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Encyclopedia > Supergravity

In theoretical physics, supergravity (supergravity theory) refers to a field theory which combines the two theories of supersymmetry and general relativity. Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... There are two types of field theory in physics: Classical field theory, the theory and dynamics of classical fields. ... This article or section is in need of attention from an expert on the subject. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Like any field theory of gravity, a supergravity theory contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries. Supergravity theories are often said to be the only consistent theories of interacting massless spin 3/2 fields[citation needed]. Gravity is a force of attraction that acts between bodies that have mass. ... In physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. ... In supersymmetry, it is proposed that every fermion should have a partner boson, known as its Superpartner. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... The gravitino is the hypothetical supersymmetric partner of the graviton, as predicted by theories combining general relativity and supersymmetry, i. ... This article or section is in need of attention from an expert on the subject. ...

### Four Dimensional SUGRA

Supergravity, also called SUGRA, was initially proposed as a four-dimensional theory in 1976 by D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, but was quickly generalized to many different theories in various numbers of dimensions. Some supergravity theories were shown to be equivalent to certain higher-dimensional supergravity theories via dimensional reduction. The resulting theories were sometimes referred to as Kaluza-Klein theories, as Kaluza and Klein constructed, nearly a century ago, a five-dimensional gravitational theory which reduces to four-dimensional electromagnetism when the fifth dimension is a circle. 1976 (MCMLXXVI) was a leap year starting on Thursday. ... In physics, compactification plays an important part in string theory. ... In physics, Kaluza-Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ...

### mSUGRA

The construction of a realistic model of particle interactions within the N = 1 supergravity framework where supersymmetry was broken by a super Higgs mechanism was carried out by Ali Chamseddine, Richard Arnowitt and Pran Nath in 1982. In these class of models collectively now known as minimal supergravity GUT models (mSUGRA), gravity mediates the breaking of SUSY. mSUGRA naturally generates the Soft SUSY breaking terms which are a consequence of the Super Higgs effect. Radiative breaking of electroweak symmetry breaking through Renormalization Group Equations (RGEs) follows as an immediate consequence. mSUGRA is one of the most widely investigated models of particle physics due to it predictive power requiring only 4 input parameters and a sign, to determine the low energy Phenomenology. This article or section is in need of attention from an expert on the subject. ... The Higgs mechanism or Anderson-Higgs mechanism, originally proposed by the British physicist Peter Higgs based on a suggestion by Philip Anderson, is the mechanism that gives mass to all elementary particles in particle physics. ... Richard Arnowitt is an American physicist known for his contributions to theoretical particle physics and to general relativity. ... Pran Nath is a physicist at Northeastern University concentrating in theoretical partical physics. ... Figure 1. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

### 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. This excitement was built on four pillars, two of which have now been largely discredited: This article or section is in need of attention from an expert on the subject. ...

• Werner Nahm showed that 11 dimensions was the largest number of dimensions consistent with a single graviton, and that a theory with more dimensions would also have particles with spins greater than 2. These problems are avoided in 12 dimensions if two of these dimensions are timelike, as has been often emphasized by Itzhak Bars[citation needed].
• Shortly afterwards, Ed Witten showed that 11 was the smallest number of dimensions that was big enough to contain the gauge groups of the Standard Model, namely SU(3) for the strong interactions and SU(2) times U(1) for the electroweak interactions. Today many techniques exist to embed the standard model gauge group in supergravity in any number of dimensions. For example, in the mid and late 1980s one often used the obligatory gauge symmetry in type I and heterotic string theories. In type II string theory they could also be obtained by compactifying on certain Calabi-Yau's. Today one may also use D-branes to engineer gauge symmetries.
• In 1978, Eugene Cremmer, Bernard Julia and Joel Scherk (CJS) of the Ecole Normale Superieure found the classical action for an 11-dimensional supergravity theory. This remains today the only known classical 11-dimensional theory with local supersymmetry and no fields of spin higher than two[citation needed]. Other 11-dimensional theories are known that are quantum-mechanically inequivalent to the CJS theory, but classically equivalent (that is, they reduce to the CJS theory when one imposes the classical equations of motion). For example, in the mid 1980s Bernard de Wit and Hermann Nicolai found an alternate theory in D=11 Supergravity with Local SU(8) Invariance. This theory, while not manifestly Lorentz-invariant, is in many ways superior to the CJS theory in that, for example, it dimensionally-reduces to the 4-dimensional theory without recourse to the classical equations of motion.
• In 1980, Peter G. O. Freund and M. A. Rubin showed that compactification from 11 dimensions preserving all the SUSY generators could occur in two ways, leaving only 4 or 7 macroscopic dimensions (the other 7 or 4 being compact). Unfortunately, the noncompact dimensions have to form an anti de Sitter space. Today it is understood that there are many possible compactifications, but that the Freund-Rubin compactifications are invariant under all of the supersymmetry transformations that preserve the action.

Thus, the first two results appeared to establish 11 dimensions uniquely, the third result appeared to specify the theory, and the last result explained why the observed universe appears to be four-dimensional. Edward Witten at Harvard University Edward Witten (born August 26, 1951) is a professor at the Institute for Advanced Study. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices, with the group operation that of matrix multiplication. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. ... In physics, a heterotic string is a peculiar mixture (or hybrid) of the bosonic string and the superstring (the adjective heterotic comes from the Greek word heterosis). ... In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. ... In physics, compactification plays an important part in string theory. ... In mathematics, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ... In theoretical physics, D-branes are a special class of p-branes, named for the mathematician Johann Dirichlet. ... 1978 (MCMLXXVIII) was a common year starting on Sunday. ... EugÃ¨ne Cremmer, born in 1942, is a French physicist. ... Bernard Julia, born in 1952, is a French theoretical physicist. ... Joel Scherk (JoÃ«l Scherk) was a physicist who studied string theory. ... The quadrangle at the main ENS building on rue dUlm is known as the Cour aux Ernests &#8211; the Ernests being the goldfish in the pond. ... This article or section is in need of attention from an expert on the subject. ... 1980 (MCMLXXX) was a leap year starting on Tuesday. ... In physics, compactification plays an important part in string theory. ... In mathematics and physics, n-dimensional anti de Sitter space, denoted , is the maximally symmetric, simply-connected, Lorentzian manifold with constant negative curvature. ... 11D supergravity contains a 3-form field C. It also contains an electric 2-brane (M2) and a magnetic 5-brane (M5) under C. These branes are BPS states and they are also black branes. ... This article or section is in need of attention from an expert on the subject. ...

Many of the details of the theory were fleshed out by Peter van Nieuwenhuizen, Sergio Ferrara and Daniel Z. Freedman. Peter van Nieuwenhuizen (born October 26, 1938) is a Dutch physicist. ...

### 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 quarks or leptons. One suggestion was to replace the compact dimensions with the 7-sphere, with the symmetry group SO(8), or the squashed 7-sphere, with symmetry group SO(5) times SU(2).
• Until recently, the physical neutrinos seen in the real world were believed to be massless, and appeared to be left-handed, a phenomenon referred to as the chirality of the Standard Model. It was very difficult to construct a chiral fermion from a compactification — the compactified manifold needed to have singularities, but physics near singularities did not begin to be understood until the advent of orbifold conformal field theories in the late 1980s.
• Supergravity models generically result in an unrealistically large cosmological constant in four dimensions, and that constant is difficult to remove, and so require fine-tuning. This is still a problem today.
• Quantization of the theory led to quantum field theory gauge anomalies 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. These are the six flavors of quarks and their most likely decay modes. ... In physics, a particle is a lepton if it has a spin of 1/2 and does not experience the strong nuclear force. ... In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. ... 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. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... Neutrinos are elementary particles denoted by the symbol Î½. Travelling close to the speed of light, lacking electric charge and able to pass through ordinary matter almost undisturbed, they are extremely difficult to detect. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ... A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Î›) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... Fine Tuning is the name of XM Satellite Radios eclectic music channel. ... In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that invalidates the gauge symmetry of a quantum field theory i. ... Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. ...

The core breakthrough for the 10-dimensional theory, known as the first superstring revolution, was a demonstration by Michael B. Green, John H. Schwarz and David Gross that there are only three supergravity models in 10 dimensions which have gauge symmetries and in which all of the anomalies cancel. These were theories built on the groups SO(32) and $E_8 times E_8$, the direct product of two copies of E8. Today we know that, using D-branes for example, gauge symmetries can be introduced in other 10-dimensional theories as well[citation needed]. In physics, the first superstring revolution is a period of important discoveries in string theory roughly between 1984 and 1986. ... Michael Boris Green (born 22 May 1946) is a physicist who is one of the pioneers of string theory. ... John Henry Schwarz John Henry Schwarz (born 1941) is an American theoretical physicist. ... David Jonathan Gross (born February 19, 1941 in Washington, D.C.) is an American particle physicist and string theorist (although hes stated to the Brazilian newspaper Folha de SÃ£o Paulo, on 09/27/2006, that the second area is included in the first one). ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, E8 is the name given to a family of closely related structures. ... In theoretical physics, D-branes are a special class of p-branes, named for the physicist Johann Dirichlet. ...

### The Second Superstring Revolution

Initial excitement about the 10d 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, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... Shing-Tung Yau (Chinese: ; pinyin: ; born April 4, 1949) is a prominent mathematician working in differential geometry, and involved in the theory of Calabi-Yau manifolds. ... 2005 (MMV) was a common year starting on Saturday of the Gregorian calendar. ... The 23rd International Solvay Conference in Physics The Quantum Structure of Space and Time was the 23rd edition of the International Solvay Conference in Physics held in Hotel MÃ©tropole, Brussels (Belgium), from December 1 to December 3, 2005. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ...

There was a comparatively quiet period at the beginning of the 1990s, during which, however, several important tools were developed. For example, it was understood that a web of dualities connects the various perturbative string theories in various regimes.

Then it all changed, in what is known as the second superstring revolution. Joseph Polchinski realized that obscure string theory objects, called D-branes, which he had discovered six years earlier, are in fact the p-branes in 10-dimensional supergravity theories. The treatment of these p-branes was not restricted by string perturbation theory; in fact, thanks to supersymmetry, p-branes in supergravity were understood well beyond the limits in which string theory was understood. The second superstring revolution refers to the intense wave of breakthroughs in string theory that took place approximately between 1994 and 1997. ... Joe Polchinski in Santa Barbara Joseph Polchinski (born on May 16, 1954 in White Plains, New York) is a physicist working on string theory. ... In theoretical physics, D-branes are a special class of p-branes, named for the physicist Johann Dirichlet. ... P-branes or branes are terms from quantum superstring theory used to refer to membrane-like structures of one to eleven dimensions that arise in equations of this heavily mathematical theory. ... P-branes or branes are terms from quantum superstring theory used to refer to membrane-like structures of one to eleven dimensions that arise in equations of this heavily mathematical theory. ... This article or section is in need of attention from an expert on the subject. ...

Armed with this new nonperturbative tool, Edward Witten and many others were able to show that all of the perturbative string theories were descriptions of different states in a single theory which he named M-theory. Furthermore he argued that the classical limit of most states in this theory are described by the 11-dimensional supergravity that had fallen out of favor with the first superstring revolution 10 years earlier. In quantum mechanics, perturbation theory is a set of approximation schemes for describing a complicated quantum system in terms of a simpler one. ... Edward Witten (born August 26, 1951) is an American mathematical physicist, Fields Medalist, and professor at the Institute for Advanced Study. ... M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ... In physics, the first superstring revolution is a period of important discoveries in string theory roughly between 1984 and 1986. ...

Nowadays, in the post matrix theory era, once again many more people work with the 10d supergravities than with the 11d theory, as a practical formulation of M-theory in an arbitrary background space-time has never been found[citation needed]. The four-dimensional theory has remained consistently popular among a large sector of the world's phenomenologists, who outnumber string theorists. It has been suggested that this article or section be merged with matrix string theory. ... This article is about the philosophical movement. ...

## Relation to superstrings

Many, if not all supergravity theories are the classical limits of superstring theory (i.e., the limit in which the string is approximated as having zero length, and treated as a dimensionless point-particle), with the exception of "maximal" 11-dimensional supergravity, which is, by definition, a classical limit of M-theory. M-theory has no strings, but has a membrane, so intuitively one may think of the supergravity limit as a limit in which the membrane size shrinks to zero. However, this doesn't necessarily mean that string theory/M-theory is the only possible UV completion of supergravity[citation needed] and supergravity is often studied even by people who are not string theorists. Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. ... Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. ... M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ... In theoretical physics, UV completion of a quantum field theory XY is another quantum field theory (or a more generalized theory) LM that reduces to XY below some energy scale (the cutoff of XY). ...

## Nomenclature

### Supermultiplets

A representation of the super-Poincaré group is called a supermultiplet, and the one which contains a graviton is called the supergravity multiplet. Supergravity theories by definition are supersymmetric, which means in particular that all fields must transform under some representation of the super-Poincaré group. Thus fields in supergravity theories come in supermultiplets, and the particle content of a particular supergravity theory is the collection of supermultiplets that is present. In theoretical physics, a supermultiplet is formally a group representation of a supersymmetry algebra. ...

The name of a supergravity theory generally includes the number of dimensions of spacetime that it inhabits, and also the number $mathcal{N}$ of gravitinos that it has. Sometimes one also includes the choices of supermultiplets in the name of theory. For example, an $mathcal{N}=2$, (9+1)-dimensional supergravity enjoys 9 spatial dimensions, one time and 2 gravitinos. While the field content of different supergravity theories varies considerably, all supergravity theories contain at least one gravitino and they all contain a single graviton. Thus every supergravity theory contains a single supergravity supermultiplet. It is still not known whether one can construct theories with multiple gravitons that are not equivalent to multiple decoupled theories with a single graviton in each[citation needed]. Maximal supergravity theories, which will be defined below, may only contain the supergravity supermultiplet. In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ... The gravitino is the hypothetical supersymmetric partner of the graviton, as predicted by theories combining general relativity and supersymmetry, i. ... In physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. ...

When all of the gravitinos are in the supergravity supermultiplet, the number $mathcal{N}$ of gravitinos is equal to the number of supercharges, which will be defined below. Therefore sometimes $mathcal{N}$ is defined to be the number of supercharges.

When the gravitino is charged under some of the vector fields in the theory, the theory is called gauged supergravity. Supergravities with neutral gravitino is sometimes called ungauged supergravity if explicit distinction is necessary. Note that ungauged supergravity may contain gauge fields and charged scalars and/or spin-1/2 fermions. Gauged supergravity is a supergravity theory with gauged gravitino, that is, the gravitino is charged with respect to some of the vector fields in the theory. ...

### Counting Gravitinos

Gravitinos are fermions, which means that according to the spin-statistics theorem they must have an odd number of spinorial indices. In fact the gravitino field has one spinor and one vector index, which means that gravitinos transform as a tensor product of a spinorial representation and the vector representation of the Lorentz group. This is a Rarita-Schwinger spinor. The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In theoretical physics, the Rarita-Schwinger equation is the field equation of spin-3/2 fermions. ...

While there is only one vector representation for each Lorentz group, in general there are several different spinorial representations. Technically these are really representations of the double cover of the Lorentz group called a spin group. In mathematics, specifically topology, a covering map is a continuous surjective map p : C &#8594; X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ...

The canonical example of a spinorial representation is the Dirac spinor, which exists in every number of space-time dimensions. However the Dirac spinor representation is not always irreducible. When calculating the number $mathcal{N}$, one always counts the number of real irreducible representations. The spinors with spins less than 3/2 that exist in each number of dimensions will be classified in the following subsection. In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

### A classification of spinors

The available spinor representations depends on k; The maximal compact subgroup of the little group of the Lorentz group that preserves the momentum of a massless particle is Spin(d-1)× Spin(d-k-1), where k is equal to the number d of spatial dimensions minus the number d-k of time dimensions. (See helicity (particle physics)) For example, in our world, this is 3-1=2. Due to the mod 8 Bott periodicity of the homotopy groups of the Lorentz group, really we only need to consider k modulo 8. In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. ... In mathematics, a symmetry group describes all symmetries of objects. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In classical mechanics, momentum (pl. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... In particle physics, helicity is the projection of the angular momentum to the direction of motion: Because the angular momentum with respect to an axis has discrete values, helicity is discrete, too. ... In mathematics, the Bott periodicity theorem is a result from homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved to be of foundational significance for much further research, in particular in K-theory. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ...

For any value of k there is a Dirac representation, which is always of real dimension $2^{1+lfloor{frac{2d-k}{2}}rfloor}$ where $lfloor xrfloor$ is the greatest integer less than or equal to x. When $-2leq kleq 2 pmod 8$ there is a real Majorana spinor representation, whose dimension is half that of the Dirac representation. When k is even there is a Weyl spinor representation, whose real dimension is again half that of the Dirac spinor. Finally when k is divisible by eight, that is, when k is zero modulo eight, there is a Majorana-Weyl spinor, whose real dimension is one quarter that of the Dirac spinor. In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ... In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ... In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

Occasionally one also considers symplectic Majorana spinor which exist when $3leq kleq 5$, which have half has many components as Dirac spinors. When k=4 these may also be Weyl, yielding Weyl symplectic Majorana spinors which have one quarter as many components as Dirac spinors.

### Choosing Chiralities

Spinors in n-dimensions are representations (really modules) not only of the n-dimensional Lorentz group, but also of a Lie algebra called the n-dimensional Clifford algebra. The most commonly used basis of the compex $2^{lfloor nrfloor}$-dimensional representation of the Clifford algebra, the representation that acts on the Dirac spinors, consists of the gamma matrices. In cognitive psychology a representation is a hypothetical internal cognitive symbol that represents external reality. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... Clifford algebras are a type of associative algebra in mathematics. ... In mathematical physics, the gamma matrices, {Î³0, Î³1, Î³2, Î³3}, also known as the Dirac matrices, form a matrix-valued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra. ...

When n is even the product of all of the gamma matrices, which is often referred to as Γ5 as it was first considered in the case n=4, is not itself a member of the Clifford algebra. However, being a product of elements of the Clifford algebra, it is in the algebra's universal cover and so has an action on the Dirac spinors.

In particular, the Dirac spinors may be decomposed into eigenspaces of Γ5 with eigenvalues equal to $pm(-1)^{-k/2}$, where k is the number of spatial minus temporal dimensions in the spacetime. The spinors in these two eigenspaces each form projective representations of the Lorentz group, known as Weyl spinors. The eigenvalue under Γ5 is known as the chirality of the spinor, which can be left or right-handed. In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...

A particle that transforms as a single Weyl spinor is said to be chiral. The CPT theorem, which is required by Lorentz invariance in Minkowski space, implies that when there is a single time direction such particles have antiparticles of the opposite chirality. CPT-symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

Recall that the eigenvalues of Γ5, whose eigenspaces are the two chiralities, are $pm(-1)^{-k/2}$. In particular, when k is equal to two modulo four the two eigenvalues are complex conjugate and so the two chiralities of Weyl representations are complex conjugate representations.

Complex conjugation in quantum theories corresponds to time inversion. Therefore the CPT theorem implies that when the number of Minkowski dimensions is divisible by four (so that k is equal to 2 modulo 4) there be an equal number of left-handed and right-handed supercharges. On the other hand, if the dimension is equal to 2 modulo 4, there can be different numbers of left and right-handed supercharges, and so often one labels the theory by a doublet $mathcal{N}=(mathcal{N}_L,mathcal{N}_R)$ where $mathcal{N}_L$ and $mathcal{N}_R$ are the number of left-handed and right-handed supercharges respectively.

### Counting Supersymmetries

All supergravity theories are invariant under transformations in the super-Poincare group, although individual configurations are not in general invariant under every transformation in this group. The super-Poincaré group is generated by the super-Poincare algebra, which is a Lie superalgebra. A Lie superalgebra is a $mathbf{Z}_2$ graded algebra in which the elements of degree zero are called bosonic and those of degree one are called fermionic. A commutator, that is an antisymmetric bracket satisfying the Jacobi identity is defined between each pair of generators of fixed degree except for pairs of fermionic generators, for which instead one defines a symmetric bracket called an anticommutator. In theoretical physics, the supersymmetry algebra is a mathematical formalism for describing the relation between bosons and fermions. ... In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ...

The fermionic generators are also called supercharges. Any configuration which is invariant under any of the supercharges is said to be BPS, and often nonrenormalization theorems demonstrate that such states are particularly easily treated because they are uneffected by many quantum corrections. BPS can stand for: The Bogomolnyi Prasad Sommerfield bound The British Psychological Society The British Pharmacological Society The Biophysical Society The additives BioPsychoSymmetrie therapy, an holistic oriented therapy Bits per second (more usually bps) Basis points one one-hundredth of a percentage point Business Planning and Simulation Battle Programmer...

The supercharges transform as spinors, and the number of irreducible spinors of these fermionic generators is equal to the number of gravitinos $mathcal{N}$ defined above. Often $mathcal{N}$ is defined to be the number of fermionic generators, instead of the number of gravitinos, because this definition extends to supersymmetric theories without gravity.

Sometimes it is convenient to characterize theories not by the number $mathcal{N}$ of irreducible representations of gravitinos or supercharges, but instead by the total Q of their dimensions. This is because some features of the theory have the same Q-dependence in any number of dimensions. For example, one is often only interested in theories in which all particles have spin less than or equal to two. This requires that Q not exceed 32, except possibly in special cases in which the supersymmetry is realized in an unconventional, nonlinear fashion with products of bosonic generators in the anticommutators of the fermionic generators. In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

## Examples

### Why fewer than 32 SUSYs?

The supergravity theories that have attracted the most interest contain no spins higher than 2. 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.

The supercharges in every super-Poincaré algebra are generated by a multiplicative basis of m fundamental supercharges, and an additive basis of the supercharges (this definition of supercharges is a bit more broad than that given above) is given by a product of any subset of these m fundamental supercharges. The number of subsets of m elements is 2m, thus the space of supercharges is 2m-dimensional.

The fields in a supersymmetric theory form representations of the super-Poincaré algebra. It can be shown that when m is greater than 5 there are no representations that contain only fields of spin less than or equal to two. Thus we are interested in the case in which m is less than or equal to 5, which means that the maximal number of supercharges is 32. A supergravity theory with precisely 32 supersymmetries is known as a maximal supergravity.

Above we saw that the number of supercharges in a spinor depends on the dimension and the signature of spacetime. The supercharges occur in spinors. Thus the above limit on the number of supercharges cannot be satisfied in a spacetime of arbitrary dimension. Below we will describe some of the cases in which it is satisfied.

### A 12-dimensional two-time theory

The highest dimension in which spinors exist with only 32 supercharges is 12. If there are 11 spatial directions and 1 time direction then there will be Weyl and Majorana spinors which both are of dimension 64, and so are too large. Although some authors have considered nonlinear actions of the supersymmetry in which higher spin fields may not appear.

If instead one considers 10 spatial direction and a second temporal dimension then there is a Majorana-Weyl spinor, which as desired has only 32 components. For an overview of two-time theories by one of their main proponents, Itzak Bars, see his paper Two-Time Physics and Two-Time Physics on arxiv.org. He considered 12-dimensional supergravity in Supergravity, p-brane duality and hidden space and time dimensions. Certain String Theories allow for a second temporal or time dimension. ...

It is widely, but not universally, believed that two-time theories do not make sense. For example the Hamiltonian-based approach to quantum mechanics would have to be modified in the presence of a second Hamiltonian for the other time. However some claim that such a theory describes low energy behavior in Cumrun Vafa's F-theory. Others claim that the 12-dimensions of F-theory are merely a bookkeeping device and should not be confused with spacetime coordinates, or that two of the dimensions are somehow dual to each other and so should not be treated independently. In physics, Hamiltonian has distinct but closely related meanings. ... Cumrun Vafa is a leading string theorist from Harvard University where he started as a Harvard Junior Fellow. ... In physics, in the context of string theory, F-theory is formally a 12-dimensional theory, but the only way to obtain an acceptable background is to compactify this theory on a two-torus. ...

### 11-dimensional maximal SUGRA

This maximal supergravity is the classical limit of M-theory. There is, classically, only one 11-dimensional supergravity theory. Like all maximal supergravities, it contains a single supermultiplet, the supergravity supermultiplet. This supermultiplet contains the graviton, a Majorana gravitino and a 3-form gauge connection often called the C-field. M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ...

It contains two p-brane solutions, a 2-brane and a 5-brane, which are electrically and magnetically charged respectively with respect to the C-field. This means that 2-brane and 5-brane charge are the violations of the Bianchi identities for the dual C-field and original C-field respectively. The supergravity 2-brane and 5-brane are the classical limits of the M2-brane and M5-brane in M-theory. P-branes or branes are terms from quantum superstring theory used to refer to membrane-like structures of one to eleven dimensions that arise in equations of this heavily mathematical theory. ...

Although the p-branes are often referred to as solitons, technically they are sources and not solitons as the field configurations are singular. However, the singularity of the supersymmetric 5-brane solution is at the end of an infinitely long throat and when defining the quantum theory of the 5-brane one usually excises a region surrounding the singularity from the spacetime. In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ...

### 10d SUGRA theories

#### Type IIA SUGRA: N=(1,1)

This maximal supergravity is the classical limit of type IIA string theory. The field content of the supergravity supermultiplet consists of a graviton, a Majorana gravitino, a Kalb-Ramond field, odd-dimensional Ramond-Ramond gauge potentials, a dilaton and a dilatino. In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. ... In theoretical physics in general and string theory in particular, the Kalb-Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i. ... In theoretical physics, Ramond-Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. ... In theoretical physics, dilaton originally referred to a theoretical scalar field; as a photon refers in one sense to the electromagnetic field. ...

The Bianchi identities of the Ramond-Ramond gauge potentials C2k − 1 can be violated by adding sources ρ, which are called D(8-2k)-branes $ddC_{2k-1}=rho. ,,,$

In the democratic formulation of type IIA supergravity there exist Ramond-Ramond gauge potentials for 0<k<6, which leads to D0-branes (also called D-particles), D2-branes, D4-branes, D6-branes and, if one includes the case k=-1, D8-branes. In addition there are fundamental strings and their electromagnetic duals, which are called NS5-branes. In theoretical physics, an NS5-brane is a five-dimensional object (a p-brane) in string theory that carries a magnetic charge under the B-field, the field under which the fundamental string is electrically charged. ...

Although obviously there are no -1-form gauge connections, the corresponding 0-form field strength, G0 may exist. This field strength is called the Romans mass and when it is not equal to zero the supergravity theory is called massive IIA supergravity or Romans IIA supergravity. From the above Bianchi identity we see that a D8-brane is a domain wall between zones of differing G0, thus in the presence of a D8-brane at least part of the spacetime will be described by the Romans theory.

#### IIA SUGRA from 11d SUGRA

IIA SUGRA is the dimensional reductions of 11-dimensional supergravity on a circle. This means that 11d supergravity on the spacetime $M^{10}times S^1,$ is equivalent to IIA supergravity on the 10-manifold $M^{10},$ where one eliminates modes with masses proportional to the inverse radius of the circle S1. In physics, Kaluza-Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ...

In particular the field and brane content of IIA supergravity can be derived via this dimensional reduction procedure. The field G0 however does not arise from the dimensional reduction, massive IIA is not known to be the dimensional reduction of any higher-dimensional theory. The 1-form Ramond-Ramond potential $C_1,$ is the usual 1-form connection that arises from the Kaluza-Klein procedure, it arises from the components of the 11-d metric that contain one index along the compactified circle. The IIA 3-form gauge potential $C_3,$ is the reduction of the 11d 3-form gauge potential components with indices that do not lie along the circle, while the IIA Kalb-Ramond 2-form B-field consists of those components of the 11-dimensional 3-form with one index along the circle. The higher forms in IIA are not independent degrees of freedom, but are obtained from the lower forms using Hodge duality.

Similarly the IIA branes descend from the 11-dimension branes and geometry. The IIA D0-brane is a Kaluza-Klein momentum mode along the compactified circle. The IIA fundamental string is an 11-dimensional membrane which wraps the compactified circle. The IIA D2-brane is an 11-dimensional membrane that does not wrap the compactified circle. The IIA D4-brane is an 11-dimensional 5-brane that wraps the compactified circle. The IIA NS5-brane is an 11-dimensional 5-brane that does not wrap the compactified circle. The IIA D6-brane is a Kaluza-Klein monopole, that is, a topological defect in the compact circle fibration. The lift of the IIA D8-brane to 11-dimensions is not known, as one side of the IIA geometry as a nontrivial Romans mass, and an 11-dimensional original of the Romans mass is unknown.

#### Type IIB SUGRA: N=(2,0)

This maximal supergravity is the classical limit of type IIB string theory. The field content of the supergravity supermultiplet consists of a graviton, a Weyl gravitino, a Kalb-Ramond field, even-dimensional Ramond-Ramond gauge potentials, a dilaton and a dilatino. In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. ... In theoretical physics in general and string theory in particular, the Kalb-Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i. ... In theoretical physics, Ramond-Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. ... In theoretical physics, dilaton originally referred to a theoretical scalar field; as a photon refers in one sense to the electromagnetic field. ...

The Ramond-Ramond fields are sourced by odd-dimensional D(2k+1)-branes, which host supersymmetric U(1) gauge theories. As in IIA supergravity, the fundamental string is an electric source for the Kalb-Ramond B-field and the NS5-brane is a magnetic source. Unlike that of the IIA theory, the NS5-brane hosts a worldvolume U(1) supersymmetric gauge theory with $mathcal N=(1,1)$ supersymmetry, although some of this supersymmetry may be broken depending on the geometry of the spacetime and the other branes that are present. In theoretical physics, an NS5-brane is a five-dimensional object (a p-brane) in string theory that carries a magnetic charge under the B-field, the field under which the fundamental string is electrically charged. ...

This theory enjoys a SL(2,R) symmetry known as S-duality that interchanges the Kalb-Ramond field and the RR 2-form and also mixes the dilaton and the RR 0-form axion. In theoretical physics, S-duality (also a strong-weak duality) is an equivalence of two quantum field theories, string theories, or M-theory. ... The axion is an exotic subatomic particle postulated by Peccei-Quinn theory to resolve the strong-CP problem in quantum chromodynamics (QCD). ...

#### Type I gauged SUGRA: N=(1,0)

These are the classical limits of type I string theory and the two heterotic string theories. There is a single Majorana-Weyl spinor of supercharges, which in 10 dimensions contains 16 supercharges. As 16 is less than 32, the maximal number of supercharges, type I is not a maximal supergravity theory. In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. ... In physics, a heterotic string is a peculiar mixture (or hybrid) of the bosonic string and the superstring (the adjective heterotic comes from the Greek word heterosis). ... In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

In particular this implies that there is more than one variety of supermultiplet. In fact, there are two. As usual, there is a supergravity supermultiplet. This is smaller than the supergravity supermultiplet in type II, it contains only the graviton, a Majorana-Weyl gravitino, a 2-form gauge potential, the dilaton and a dilatino. Whether this 2-form is considered to be a Kalb-Ramond field or Ramond-Ramond field depends on whether one considers the supergravity theory to be a classical limit of a heterotic string theory or type I string theory. There is also a vector supermultiplet, which contains a one-form gauge potential called a gluon and also a Majorana-Weyl gluino. In physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. ... The gravitino is the hypothetical supersymmetric partner of the graviton, as predicted by theories combining general relativity and supersymmetry, i. ... In theoretical physics, dilaton originally referred to a theoretical scalar field; as a photon refers in one sense to the electromagnetic field. ... In theoretical physics in general and string theory in particular, the Kalb-Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i. ... In theoretical physics, Ramond-Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. ... In physics, a heterotic string is a peculiar mixture (or hybrid) of the bosonic string and the superstring (the adjective heterotic comes from the Greek word heterosis). ... In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... A gluino is a subatomic particle, the fermion superpartner of the gluon predicted by supersymmetry. ...

Unlike type IIA and IIB supergravities, for which the classical theory is unique, as a classical theory $mathcal{N}=1$ supergravity is consistent with a single supergravity supermultiplet and any number of vector multiplets. It is also consistent without the supergravity supermultiplet, but then it would contain no graviton and so would not be a supergravity theory. While one may add multiple supergravity supermultiplets, it is not known if they may consistently interact. One is free not only to determine the number, if any, of vector supermultiplets, but also there is some freedom in determining their couplings. They must describe a classical super Yang-Mills gauge theory, but the choice of gauge group is arbitrary. In addition one is free to make some choices of gravitational couplings in the classical theory. In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

While there are many varieties of classical $mathcal{N}=1$ supergravities, not all of these varieties are the classical limits of quantum theories. Generically the quantum versions of these theories suffer from various anomalies, as can be seen already at 1-loop in the hexagon Feynman diagrams. In 1984 and 1985 Michael Green and John H. Schwarz have shown that if one includes precisely 496 vector supermultiplets and chooses certain couplings of the 2-form and the metric then the gravitational anomalies cancel. This is called the Green-Schwarz anomaly cancellation mechanism. A regular hexagon. ... A Feynman diagram is a bookkeeping device for performing calculations in quantum field theory, invented by American physicist Richard Feynman. ... There are several people called Michael Green, including: Michael Green (cricketer) Michael Green (field hockey) - Field hockey player from Germany Michael Green (humorist) - humorous author Michael Green (physicist) - involved with string theory Michael Green (political expert) - Asia expert on the National Security Council; see NK news article Michael Green (runner... John Henry Schwarz John Henry Schwarz (born 1941) is an American theoretical physicist. ... Anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams In theoretical physics, a gravitational anomaly is an example of an 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 some... In physics, the Green-Schwarz mechanism is the main discovery that started the first superstring revolution in superstring theory in 1984. ...

In addition, anomaly cancellation requires one to cancel the gauge anomalies. This fixes the gauge symmetry algebra to be either $mathfrak{so}(32)$, $mathfrak{e}_8 oplus mathfrak{e}_8$, $mathfrak{e}_8 oplus 248mathfrak{u}(1)$ or $496mathfrak{u}(1)$. However, only the first two Lie algebras can be gotten from superstring theory[citation needed]. Quantum theories with at least 8 supercharges tend to have continuous moduli spaces of vacua. In compactifications of these theories, which have 16 supercharges, there exist degenerate vacua with different values of various Wilson loops. Such Wilson loops may be used to break the gauge symmetries to various subgroups. In particular the above gauge symmetries may be broken to obtain not only the standard model gauge symmetry but also symmetry groups such as SO(10) and SU(5) that are popular in GUT theories. In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that invalidates the gauge symmetry of a quantum field theory i. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ... In physics, compactification plays an important part in string theory. ... It has been suggested that this article or section be merged into Unified field theory. ...

### 9d SUGRA theories

In 9-dimensional Minkowski space the only irreducible spinor representation is the Majorana spinor, which has 16 components. Thus supercharges inhabit Majorana spinors of which there are at most two. In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

#### Maximal 9d SUGRA from 10d

In particular, if there are two Majorana spinors then one obtains the 9-dimensional maximal supergravity theory. Recall that in 10 dimensions there were two inequivalent maximal supergravity theories, IIA and IIB. The dimensional reduction of either IIA or IIB on a circle is the unique 9-dimensional supergravity. In other words, IIA or IIB on the product of a 9-dimensional space M9 and a circle is equivalent to the 9-dimension theory on M9, with Kaluza-Klein modes if one does not take the limit in which the circle shrinks to zero. In statistics, dimensionality reduction can be divided into two categories: feature selection and feature extraction. ...

#### T-duality

More generally one could consider the 10-dimensional theory on a nontrivial circle bundle over M9. Dimensional reduction still leads to a 9-dimensional theory on M9, but with a 1-form gauge potential equal to the connection of the circle bundle and a 2-form field strength which is equal to the Chern class of the old circle bundle. One may then lift this theory to the other 10-dimensional theory, in which case one finds that the 1-form gauge potential lifts to the Kalb-Ramond field. Similarly, the connection of the fibration of the circle in the second 10-dimensional theory is the integral of the Kalb-Ramond field of the original theory over the compactified circle. In mathematics, a circle bundle is a fiber bundle where the fiber is the circle , or more precisely, a principal U(1)-bundle with fiber U(1). ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... ÊIn physics, the field strength of a field is the magnitude of its vector (spatial) value. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In theoretical physics in general and string theory in particular, the Kalb-Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i. ...

This transformation between the two 10-dimensional theories is known as T-duality. While T-duality in supergravity involves dimensional reduction and so loses information, in the full quantum string theory the extra information is stored in string winding modes and so T-duality is a duality between the two 10-dimensional theories. The above construction can be used to obtain the relation between the circle bundle's connection and dual Kalb-Ramond field even in the full quantum theory. T-duality is a symmetry of string theory, relating type IIA and type IIB string theory, and the two heterotic string theories. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ...

#### N=1 Gauged SUGRA

As was the case in the parent 10-dimensional theory, 9-dimensional N=1 supergravity contains a single supergravity multiplet and an arbitrary number of vector multiplets. These vector multiplets may be coupled so as to admit arbitrary gauge theories, although not all possibilities have quantum completions. Unlike the 10-dimensional theory, as was described in the previous subsection, the supergravity multiplet itself contains a vector and so there will always be at least a U(1) gauge symmetry, even in the N=2 case.

### 4d SUGRA theories

#### 4-dimensional N=1 SUGRA

General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Fig. ... This article or section is in need of attention from an expert on the subject. ... In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... Results from FactBites:

 Supergravity (229 words) Supergravity is a quantum field theory of elemental particles based on Supersymmetry theories, which is consistent with General Relativity on local scales (as opposed to global scales). Supergravity theories predict the existence of a boson (a force-carrying particle known as the graviton [spin-2] and its fermionic superpartner, the gravitino (spin 3/2), both of which are massless. Supergravity theories have fallen out of favor in recent years due to the necessity of an unrealistically large cosmological constant in four dimensions, thus requires fine-tuning without at least another seven dimensions (as in 11-dimensional string theories).
 7.3 Divergence properties of N=8 supergravity (599 words) In all cases the linearized divergences take the form of derivatives acting on a particular contraction of Riemann tensors, which in four dimensions is equivalent to the square of the Bel-Robinson tensor [6, 37, 38]. This operator appears in the first set of corrections to the N =8 supergravity Lagrangian, in the inverse string-tension expansion of the effective field theory for the type II superstring . The explicit form of the linearized N =1, D =11 counterterm expressed as derivatives acting on Riemann tensors along with a more general discussion of supergravity divergences may be found in Ref. .
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