Supergravity

# Supergravity

Discussion

Encyclopedia
In theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, supergravity (supergravity theory) is a field theory that combines the principles of supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. Together, these imply that, in supergravity, the supersymmetry is a local symmetry
Local symmetry
In physics, a local symmetry is symmetry of some physical quantity, which smoothly depends on the point of the base manifold. Such quantities can be for example an observable, a tensor or the Lagrangian of a theory....

(in contrast to non-gravitational supersymmetric theories, such as the Minimal Supersymmetric Standard Model
Minimal Supersymmetric Standard Model
The Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...

). Since the generators of supersymmetry (SUSY) are convoluted with the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

to form a Super-Poincaré algebra it is very natural to see that supergravity follows naturally from supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

.

## Gravitons

Like any field theory of gravity, a supergravity theory contains a spin-2 field whose quantum is the graviton
Graviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...

. Supersymmetry requires the graviton field to have a superpartner
Superpartner
In particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....

. This field has spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

3/2 and its quantum is the gravitino
Gravitino
The gravitino is the supersymmetric partner of the graviton, as predicted by theories combining general relativity and supersymmetry; i.e. supergravity theories...

. The number of gravitino fields is equal to the number of supersymmetries
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

. Supergravity theories are often said to be the only consistent theories of interacting massless spin 3/2 fields.

### Four-dimensional SUGRA

SUGRA, or SUper GRAvity, was initially proposed as a four-dimensional theory in 1976 by Daniel Z. Freedman
Daniel Z. Freedman
Daniel Z. Freedman is an American theoretical physicist. He is a Professor of Physics and Applied Mathematics at the Massachusetts Institute of Technology . He is known for his work in supergravity.-Education:...

, Peter van Nieuwenhuizen
Peter van Nieuwenhuizen
Peter van Nieuwenhuizen is a Dutch physicist. He is now a distinguished Professor at Stony Brook University in the United States. Van Nieuwenhuizen is best-known for his discovery of supergravity with Sergio Ferrara and Daniel Z...

and Sergio Ferrara
Sergio Ferrara
Sergio Ferrara is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles Sergio Ferrara (born 1945) is an Italian physicist working on...

at Stony Brook University, but was quickly generalized to many different theories in various numbers of dimensions and greater number (N) of supersymmetry charges. Supergravity theories with N>1 are usually referred to as extended supergravity (SUEGRA). Some supergravity theories were shown to be equivalent to certain higher-dimensional supergravity theories via dimensional reduction
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....

(e.g. N = 1 11-dimensional supergravity is dimensionally reduced on S7 to N = 8, d = 4 SUGRA). 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, that when dimensionally reduced on circle, its 4-dimensional non-massive modes describe electromagnetism coupled to gravity.

### mSUGRA

mSUGRA means minimal SUper GRAvity.
The construction of a realistic model of particle interactions
within the N = 1 supergravity framework where supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

(SUSY) is
broken by a super Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....

was carried out by
Ali Chamseddine, Richard Arnowitt
Richard Arnowitt
Richard Lewis Arnowitt is an American physicist known for his contributions to theoretical particle physics and to general relativity.Arnowitt is a Distinguished Professor at Texas A&M University, where he is a member of the Department of Physics....

and Pran Nath
Pran Nath
Pran Nath is a theoretical physicist working at Northeastern University, with research focus in elementary particle physics. He holds a Matthews Distinguished University Professor chair.-Research:...

in 1982. In these classes of models collectively now known as minimal supergravity Grand Unification Theories (mSUGRA GUT), gravity mediates the breaking of SUSY through the existence of a hidden sector. mSUGRA naturally generates the Soft SUSY breaking terms which are a consequence of the Super Higgs effect. Radiative breaking of electroweak symmetry through Renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

Group Equations (RGEs) follows as an immediate consequence. mSUGRA is one of the most widely investigated models of particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

due to it predictive power requiring only four input parameters and a sign, to determine the low energy Phenomenology from the scale of Grand Unification.

### 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
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....

. 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
Itzhak Bars
Itzhak Bars is a theoretical physicist at the University of Southern California in Los Angeles. In 2007, Bars presented a theory concerning the fundamental laws of physics. According to Bars' theory, time may not have only one dimension , but may have two separate dimensions instead.Humans normally...

.

• In 1981, Ed Witten showed that 11 was the smallest number of dimensions that was big enough to contain the gauge groups of the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

, 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
Type I string theory
In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...

and heterotic string theories. In type II string theory
Type II string theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. These account for two of the five consistent superstring theories in ten dimensions. Both theories have the maximal amount of supersymmetry — namely 32 supercharges...

they could also be obtained by compactifying
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....

on certain Calabi-Yau manifold
Calabi-Yau manifold
A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics...

s. Today one may also use D-brane
D-brane
In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Hořava in 1989...

s to engineer gauge symmetries.

• In 1978, Eugene Cremmer
Eugène Cremmer
Eugène Cremmer, born in 1942, is a French physicist. He is directeur de recherche at the Ecole Normale Supérieure. In 1978, together with Bernard Julia and Joël Scherk, he co-developed 11 dimensional supergravity theory and proposed a mechanism of spontaneous compactification in field theory.-...

, Bernard Julia
Bernard Julia
Bernard Julia is a French theoretical physicist who has made contributions to the theory of supergravity. He graduated from Université Paris-Sud in 1978,and is directeur de recherche at the Ecole Normale Supérieure...

and Joel Scherk
Joël Scherk
Joël Scherk was a French theoretical physicist who studied string theory and supergravity. Together with John H. Schwarz, he figured out that string theory was a theory of quantum gravity in 1974...

(CJS) found the classical action for an 11-dimensional supergravity theory. This remains today the only known classical 11-dimensional theory with local supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

and no fields of spin higher than two. 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
Bernard de Wit
Bernard de Wit is a Dutch theoretical physicist specialized in supergravity and particle physics.Bernard de Wit studied theoretical physics at Utrecht University, where he got his PhD under supervision of Nobel Prize laureate Martinus Veltman in 1973...

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
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....

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 compactification
Freund-Rubin compactification
11D supergravity contains a 3-form field C. It also contains an electric 2-brane and a magnetic 5-brane under C. These branes are BPS states and they are also black branes. Their near horizon metrics are given by AdS4×S7 and AdS7× S4 respectively. These geometries contain a nonzero...

s are invariant under all of the supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

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
Peter van Nieuwenhuizen
Peter van Nieuwenhuizen is a Dutch physicist. He is now a distinguished Professor at Stony Brook University in the United States. Van Nieuwenhuizen is best-known for his discovery of supergravity with Sergio Ferrara and Daniel Z...

, Sergio Ferrara
Sergio Ferrara
Sergio Ferrara is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles Sergio Ferrara (born 1945) is an Italian physicist working on...

and Daniel Z. Freedman
Daniel Z. Freedman
Daniel Z. Freedman is an American theoretical physicist. He is a Professor of Physics and Applied Mathematics at the Massachusetts Institute of Technology . He is known for his work in supergravity.-Education:...

.

### The end of the SUGRA era

The initial excitement over 11-dimensional supergravity soon waned, as various failings were discovered, and attempts to repair the model failed as well. Problems included:
• The compact manifolds which were known at the time and which contained the standard model were not compatible with supersymmetry, and could not hold quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s or lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...

s. One suggestion was to replace the compact dimensions with the 7-sphere, with the symmetry group SO(8)
SO(8)
In mathematics, SO is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.-Spin:...

, or the squashed 7-sphere, with symmetry group SO(5)
SO(5)
In mathematics, SO, also denoted SO5 or SO, is the special orthogonal group of degree 5 over the field R of real numbers, i.e...

times SU(2).

• Until recently, the physical neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

s seen in the real world were believed to be massless, and appeared to be left-handed, a phenomenon referred to as the chirality
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...

of the Standard Model. It was very difficult to construct a chiral fermion from a compactification
Compactification
Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...

— the compactified manifold needed to have singularities, but physics near singularities did not begin to be understood until the advent of orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

conformal field theories
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...

in the late 1980s.

• Supergravity models generically result in an unrealistically large cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

in four dimensions, and that constant is difficult to remove, and so require fine-tuning
Fine-tuning
In theoretical physics, fine-tuning refers to circumstances when the parameters of a model must be adjusted very precisely in order to agree with observations. Theories requiring fine-tuning are regarded as problematic in the absence of a known mechanism to explain why the parameters happen to...

. This is still a problem today.

• Quantization of the theory led to quantum field theory gauge anomalies
Gauge anomaly
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.e...

rendering the theory inconsistent. In the intervening years physicists have learned how to cancel these anomalies.

Some of these difficulties could be avoided by moving to a 10-dimensional theory involving superstrings. However, by moving to 10 dimensions one loses the sense of uniqueness of the 11-dimensional theory.

The core breakthrough for the 10-dimensional theory, known as the first superstring revolution, was a demonstration by Michael B. Green, John H. Schwarz and David Gross
David Gross
David Jonathan Gross is an American particle physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. He is currently the director and holder of the Frederick W...

that there are only three supergravity models in 10 dimensions which have gauge symmetries and in which all of the gauge and gravitational anomalies cancel. These were theories built on the groups SO(32) and , the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

of two copies of E8
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

. 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 10-dimensional theories, and the string theories that provide their quantum completion, died by the end of the 1980s. There were too many Calabi-Yaus to compactify
Compactification
Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...

on, many more than Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....

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
Perturbation theory
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; however, several important tools were developed. For example, it became apparent that the various superstring theories were related by "string dualities", some of which relate weak string-coupling (i.e. perturbative) physics in one model with strong string-coupling (i.e. non-perturbative) in another.

Then it all changed, in what is known as the second superstring revolution. Joseph Polchinski
Joseph Polchinski
Joseph Polchinski is a physicist working on string theory. He graduated from Canyon del Oro High School in Tucson, Arizona in 1971, obtained his B.S. degree from Caltech in 1975, and his Ph.D. from the University of California, Berkeley in 1980 under the supervision of Stanley Mandelstam...

realized that obscure string theory objects, called D-branes, which he had discovered six years earlier, are stringy versions of the p-branes that were known in supergravity theories. The treatment of these p-branes was not restricted by string perturbation theory; in fact, thanks to supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

, p-branes in supergravity were understood well beyond the limits in which string theory was understood.

Armed with this new nonperturbative tool, Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....

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
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...

. Furthermore he argued that the long wavelength limit
Long Wavelength Limit
In electricity and magnetism, the long wavelength limit is the limiting case when the wavelength is much larger than the system size. This corresponds to the quasi-static case, and reduces to electrostatics and magnetostatics....

* of M-theory should be described by the 11-dimensional supergravity that had fallen out of favor with the first superstring revolution 10 years earlier, accompanied by the 2- and 5-branes. [*= i.e. when the quantum wavelength associated to objects in the theory are much larger than the size of the 11th dimension].

Historically, then, supergravity has come "full circle". It is a commonly used framework in understanding features of string theories, M-theory and their compactifications
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic....

to lower spacetime dimensions.

## Relation to superstrings

Particular 10-dimensional supergravity theories are considered "low energy limits" of the 10-dimensional superstring theories; more precisely, these arise as the massless, tree-level approximation of string theories. True effective field theories of string theories, rather than truncations, are rarely available. Due to string dualities, the conjectured 11-dimensional M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...

is required to have 11-dimensional supergravity as a "low energy limit". However, this doesn't necessarily mean that string theory/M-theory is the only possible UV completion
UV Completion
In theoretical physics, ultraviolet completion, or UV completion, of a quantum field theory is the passing from a lower energy quantum field theory to a more general quantum field theory above a threshold value known as the cutoff...

of supergravity; supergravity research is useful independent of those relations.

## 4D N = 1 SUGRA

Before we move on to SUGRA proper, let's recapitulate some important details about general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. We have a 4D differentiable manifold M with a Spin(3,1) principal bundle over it. This principal bundle represents the local Lorentz symmetry. In addition, we have a vector bundle T over the manifold with the fiber having four real dimensions and transforming as a vector under Spin(3,1).
We have an invertible linear map from the tangent bundle TM to T. This map is the vierbein. The local Lorentz symmetry has a gauge connection associated with it, the spin connection
Spin connection
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the Levi-Civita connection...

.

The following discussion will be in superspace notation, as opposed to the component notation, which isn't manifestly covariant under SUSY. There are actually many different versions of SUGRA out there which are inequivalent in the sense that their actions and constraints upon the torsion tensor are different, but ultimately equivalent in that we can always perform a field redefinition of the supervierbeins and spin connection to get from one version to another.

In 4D N=1 SUGRA, we have a 4|4 real differentiable supermanifold M, i.e. we have 4 real bosonic dimensions and 4 real fermionic dimensions. As in the nonsupersymmetric case, we have a Spin(3,1) principal bundle over M. We have an R4|4 vector bundle T over M. The fiber of T transforms under the local Lorentz group as follows; the four real bosonic dimensions transform as a vector and the four real fermionic dimensions transform as a Majorana spinor. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handed Weyl spinor (they're not independent of each other). We also have a spin connection as before.

We will use the following conventions; the spatial (both bosonic and fermionic) indices will be indicated by M, N, ... . The bosonic spatial indices will be indicated by μ, ν, ..., the left-handed Weyl spatial indices by α, β,..., and the right-handed Weyl spatial indices by , , ... . The indices for the fiber of T will follow a similar notation, except that they will be hatted like this: . See van der Waerden notation
Van der Waerden notation
In theoretical physics, van der Waerden notation refers to the usage of two-component spinors in four spacetime dimensions. This is standard in twistor theory and supersymmetry....

for more details. . The supervierbein is denoted by , and the spin connection by . The inverse supervierbein is denoted by .

The supervierbein and spin connection are real in the sense that they satisfy the reality conditions where , , and and .

The covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

is defined as.

The covariant exterior derivative as defined over supermanifolds needs to be super graded. This means that every time we interchange two fermionic indices, we pick up a +1 sign factor, instead of -1.

The presence or absence of R symmetries
R-symmetry
In theoretical physics, the R-symmetry is the symmetry transforming different supercharges in a theory with supersymmetry into each other. In the simplest case of the N=1 supersymmetry, such an R-symmetry is isomorphic to a global U group or its discrete subgroup...

is optional, but if R-symmetry exists, the integrand over the full superspace has to have an R-charge of 0 and the integrand over chiral superspace has to have an R-charge of 2.

A chiral superfield X is a superfield which satisfies . In order for this constraint to be consistent, we require the integrability conditions that for some coefficients c.

Unlike nonSUSY GR, the torsion
Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...

has to be nonzero, at least with respect to the fermionic directions. Already, even in flat superspace, .
In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:
Here, is a shorthand notation to mean the index runs over either the left or right Weyl spinors.

The superdeterminant of the supervierbein, , gives us the volume factor for M. Equivalently, we have the volume 4|4-superform .

If we complexify the superdiffeomorphisms, there is a gauge where , and . The resulting chiral superspace has the coordinates x and Θ.

R is a scalar valued chiral superfield derivable from the supervielbeins and spin connection. If f is any superfield, is always a chiral superfield.

The action for a SUGRA theory with chiral superfields X, is given by
where K is the Kähler potential and W is the superpotential
Superpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...

, and is the chiral volume factor. Unlike the case for flat superspace, adding a constant to either the Kähler or superpotential is now physical. A constant shift to the Kähler potential changes the effective Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

, while a constant shift to the superpotential changes the effective cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

. As the effective Planck constant now depends upon the value of the chiral superfield X, we need to rescale the supervierbeins (a field redefinition) to get a constant Planck constant. This is called the Einstein frame.

## Higher-dimensional SUGRA

See the article higher-dimensional supergravity for more details.

• M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...

• General relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

• Supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

• Grand Unified Theory
• Super-Poincaré algebra
• Quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

• Supermanifold
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.- Physics :...

• String Theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

### Historical

• D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, "Progress Toward A Theory Of Supergravity", Physical Review D13 (1976) pp 3214–3218.
• E. Cremmer, B. Julia and J. Scherk, "Supergravity theory in eleven dimensions", Physics Letters B76 (1978) pp 409–412. scanned version
• P. Freund and M. Rubin, "Dynamics of dimensional reduction", Physics Letters B97 (1980) pp 233–235.
• Ali H. Chamseddine, R. Arnowitt, Pran Nath, "Locally Supersymmetric Grand Unification", " Phys. Rev.Lett.49:970,1982"
• Michael B. Green, John H. Schwarz, "Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory", Physics Letters B149 (1984) pp117–122.