In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, supergravity (supergravity theory; SUGRA for short) is a modern
field theory that combines the principles of
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
; this is in contrast to non-gravitational supersymmetric theories such as the
Minimal Supersymmetric Standard Model
The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the inimumnumber of new particle states and new interactions con ...
. Supergravity is the
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the
Poincaré algebra a
superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
Th ...
, called the
super-Poincaré algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symme ...
, supersymmetry as a gauge theory makes gravity arise in a natural way.
Gravitons
Like any field theory of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, a supergravity theory contains a spin-2 field whose quantum is the
graviton
In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...
. Supersymmetry requires the graviton field to have a
superpartner
In particle physics, a superpartner (also sparticle) is a class of hypothetical elementary particles predicted by supersymmetry, which, among other applications, is one of the well-studied ways to extend the standard model of high-energy physics.
...
. This field has
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
3/2 and its quantum is the
gravitino
In supergravity theories combining general relativity and supersymmetry, the gravitino () is the gauge fermion supersymmetric partner of the hypothesized graviton. It has been suggested as a candidate for dark matter.
If it exists, it is a f ...
. The number of gravitino fields is equal to the number of
supersymmetries.
History
Gauge supersymmetry
The first theory of local supersymmetry was proposed by
Dick Arnowitt and
Pran Nath in 1975
and was called gauge supersymmetry.
Supergravity
The first model of 4-dimensional supergravity (without this denotation) was formulated by Dmitri Vasilievich Volkov and Vyacheslav A. Soroka in 1973,
emphasizing the importance of spontaneous supersymmetry breaking for the possibility of a realistic model. The minimal version of 4-dimensional supergravity (with unbroken local supersymmetry) was constructed in detail in 1976 by
Dan Freedman
Dan or DAN may refer to:
People
* Dan (name), including a list of people with the name
** Dan (king), several kings of Denmark
* Dan people, an ethnic group located in West Africa
**Dan language, a Mande language spoken primarily in Côte d'Ivoir ...
,
Sergio Ferrara
Sergio Ferrara (born May 2, 1945) is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles ...
and
Peter van Nieuwenhuizen
Peter van Nieuwenhuizen (; born October 26, 1938) is a Dutch physicist. He is now a distinguished Professor at Stony Brook University in the United States. Van Nieuwenhuizen is best known for his discovery of supergravity with Sergio Ferrara ...
.
In 2019 the three were awarded a special
Breakthrough Prize in Fundamental Physics
The Breakthrough Prize in Fundamental Physics is one of the Breakthrough Prizes, awarded by the Breakthrough Prize Board. Initially named Fundamental Physics Prize, it was founded in July 2012 by Russia-born Israeli entrepreneur, venture capita ...
for the discovery. The key issue of whether or not the spin 3/2 field is consistently coupled was resolved in the nearly simultaneous paper, by
Deser and
Zumino,
which independently proposed the minimal 4-dimensional model. It was quickly generalized to many different theories in various numbers of
dimensions
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
and involving additional (N) supersymmetries. Supergravity theories with N>1 are usually referred to as extended supergravity (SUEGRA). Some supergravity theories were shown to be related to certain
higher-dimensional supergravity theories via
dimensional reduction
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in ''D'' spacetime dimensions can be redefined in a lower number of dimensions ''d'', by taking all the fields ...
(e.g. N=1, 11-dimensional supergravity is dimensionally reduced on T
7 to 4-dimensional, ungauged, ''N'' = 8 supergravity). The resulting theories were sometimes referred to as
Kaluza–Klein theories as Kaluza and Klein constructed in 1919 a 5-dimensional gravitational theory, that when dimensionally reduced on a circle, its 4-dimensional non-massive modes describe
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
coupled to
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
.
mSUGRA
mSUGRA means minimal SUper GRAvity. The construction of a realistic model of particle interactions within the ''N'' = 1 supergravity framework where
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
(SUSY) breaks by a super
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
carried out by
Ali Chamseddine
Ali H. Chamseddine ( ar, علي شمس الدين, link=no, born 20 February 1953) is a Lebanese physicist known for his contributions to particle physics, general relativity and mathematical physics. , Chamseddine is a physics Professor at t ...
,
Richard Arnowitt
Richard Lewis Arnowitt (May 3, 1928 – June 12, 2014) was an American physicist known for his contributions to theoretical particle physics and to general relativity.
Arnowitt was a Distinguished Professor (Emeritus) at Texas A&M University, whe ...
and
Pran Nath in 1982. Collectively now known as minimal supergravity Grand Unification Theories (mSUGRA GUT), gravity mediates the breaking of SUSY through the existence of a
hidden sector
In particle physics, the hidden sector, also known as the dark sector, is a hypothetical collection of yet-unobserved quantum fields and their corresponding hypothetical particles. The interactions between the hidden sector particles and the Sta ...
. mSUGRA naturally generates the Soft SUSY breaking terms which are a consequence of the Super Higgs effect. Radiative breaking of electroweak symmetry through
Renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
Group Equations (RGEs) follows as an immediate consequence.
Due to its predictive power, requiring only four input parameters and a sign to determine the low energy phenomenology from the scale of Grand Unification, its interest is a widely investigated model of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
11D: the maximal SUGRA
One of these supergravities, the 11-dimensional theory, generated considerable excitement as the first potential candidate for the
theory of everything
A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp ...
. This excitement was built on four pillars, two of which have now been largely discredited:
*
Werner Nahm
Werner Nahm (; born 21 March 1949) is a German theoretical physicist, with the status of professor. He has made contributions to mathematical physics and fundamental theoretical physics.
Life and work
Werner Nahm attended Gymnasium Philipp ...
showed 11 dimensions as the largest number of dimensions consistent with a single graviton, and more dimensions will show particles with spins greater than 2. However, if two of these dimensions are time-like, these problems are avoided in 12 dimensions.
Itzhak Bars Itzhak Bars (born 31 August 1943, İzmir, Turkey) is a theoretical physicist at the University of Southern California in Los Angeles.
Education
After receiving his B.S. from Robert College in physics in 1967, Bars obtained his Ph.D. under the super ...
gives this emphasis.
* In 1981
Ed Witten showed 11 as the smallest number of dimensions big enough to contain the
gauge group
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
s of the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
, namely
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
for the
strong interactions and
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
times
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
for the
electroweak
In particle physics, the electroweak interaction or electroweak force is the unified field theory, unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two force ...
interactions. Many techniques exist to embed the standard model gauge group in supergravity in any number of dimensions like the obligatory gauge symmetry in
type I and
heterotic string theories, and obtained in
type II string theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories ...
by
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
on certain
Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
s. The
D-brane
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
s engineer gauge symmetries too.
* In 1978
Eugène Cremmer
Eugène Cremmer (7 February 1942, in Paris – 30 October 2019, in Paris) was a French theoretical physicist. He was directeur de recherche at the CNRS working at the École Normale Supérieure. Cremmer was a postdoc at CERN from 1971–72. In ...
,
Bernard Julia and
Joël Scherk
Joël Scherk (; 27 May 1946 – 16 May 1980) was a French theoretical physicist who studied string theory and supergravity.
Work
Scherk studied in Paris at the École Normale Supérieure (ENS). In 1969 he received his diploma (Thèse de troisièm ...
(CJS) found the classical action for an 11-dimensional supergravity theory. This remains today the only known classical 11-dimensional theory with local
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
and no fields of spin higher than two. Other 11-dimensional theories known and quantum-mechanically inequivalent reduce to the CJS theory when one imposes the classical equations of motion. However, in the mid 1980s
Bernard de Wit
Bernard Quirinus Petrus Joseph de Wit (born 1945 in Bergen op Zoom) is a Dutch theoretical physicist specializing in supergravity and particle physics.
Bernard de Wit studied theoretical physics at Utrecht University, where he got his PhD under ...
and
Hermann Nicolai found an alternate theory i
D=11 Supergravity with Local SU(8) Invariance While not manifestly Lorentz-invariant, it is in many ways superior, because it dimensionally-reduces to the 4-dimensional theory without recourse to the classical equations of motion.
* In 1980
Peter Freund
Peter George Oliver Freund (7 September 1936, Timișoara – 6 March 2018, Chicago) was a professor of theoretical physics at the University of Chicago. He made important contributions to particle physics and string theory. He was also active as ...
and
M. A. Rubin showed that
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
from 11 dimensions preserving all the SUSY generators could occur in two ways, leaving only 4 or 7 macroscopic dimensions, the others compact. The noncompact dimensions have to form an
anti-de Sitter space
In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872 ...
. There are many possible compactifications, but the
Freund-Rubin compactification's invariance under all of the
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
transformations preserves the action.
Finally, the first two results each appeared to establish 11 dimensions, the third result appeared to specify the theory, and the last result explained why the observed universe appears to be four-dimensional.
Many of the details of the theory were fleshed out by
Peter van Nieuwenhuizen
Peter van Nieuwenhuizen (; born October 26, 1938) is a Dutch physicist. He is now a distinguished Professor at Stony Brook University in the United States. Van Nieuwenhuizen is best known for his discovery of supergravity with Sergio Ferrara ...
,
Sergio Ferrara
Sergio Ferrara (born May 2, 1945) is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles ...
and
Daniel Z. Freedman
Daniel Zissel Freedman (born 1939 in Hartford, Connecticut) is an American theoretical physicist. He is an Emeritus Professor of Physics and Applied Mathematics at the Massachusetts Institute of Technology (MIT), and is currently a visiting profe ...
.
The end of the SUGRA era
The initial excitement over 11-dimensional supergravity soon waned, as various failings were discovered, and attempts to repair the model failed as well. Problems included:
* The compact manifolds which were known at the time and which contained the standard model were not compatible with supersymmetry, and could not hold
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s or
lepton
In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...
s. One suggestion was to replace the compact dimensions with the 7-sphere, with the symmetry group
SO(8)
In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.
Spin(8)
Like all special orthogonal groups of n > 2, SO(8) is n ...
, or the squashed 7-sphere, with symmetry group
SO(5)
In mathematics, SO(5), also denoted SO5(R) or SO(5,R), is the special orthogonal group of degree 5 over the field R of real numbers, i.e. (isomorphic to) the group of orthogonal matrix, orthogonal 5×5 Matrix (mathematics), matrices of determi ...
times
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
.
* Until recently, the physical
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s seen in experiments were believed to be massless, and appeared to be left-handed, a phenomenon referred to as the
chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
of the Standard Model. It was very difficult to construct a chiral fermion from a
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
— the compactified manifold needed to have singularities, but physics near singularities did not begin to be understood until the advent of
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
conformal field theories
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
in the late 1980s.
* Supergravity models generically result in an unrealistically large
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
in four dimensions, and that constant is difficult to remove, and so require
fine-tuning
In theoretical physics, fine-tuning is the process in which parameters of a model must be adjusted very precisely in order to fit with certain observations. This had led to the discovery that the fundamental constants and quantities fall into suc ...
. This is still a problem today.
* Quantization of the theory led to quantum field theory
gauge anomalies
In theoretical physics, a gauge anomaly is an example of an anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory
In theoretical physics, quantum field theo ...
rendering the theory inconsistent. In the intervening years physicists have learned how to cancel these anomalies.
Some of these difficulties could be avoided by moving to a 10-dimensional theory involving
superstrings. However, by moving to 10 dimensions one loses the sense of uniqueness of the 11-dimensional theory.
The core breakthrough for the 10-dimensional theory, known as the
first superstring revolution
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 ...
, was a demonstration by
Michael B. Green,
John H. Schwarz
John Henry Schwarz (; born November 22, 1941) is an American theoretical physicist. Along with Yoichiro Nambu, Holger Bech Nielsen, Joël Scherk, Gabriele Veneziano, Michael Green, and Leonard Susskind, he is regarded as one of the founders of s ...
and
David Gross
David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. Gr ...
that there are only three supergravity models in 10 dimensions which have gauge symmetries and in which all of the gauge and
gravitational anomalies
In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics — usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with som ...
cancel. These were theories built on the groups
SO(32)
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
and
, the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of two copies of
E8. Today we know that, using
D-branes
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
for example, gauge symmetries can be introduced in other 10-dimensional theories as well.
[
]
The second superstring revolution
Initial excitement about the 10-dimensional theories, and the string theories that provide their quantum completion, died by the end of the 1980s. There were too many
Calabi–Yaus to
compactify on, many more than
Yau had estimated, as he admitted in December 2005 at the
23rd International Solvay Conference in Physics. None quite gave the standard model, but it seemed as though one could get close with enough effort in many distinct ways. Plus no one understood the theory beyond the regime of applicability of string
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.
There was a comparatively quiet period at the beginning of the 1990s; however, several important tools were developed. For example, it became apparent that the various superstring theories were related by "
string dualities
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* Strings (1991 film), ''Strings'' (1991 fi ...
", some of which relate weak string-coupling - perturbative - physics in one model with strong string-coupling - non-perturbative - in another.
Then the
second superstring revolution
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 ...
occurred.
Joseph Polchinski
Joseph Gerard Polchinski Jr. (; May 16, 1954 – February 2, 2018) was an American theoretical physicist and string theorist.
Biography
Polchinski was born in White Plains, New York, the elder of two children to Joseph Gerard Polchinski Sr. (1929 ...
realized that obscure string theory objects, called
D-branes
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
, which he discovered six years earlier, equate to stringy versions of the
p-branes known in supergravity theories. String theory perturbation didn't restrict these
p-branes. Thanks to
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
, p-branes in supergravity gained understanding well beyond the limits of string theory.
Armed with this new
nonperturbative
In mathematics and physics, a non-perturbative function (mathematics), function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every c ...
tool,
Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
and many others could show all of the perturbative string theories as descriptions of different states in a single theory that Witten named
M-theory
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 1995. Witten's ...
. Furthermore, he argued that M-theory's
long wavelength limit
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase (waves), phase on the wave, such as two adjac ...
, i.e. when the quantum wavelength associated to objects in the theory appear much larger than the size of the 11th dimension, needs 11-dimensional supergravity descriptors that fell out of favor with the
first superstring revolution
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 ...
10 years earlier, accompanied by the 2- and 5-branes.
Therefore, supergravity comes full circle and uses a common framework in understanding features of string theories, M-theory, and their
compactifications to lower spacetime dimensions.
Relation to superstrings
The term "low energy limits" labels some 10-dimensional supergravity theories. These arise as the massless,
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
-level approximation of string theories. True
effective field theories of string theories, rather than truncations, are rarely available. Due to string dualities, the conjectured 11-dimensional
M-theory
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 1995. Witten's ...
is required to have 11-dimensional supergravity as a "low energy limit". However, this doesn't necessarily mean that string theory/M-theory is the only possible
UV completion of supergravity; supergravity research is useful independent of those relations.
4D ''N'' = 1 SUGRA
Before we move on to SUGRA proper, let's recapitulate some important details about
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. We have a 4D differentiable manifold M with a Spin(3,1) principal bundle over it. This principal bundle represents the local Lorentz symmetry. In addition, we have a vector bundle T over the manifold with the fiber having four real dimensions and transforming as a vector under Spin(3,1).
We have an invertible linear map from the tangent bundle TM to T. This map is the
vierbein
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent ...
. The local Lorentz symmetry has a
gauge connection associated with it, the
spin connection
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz tran ...
.
The following discussion will be in superspace notation, as opposed to the component notation, which isn't manifestly covariant under SUSY. There are actually ''many'' different versions of SUGRA out there which are inequivalent in the sense that their actions and constraints upon the torsion tensor are different, but ultimately equivalent in that we can always perform a field redefinition of the supervierbeins and spin connection to get from one version to another.
In 4D N=1 SUGRA, we have a 4, 4 real differentiable supermanifold M, i.e. we have 4 real bosonic dimensions and 4 real fermionic dimensions. As in the nonsupersymmetric case, we have a Spin(3,1) principal bundle over M. We have an R
4, 4 vector bundle T over M. The fiber of T transforms under the local Lorentz group as follows; the four real bosonic dimensions transform as a vector and the four real fermionic dimensions transform as a
Majorana spinor
In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this e ...
. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handed
Weyl spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
(they're not independent of each other). We also have a spin connection as before.
We will use the following conventions; the spatial (both bosonic and fermionic) indices will be indicated by M, N, ... . The bosonic spatial indices will be indicated by μ, ν, ..., the left-handed Weyl spatial indices by α, β,..., and the right-handed Weyl spatial indices by
,
, ... . The indices for the fiber of T will follow a similar notation, except that they will be hatted like this:
. See
van der Waerden notation for more details.
. The supervierbein is denoted by
, and the spin connection by
. The ''inverse'' supervierbein is denoted by
.
The supervierbein and spin connection are real in the sense that they satisfy the reality conditions
:
where
,
, and
and
.
The
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
is defined as
:
.
The
covariant exterior derivative as defined over supermanifolds needs to be super graded. This means that every time we interchange two fermionic indices, we pick up a +1 sign factor, instead of -1.
The presence or absence of
R symmetries is optional, but if R-symmetry exists, the integrand over the full superspace has to have an R-charge of 0 and the integrand over chiral superspace has to have an R-charge of 2.
A chiral superfield ''X'' is a superfield which satisfies
. In order for this constraint to be consistent, we require the integrability conditions that
for some coefficients ''c''.
Unlike nonSUSY GR, the
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
has to be nonzero, at least with respect to the fermionic directions. Already, even in flat superspace,
.
In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:
:
:
:
:
:
:
Here,
is a shorthand notation to mean the index runs over either the left or right Weyl spinors.
The
superdeterminant In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrix, supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when ...
of the supervierbein,
, gives us the volume factor for M. Equivalently, we have the volume 4, 4-superform
.
If we complexify the superdiffeomorphisms, there is a gauge where
,
and
. The resulting chiral superspace has the coordinates x and Θ.
''R'' is a scalar valued chiral superfield derivable from the supervielbeins and spin connection. If ''f'' is any superfield,
is always a chiral superfield.
The action for a SUGRA theory with chiral superfields ''X'', is given by
:
where ''K'' is the
Kähler potential Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
and ''W'' is the
superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
, and
is the chiral volume factor.
Unlike the case for flat superspace, adding a constant to either the Kähler or superpotential is now physical. A constant shift to the Kähler potential changes the effective
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, while a constant shift to the superpotential changes the effective
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
. As the effective Planck constant now depends upon the value of the chiral superfield ''X'', we need to rescale the supervierbeins (a field redefinition) to get a constant Planck constant. This is called the Einstein frame.
''N'' = 8 supergravity in 4 dimensions
''N'' = 8 supergravity is the most
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
quantum field theory which involves gravity and a finite number of fields. It can be found from a dimensional reduction of 11D supergravity by making the size of 7 of the dimensions go to zero. It has 8 supersymmetries which is the most any gravitational theory can have since there are 8 half-steps between spin 2 and spin −2. (A graviton has the highest spin in this theory which is a spin 2 particle). More supersymmetries would mean the particles would have superpartners with spins higher than 2. The only theories with spins higher than 2 which are consistent involve an infinite number of particles (such as string theory and higher-spin theories).
Stephen Hawking in his ''A Brief History of Time'' speculated that this theory could be the
Theory of Everything
A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp ...
. However, in later years this was abandoned in favour of string theory. There has been renewed interest in the 21st century with the possibility that this theory may be finite.
Higher-dimensional SUGRA
Higher-dimensional SUGRA is the higher-dimensional, supersymmetric generalization of general relativity. Supergravity can be formulated in any number of dimensions up to eleven. Higher-dimensional SUGRA focuses upon supergravity in greater than four dimensions.
The number of supercharges in a
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
depends on the dimension and the signature of spacetime. The supercharges occur in spinors. Thus the limit on the number of supercharges cannot be satisfied in a spacetime of arbitrary dimension. Some theoretical examples in which this is satisfied are:
* 12-dimensional two-time theory
* 11-dimensional maximal SUGRA
* 10-dimensional SUGRA theories
** Type IIA SUGRA: N = (1, 1)
** IIA SUGRA from 11d SUGRA
** Type IIB SUGRA: N = (2, 0)
** Type I gauged SUGRA: N = (1, 0)
* 9d SUGRA theories
** Maximal 9d SUGRA from 10d
** T-duality
** N = 1 Gauged SUGRA
The supergravity theories that have attracted the most interest contain no spins higher than two. This means, in particular, that they do not contain any fields that transform as symmetric tensors of rank higher than two under Lorentz transformations. The consistency of interacting higher spin field theories is, however, presently a field of very active interest.
See also
*
General relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
*
Grand Unified Theory
A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this ...
*
M-theory
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 1995. Witten's ...
*
''N'' = 8 supergravity
*
Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
*
String theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
*
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is com ...
*
Super-Poincaré algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symme ...
*
Supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
*
Supermetric
Notes
References
Historical
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General
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Further reading
* Dall'Agata, G., Zagermann, M., ''Supergravity: From First Principles to Modern Applications'', Springer, (2021).
* Freedman, D. Z., Van Proeyen, A., ''Supergravity'', Cambridge University Press, Cambridge, (2012).
* Lauria, E., Van Proeyen, A., ''N = 2 Supergravity in D = 4, 5, 6 Dimensions'', Springer, (2020).
* Nath, P., ''Supersymmetry, Supergravity, and Unification'', Cambridge University Press, Cambridge, (2016)
* Tanii, Y., ''Introduction to Supergravity'', Springer, (2014).
* Rausch de Traubenberg, M., Valenzuela, M., ''A Supergravity Primer'', World Scientific Press, Singapore, (2019).
* Wess, P., ''Introduction To Supersymmetry And Supergravity'', World Scientific Press, Singapore, (1990).
* Wess, P., Bagger, J., ''Supersymmetry and Supergravity'', Princeton University Press, Princeton, (1992).
External links
*
{{portal bar , Physics , Science
Theories of gravity
Supersymmetry
Physics beyond the Standard Model