer of 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution.Prior to Witten's announcement, string theorists had identified five different versions of superstring theory. Although these theories appeared at first to be very different, work by a number of different physicists including Ashoke Sen, Chris Hull, Paul Townsend, and Michael Duff showed that the theories were related in intricate and nontrivial ways. In particular, physicists found that apparently distinct theories were identified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a gravitational theory called eleven-dimensional supergravity.
Although a complete formulation of M-theory is not known, the theory should describe two- and five-dimensional objects called branes and should be approximated by eleven-dimensional supergravity at low energies. Modern attempts to formulate M-theory are typically based on matrix theory or the AdS/CFT correspondence. According to Witten, the M in M-theory can stand for "magic", "mystery", or "matrix" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.
Investigations of the mathematical structure of M-theory have spawned a number of important theoretical results in physics and mathematics. More speculatively, M-theory may provide a framework for developing a unified theory of all of the fundamental forces of nature. Attempts to connect M-theory to experiment typically focus on compactifying its extra dimensions to construct models of our four-dimensional world.
Background
Quantum gravity and strings
Main articles: Quantum gravity and String theory
The fundamental objects of string theory are open and closed strings.
One of the deepest problems in modern physics is the problem of quantum gravity. Our current understanding of gravity is based on Albert Einstein's general theory of relativity, which is formulated within the framework of classical physics. However, nongravitational forces are described within the framework of quantum mechanics, a radically different formalism for describing physical phenomena based on probability.[1] A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics,[2] but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.[3]
String theory is a theoretical framework that attempts to reconcile gravity and quantum mechanics. In string theory, the point-like particles of particle physics are replaced by one-dimensional objects called strings. These strings look like small segments or loops of ordinary string. String theory describes how strings propagate through space and interact with each other. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. One of the vibrational states of a string gives rise to the graviton, a quantum mechanical particle that mediates gravitational interactions.[4]
There are several versions of string theory known as type I, type IIA, type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while the type II theories include only closed strings.[5] Each of these five string theories arises as a special limiting case of M-theory. This theory, like its string theory predecessors, is an example of a quantum theory of gravity. It describes a force just like the familiar gravitational force subject to the rules of quantum mechanics.[6]
Number of dimensions
An example of compactification: At large distances, a two dimensional surface with one circular dimension looks one-dimensional.
In everyday life, there are three familiar dimensions of space (up/down, left/right, and forward/backward), and there is one dimension of time (later/earlier). Thus, in the language of modern physics, one says that spacetime is four-dimensional.[7]
Despite the obvious relevance of four-dimensional spacetime for describing the physical world, there are several reasons why physicists often consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily.[8] There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in condensed matter physics.[9] Finally, there exist scenarios in which there could actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.[10]
One notable feature of string theory and M-theory is that these theories require extra dimensions of spacetime for their mathematical consistency. In string theory, spacetime is ten-dimensional, while in M-theory it is eleven-dimensional. In order to describe real physical phenomena using these theories, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.[11]
Compactification is one way of modifying the number of dimensions in a physical theory.[12] In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.[13] In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.[14]
Dualities
Theories that arise as different limits of M-theory turn out to be related in highly nontrivial ways. One of the relationships that can exist between these different physical theories is called S-duality. This is a relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. For example, type I string theory turns out to be equivalent by S-duality to the SO(32) heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality.[15]
Another relationship between different string theories is T-duality. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius
is equivalent to a string propagating around a circle of radius 
in the sense that all observable quantities in one description are
identified with quantities in the dual description. For example, a
string has momentum
as it propagates around a circle, and it can also wind around the
circle one or more times. The number of times the string winds around a
circle is called the winding number. If a string has momentum 
and winding number 
in one description, it will have momentum and winding number
in the dual description. For example, type IIA string theory is
equivalent to type IIB string theory via T-duality, and the two versions
of heterotic string theory are also related by T-duality.[15]In general, the term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.[16]
Supersymmetry
Another important theoretical idea that plays a role in M-theory is supersymmetry. This is a mathematical relation that exists in certain physical theories between a class of particles called bosons and a class of particles called fermions. Roughly speaking, bosons are the constituents of radiation, while fermions are the constituents of matter. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa. When supersymmetry is imposed as a local symmetry, one automatically obtains a quantum mechanical theory that includes gravity. Such a theory is called a supergravity theory.[17]A theory of strings that incorporates the idea of supersymmetry is called a superstring theory. There are several different versions of superstring theory which are all subsumed within the M-theory framework. At low energies, the superstring theories are approximated by supergravity in ten spacetime dimensions. Similarly, M-theory is approximated at low energies by supergravity in eleven dimensions.[6]
Branes
In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A p-brane sweeps out a (p+1)-dimensional volume in spacetime called its worldvolume. Physicists often study fields analogous to the electromagnetic field which live on the worldvolume of a brane. The word brane comes from the word "membrane" which refers to a two-dimensional brane.[18]In string theory, the fundamental objects that give rise to elementary particles are the one-dimensional strings. Although the physical phenomena described by M-theory are still poorly understood, physicists know that the theory describes two- and five-dimensional branes. Much of the current research in M-theory attempts to better understand the properties of these branes.[19]
History and development
Early work on supergravity
General relativity does not place any limits on the possible dimensions of spacetime. Although the theory is typically formulated in four dimensions, one can write down the same equations for the gravitational field in any number of dimensions. Supergravity is more restrictive because it places an upper limit on the number of dimensions.[17] In 1978, work of Werner Nahm showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven.[20] In the same year, Eugene Cremmer, Bernard Julia, and Joel 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.[21][22]
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 and weak nuclear forces, and gravity. Interest in eleven-dimensional supergravity soon waned, however, as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between left and right, a phenomenon known as chirality. As emphasized by Edward Witten and others, this chirality property cannot be readily derived by compactifying from eleven dimensions.[22]
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.[22] 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. In string theory, the possibilities are much more constrained, and there are only a few consistent formulations of the theory. Indeed, by the 1990s, physicists had identified five consistent supersymmetric versions of the theory.[22]
Relationships between string theories
Although there was only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation.[22] However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways.[23]In the late 1970s, Claus Montonen and David Olive, had conjectured a special property of certain physical theories.[24] A sharpened version of their conjecture concerns a theory called N = 4 supersymmetric Yang–Mills theory, which describes particles similar to the quarks and gluons that make up atomic nuclei. The strength with which the particles of this theory interact is measured by a number called the coupling constant. The result of Montonen and Olive, now known as Montonen–Olive duality, states that N=4 supersymmetric Yang–Mills theory with coupling constant
is equivalent to the same theory with coupling constant 
.
In other words, a system of strongly interacting particles (large
coupling constant) has an equivalent description as a system of weakly
interacting particles (small coupling constant) and vice versa.[25]In the 1990s, several theorists generalized Montonen–Olive duality to a relationship called S-duality which connects different string theories. Ashoke Sen studied the heterotic theory compactified down to four dimensions.[26][27] Chris Hull and Paul Townsend showed that type IIB string theory with a large coupling constant is equivalent via S-duality to the same theory with small coupling constant.[28] Theorists also found that different string theories may be related by a totally different kind of duality known as T-duality. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent.[29]
Membranes and fivebranes
String theory extends ordinary quantum field theory by promoting zero-dimensional point particles to one-dimensional objects called strings. In the late 1980s, it was natural for theorists to attempt to formulate other extensions of quantum field theory in which particles are replaced by two-dimensional supermembranes or by higher-dimensional objects called branes. Such objects had been considered as early as 1962 by Paul Dirac, and they were reconsidered by a small but enthusiastic group of physicists in the 1980s.[22]
Supersymmetry severely restricts the possible number of dimensions of a brane. In 1987, Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes.[30] Intuitively, these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after this discovery, Michael Duff, Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle.[31] In this setting, one can imagine the membrane wrapping around the circular dimension. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. In fact, Duff and his collaborators showed that this construction reproduces exactly the strings appearing in type IIA superstring theory.[25]
In 1990, Andrew Strominger published a similar result which suggested that strongly interacting strings in ten dimensions might have an equivalent description in terms of weakly interacting five-dimensional branes.[32] Initially, physicists were unable to prove this relationship for two important reasons. On the one hand, the Montonen–Olive duality was still unproven, and so Strominger's conjecture was even more tenuous. On the other hand, there were many technical issues related to the quantum properties of five-dimensional branes.[33] The first of these problems was solved in 1993 when Ashoke Sen established that certain physical theories require the existence of objects with both electric and magnetic charge which were predicted by the work of Montonen and Olive.[34]
In spite of this progress, the relationship between strings and five-dimensional branes remained conjectural because theorists were unable to quantize the branes. Starting in 1991, a team of researchers including Michael Duff, Ramzi Khuri, Jianxin Lu, and Ruben Minasian considered a special compactification of string theory in which four of the ten dimensions curl up. If one considers a five-dimensional brane wrapped around these extra dimensions, then the brane looks just like a one-dimensional string. In this way, the conjectured relationship between strings and branes was reduced to a relationship between strings and strings, and the latter could be tested using already established theoretical techniques.[29]
Second superstring revolution
A schematic illustration of the relationship between M-theory, the five superstring theories, and eleven-dimensional supergravity.
The shaded region represents a family of different physical scenarios
that are possible in M-theory, and these last six theories arise as
special limiting cases.
Speaking at the string theory conference at the University of Southern California in 1995, Edward Witten of the Institute for Advanced Study made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions. Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of two- and five-dimensional branes in string theory.[35] In the months following Witten's announcement, hundreds of new papers appeared on the Internet confirming that the new theory involved membranes in an important way.[36] Today this flurry of work is known as the second superstring revolution.[37]
One of the important developments following Witten's announcement was Witten's work in 1996 with string theorist Petr Hořava.[38][39] Witten and Hořava studied M-theory on a special spacetime geometry with two ten-dimensional boundary components. Their work shed light on the mathematical structure of M-theory and suggested possible ways of connecting M-theory to real world physics.[40]
Origin of the term
Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In a paper from 1996, Hořava and Witten wroteAs it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes.[38]In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the M should stand for "magic", "mystery", or "matrix" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.[41]
Matrix theory
BFSS matrix model
A matrix is a rectangular array of numbers or other data.
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes how a set of matrices evolves in time according to the rules of quantum mechanics.[42][43]
One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to conjecture that the BFSS matrix model is exactly equivalent to M-theory. It can therefore be used to describe M-theory and investigate its properties in a relatively simple setting.[42]
Noncommutative geometry
In geometry, it is often useful to introduce coordinates. For example, in order to study the geometry of the Euclidean plane, one defines the coordinates
and 
as the distances between any point in the plane and a pair of axes.
In ordinary geometry, the coordinates of a point are numbers, so they
can be multiplied, and the product of two coordinates does not depend on
the order of multiplication. That is, 
. This property of multiplication is known as the commutative law, and this relationship between geometry and the commutative algebra of coordinates is the starting point for much of modern geometry.[44]
Noncommutative geometry is a branch of mathematics that attempts to generalize this situation. Rather than working with ordinary numbers, one considers some similar objects, such as matrices, whose multiplication does not satisfy the commutative law (that is, objects for which
is not necessarily equal to 
).
One imagines that these noncommuting objects are coordinates on some
more general notion of "space" and proves theorems about these
generalized spaces by exploiting the analogy with ordinary geometry.[45] Noncommutative geometry also has many applications in modern theoretical physics.[46]In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property.[43] This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. In 1999, Nathan Seiberg and Edward Witten described further relations between string theory and noncommutative geometry.[47]
AdS/CFT correspondence
Overview
A tessellation of the hyperbolic plane by triangles and squares.
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.[48] In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. Quantum field theories are also used throughout condensed matter physics to model particle-like objects called quasiparticles.[49]
One approach to formulating M-theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. Proposed by Juan Maldacena in late 1997, the AdS/CFT correspondence is a theoretical result which implies that M-theory is in some cases equivalent to a quantum field theory.[50] In addition to providing insights into the mathematical structure of string and M-theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.[51]
In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain vacuum solution of Einstein's equation called anti-de Sitter space.[52] In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the metric) is different from the notion of distance in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be viewed as a disk as illustrated on the left.[53] This image shows a tessellation of a disk by triangles and squares. 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.[54]
Three-dimensional anti-de Sitter space is like a stack of hyperbolic disks, each one representing the state of the universe at a given time. The resulting spacetime looks like a solid cylinder.
Now 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.[53] It looks like a solid cylinder in which any 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 curved in such a way that any point in the interior is actually infinitely far from this boundary surface.[54]
This construction describes a hypothetical universe with only two space 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.[53]
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, locally around any point, it looks just like Minkowski space, the model of spacetime used in nongravitational physics.[55] 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 the gravitational theory on the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating calculations in one theory into calculations in the other. Every entity in one theory has a counterpart 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.[56]
6D (2,0) superconformal field theory
The six-dimensional (2,0)-theory has been used to understand results from the mathematical theory of knots.
One particular realization of the AdS/CFT correspondence states that M-theory on the product space
is equivalent to the so-called (2,0)-theory on the six-dimensional boundary.[50] In this example, the spacetime of the gravitational theory is effectively seven-dimensional (hence the notation 
), and there are four additional "compact" dimensions (encoded by the 
factor). In the real world, spacetime is four-dimensional, at least
macroscopically, so this version of the correspondence does not provide a
realistic model of gravity. Likewise, the dual theory is not a viable
model of any real-world system since it describes a world with six
spacetime dimensions.[57]Nevertheless, the (2,0)-theory has proven to be important for studying the general properties of quantum field theories. Indeed, this theory subsumes a large number of mathematically interesting effective quantum field theories and points to new dualities relating these theories. For example, Luis Alday, Davide Gaiotto, and Yuji Tachikawa showed that by compactifying this theory on a surface, one obtains a four-dimensional quantum field theory, and there is a duality known as the AGT correspondence which relates the physics of this theory to certain physical concepts associated with the surface itself.[58] More recently, theorists have extended these ideas to study the theories obtained by compactifying down to three dimensions.[59]
In addition to its applications in quantum field theory, the (2,0)-theory has spawned a number of important results in pure mathematics. For example, the existence of the (2,0)-theory was used by Witten to give a "physical" explanation for a conjectural relationship in mathematics called the geometric Langlands correspondence.[60] In subsequent work, Witten showed that the (2,0)-theory could be used to understand a concept in mathematics called Khovanov homology.[61] Developed by Mikhail Khovanov around 2000, Khovanov homology provides a tool in knot theory, the branch of mathematics that studies and classifies the different shapes of knots.[62] Another application of the (2,0)-theory in mathematics is the work of Davide Gaiotto, Greg Moore, and Andrew Neitzke, which used physical ideas to derive new results in hyperkähler geometry.[63]
ABJM superconformal field theory
Another realization of the AdS/CFT correspondence states that M-theory on
is equivalent to a quantum field theory called the ABJM theory
in three dimensions. In this version of the correspondence, seven of
the dimensions of M-theory are curled up, leaving four non-compact
dimensions. Since the spacetime of our universe is four-dimensional,
this version of the correspondence provides a somewhat more realistic
description of gravity.[64]
The ABJM theory appearing in this version of the correspondence is also interesting for a variety of reasons. Introduced by Aharony, Bergman, Jafferis, and Maldacena, it is closely related to another quantum field theory called Chern-Simons theory. The latter theory was popularized by Witten in the late 1980s because of its applications to knot theory.[65] In addition, the ABJM theory serves as a useful toy model for solving problems that arise in condensed matter physics.[64]
Phenomenology
Overview
A cross section of a Calabi–Yau manifold
In addition to being an idea of considerable theoretical interest, M-theory provides a framework for constructing models of real world physics that combine general relativity with the standard model of 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 models of particle physics based on string and M-theory.[66]
Typically, such models are based on the idea of compactification.[67] 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.[68] 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. This is a special kind of geometric object named after mathematicians Eugenio Calabi and Shing-Tung Yau.[69] Calabi–Yau manifolds offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct realistic models of our four-dimensional world based on M-theory.[70]
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.[71]
Compactification on G2 manifolds
In one approach to M-theory phenomenology, theorists assume that the seven extra dimensions of M-theory are shaped like a G2 manifold. This is a special kind of seven-dimensional shape constructed by mathematician Dominic Joyce of the University of Oxford.[72] These G2 manifolds are still poorly understood mathematically, and this fact has made it difficult for physicists to fully develop this approach to phenomenology.[73]For example, physicists and mathematicians often assume that space has a mathematical property called smoothness, but this property cannot be assumed in the case of a G2 manifold if one wishes to recover the physics of our four-dimensional world. Another problem is that G2 manifolds are not complex manifolds, so theorists are unable to use tools from the branch of mathematics known as complex analysis. Finally, there are many open questions about the existence, uniqueness, and other mathematical properties of G2 manifolds, and mathematicians lack a systematic way of searching for these manifolds.[73]
