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In theoretical physics, a supergravity theory is a field theory combining supersymmetry and general relativity.

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.

History

Four Dimensional SUGRA

Supergravity, also called SUGRA, was initially proposed as a four-dimensional theory in 1976, 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.

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:

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.

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. Quantum-mechanically, inequivalent 11-dimensional theories are known which reduce to the CJS theory in the classical limit, that is 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.

Many of the details of the theory were fleshed out by Peter van Nieuwenhuizen, Sergio Ferrara and Daniel Z. Freedman.

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 we 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, 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.

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.

There was a long, slow period at the beginning of the 1990s as the field faded into obscurity. During this time, 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.

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.

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. The four-dimensional theory has remained consistently popular among a large sector of the world's phenomenologists, who outnumber string theorists.

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 and supergravity is often studied even by people who are not string theorists.

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. When a supergravity theory contains a supermultiplet which contains a field that transforms as a vector under the super-Poincaré group, the theory is said to be a gauged supergravity.

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. Maximal supergravity theories, which will be defined below, may only contain the supergravity supermultiplet.

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.

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.

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.

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.

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.

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 $\lfloor x\rfloor$ is the greatest integer less than or equal to x. When $-2\leq k\leq 2{\pmod {8}}$ 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.

Occasionally one also considers symplectic Majorana spinor which exist when $3\leq k\leq 5$ , which have half has many compenents 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 $2^{\lfloor n\rfloor }$ -dimensional representation of the Clifford algebra, the representation that acts on the Dirac spinors, consists of the gamma matrices.

When n is even the product of all of the gamma matrices, which is often referred to as $\Gamma _{5}$ 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 $\Gamma _{5}$ 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 $\Gamma _{5}$ is known as the chirality of the spinor, which can be left or right-handed.

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.

Recall that the eigenvalues of $\Gamma _{5}$ , 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.

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.

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 be 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.

Examples

Why less 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 $2^{m}$ , thus the space of supercharges is $2^{m}$ -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. He considered 12-dimensional supergravity in Supergravity, p-brane duality and hidden space and time dimensions.

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.

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.

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.

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.

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.

The Bianchi identities of the Ramond-Ramond gauge potentials $C_{2k-1}$ can be violated by adding sources $\rho$ , 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.

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 S.

In particular the field and brane content of IIA supergravity can be derived via this dimensional reduction procedure. The field $G_{0}$ 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.

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.

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.

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 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.

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.

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 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.

In addition, anomaly cancellation requires one to cancel the gauge anomalies. This fixes the gauge symmetry algebra to be either ${\mathfrak {so}}(32)$ , ${\mathfrak {e}}_{8}\oplus {\mathfrak {e}}_{8}$ , ${\mathfrak {e}}_{8}\oplus 248{\mathfrak {u}}(1)$ or $496{\mathfrak {u}}(1)$ . However, only the first two Lie algebras can be gotten from superstring theory. 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.

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.

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 M and a circle is equivalent to the 9-dimension theory on M, with Kaluza-Klein modes if one does not take the limit in which the circle shrinks to zero.

T-duality

More generally one could consider the 10-dimensional theory on a nontrivial circle bundle over M. Dimensional reduction still leads to a 9-dimensional theory on M, 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.

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.

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

Maximal SUGRA from Freund-Rubin

4-dimensional N=1 SUGRA

References

Historical

E. Cremmer, B. Julia and J. Scherk, "Supergravity theory in eleven dimensions", Physics Letters B76 (1978) pp 409-412.
P. Freund and M. Rubin, "Dynamics of dimensional reduction", Physics Letters B97 (1980) pp 233-235.
Michael B. Green, John H. Schwarz, "Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory", Physics Letters B149 (1984) pp117-122.

General

Adel Bilal, "Introduction to supersymmetry" (2001) ArXiv hep-th/0101055. (a comprehensive introduction to supersymmetry.)

Friedemann Brandt, "Lectures on supergravity" (2002) ArXiv hep-th/0204035. (an introduction to 4-dimensional N=1 supergravity.)

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