In theoretical physics, supergravity (supergravity
theory; SUGRA for short) is a modern field
theory that combines the principles of supersymmetry
and general relativity where supersymmetry
obeys locality; in contrast to non-gravitational
supersymmetric theories such as the Minimal
Supersymmetric Standard Model. Since the generators
of supersymmetry (SUSY) use the Poincaré
group to form a super-Poincaré algebra, gravity
follows naturally from local supersymmetry.
== Gravitons ==
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.
== History ==
=== Gauge supersymmetry ===
The first theory of local supersymmetry was
proposed by Dick Arnowitt and Pran Nath in
1975 and was called gauge supersymmetry.
=== Supergravity ===
The minimal version of four-dimensional Supergravity
was discovered in 1976 by Dan Freedman, Sergio
Ferrara and Peter van Nieuwenhuizen, and it
was quickly generalized to many different
theories in various numbers of dimensions
and involving additional (N) supersymmetries.
Supergravity theories with N>1 are usually
referred to as extended supergravity (SUEGRA).
Some supergravity theories were shown to be
related to certain higher-dimensional supergravity
theories via dimensional reduction (e.g. N=1,
11-dimensional supergravity is dimensionally
reduced on T7 to four-dimensional, ungauged,
N=8 Supergravity). The resulting theories
were sometimes referred to as Kaluza–Klein
theories as Kaluza and Klein constructed in
1919 a 5-dimensional gravitational theory,
that when dimensionally reduced on a circle,
its 4-dimensional non-massive modes describe
electromagnetism 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 (SUSY) breaks by a super Higgs
mechanism carried out by Ali Chamseddine,
Richard Arnowitt and Pran Nath in 1982. Collectively
now known as minimal supergravity Grand Unification
Theories (mSUGRA GUT), gravity mediates the
breaking of SUSY through the existence of
a hidden sector. mSUGRA naturally generates
the Soft SUSY breaking terms which are a consequence
of the Super Higgs effect. Radiative breaking
of electroweak symmetry through Renormalization
Group Equations (RGEs) follows as an immediate
consequence.
Due to its predictive power, requiring only
four input parameters and a sign to determine
the low energy phenomenology from the scale
of Grand Unification, its interest is a widely
investigated model of particle physics
=== 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 11 dimensions as the largest
number of dimensions consistent with a single
graviton, and more dimensions will show particles
with spins greater than 2. However, if two
of these dimensions are time-like, these problems
are avoided in 12 dimensions. Itzhak Bars
gives this emphasis.
In 1981 Ed Witten showed 11 as the smallest
number of dimensions 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.
Many techniques exist to embed the standard
model gauge group in supergravity in any number
of dimensions like the obligatory gauge symmetry
in type I and heterotic string theories, and
obtained in type II string theory by compactification
on certain Calabi–Yau manifolds. The D-branes
engineer gauge symmetries too.
In 1978 Eugène Cremmer, Bernard Julia and
Joël Scherk (CJS) 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. Other
11-dimensional theories known and quantum-mechanically
inequivalent reduce to the CJS theory when
one imposes the classical equations of motion.
However, in the mid 1980s Bernard de Wit and
Hermann Nicolai found an alternate theory
in D=11 Supergravity with Local SU(8) Invariance.
While not manifestly Lorentz-invariant, it
is in many ways superior, because it dimensionally-reduces
to the 4-dimensional theory without recourse
to the classical equations of motion.In 1980
Peter Freund 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 others
compact. The noncompact dimensions have to
form an anti-de Sitter space. There are many
possible compactifications, but the Freund-Rubin
compactification's invariance under all of
the supersymmetry transformations preserves
the action.Finally, the first two results
each appeared to establish 11 dimensions,
the third result appeared to specify the theory,
and the last result explained why the observed
universe appears to be four-dimensional.
Many of the details of the theory were fleshed
out by Peter van Nieuwenhuizen, 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 experiments 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.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 gauge and gravitational
anomalies cancel. These were theories built
on the groups SO(32) and
E
8
×
E
8
{\displaystyle 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 10-dimensional
theories, and the string theories that provide
their quantum completion, died by the end
of the 1980s. There were too many Calabi–Yaus
to compactify on, many more than Yau had estimated,
as he admitted in December 2005 at the 23rd
International Solvay Conference in Physics.
None quite gave the standard model, but it
seemed as though one could get close with
enough effort in many distinct ways. Plus
no one understood the theory beyond the regime
of applicability of string perturbation theory.
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
- perturbative - physics in one model with
strong string-coupling - non-perturbative
- in another.
Then the second superstring revolution occurred.
Joseph Polchinski realized that obscure string
theory objects, called D-branes, which he
discovered six years earlier, equate to stringy
versions of the p-branes known in supergravity
theories. String theory perturbation didn't
restrict these p-branes. Thanks to supersymmetry,
p-branes in supergravity gained understanding
well beyond the limits of string theory.
Armed with this new nonperturbative tool,
Edward Witten and many others could show all
of the perturbative string theories as descriptions
of different states in a single theory that
Witten named M-theory. Furthermore, he argued
that M-theory's long wavelength limit, i.e.
when the quantum wavelength associated to
objects in the theory appear much larger than
the size of the 11th dimension, need 11-dimensional
supergravity descriptors that fell out of
favor with the first superstring revolution
10 years earlier, accompanied by the 2- and
5-branes.
Therefore, supergravity comes full circle
and uses a common framework in understanding
features of string theories, M-theory, and
their compactifications to lower spacetime
dimensions.
== Relation to superstrings ==
The term "low energy limits" labels some 10-dimensional
supergravity theories. These arise as the
massless, tree-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 is
required to have 11-dimensional supergravity
as a "low energy limit". However, this doesn't
necessarily mean that string theory/M-theory
is the only possible UV completion of supergravity;
supergravity research is useful independent
of those relations.
== 4D N = 1 SUGRA ==
Before we move on to SUGRA proper, let's recapitulate
some important details about general relativity.
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.
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
α
˙
{\displaystyle {\dot {\alpha }}}
,
β
˙
{\displaystyle {\dot {\beta }}}
, ... . The indices for the fiber of T will
follow a similar notation, except that they
will be hatted like this:
M
^
,
α
^
{\displaystyle {\hat {M}},{\hat {\alpha }}}
. See van der Waerden notation for more details.
M
=
(
μ
,
α
,
α
˙
)
{\displaystyle M=(\mu ,\alpha ,{\dot {\alpha
}})}
. The supervierbein is denoted by
e
N
M
^
{\displaystyle e_{N}^{\hat {M}}}
, and the spin connection by
ω
M
^
N
^
P
{\displaystyle \omega _{{\hat {M}}{\hat {N}}P}}
. The inverse supervierbein is denoted by
E
M
^
N
{\displaystyle E_{\hat {M}}^{N}}
.
The supervierbein and spin connection are
real in the sense that they satisfy the reality
conditions
e
N
M
^
(
x
,
θ
¯
,
θ
)
∗
=
e
N
∗
M
^
∗
(
x
,
θ
,
θ
¯
)
{\displaystyle e_{N}^{\hat {M}}(x,{\overline
{\theta }},\theta )^{*}=e_{N^{*}}^{{\hat {M}}^{*}}(x,\theta
,{\overline {\theta }})}
where
μ
∗
=
μ
{\displaystyle \mu ^{*}=\mu }
,
α
∗
=
α
˙
{\displaystyle \alpha ^{*}={\dot {\alpha }}}
, and
α
˙
∗
=
α
{\displaystyle {\dot {\alpha }}^{*}=\alpha
}
and
ω
(
x
,
θ
¯
,
θ
)
∗
=
ω
(
x
,
θ
,
θ
¯
)
{\displaystyle \omega (x,{\overline {\theta
}},\theta )^{*}=\omega (x,\theta ,{\overline
{\theta }})}
.The covariant derivative is defined as
D
M
^
f
=
E
M
^
N
(
∂
N
f
+
ω
N
[
f
]
)
{\displaystyle D_{\hat {M}}f=E_{\hat {M}}^{N}\left(\partial
_{N}f+\omega _{N}[f]\right)}
.The covariant exterior derivative as defined
over supermanifolds needs to be super graded.
This means that every time we interchange
two fermionic indices, we pick up a +1 sign
factor, instead of -1.
The presence or absence of R symmetries is
optional, but if R-symmetry exists, the integrand
over the full superspace has to have an R-charge
of 0 and the integrand over chiral superspace
has to have an R-charge of 2.
A chiral superfield X is a superfield which
satisfies
D
¯
α
˙
^
X
=
0
{\displaystyle {\overline {D}}_{\hat {\dot
{\alpha }}}X=0}
. In order for this constraint to be consistent,
we require the integrability conditions that
{
D
¯
α
˙
^
,
D
¯
β
˙
^
}
=
c
α
˙
^
β
˙
^
γ
˙
^
D
¯
γ
˙
^
{\displaystyle \left\{{\overline {D}}_{\hat
{\dot {\alpha }}},{\overline {D}}_{\hat {\dot
{\beta }}}\right\}=c_{{\hat {\dot {\alpha
}}}{\hat {\dot {\beta }}}}^{\hat {\dot {\gamma
}}}{\overline {D}}_{\hat {\dot {\gamma }}}}
for some coefficients c.
Unlike nonSUSY GR, the torsion has to be nonzero,
at least with respect to the fermionic directions.
Already, even in flat superspace,
D
α
^
e
α
˙
^
+
D
¯
α
˙
^
e
α
^
≠
0
{\displaystyle D_{\hat {\alpha }}e_{\hat {\dot
{\alpha }}}+{\overline {D}}_{\hat {\dot {\alpha
}}}e_{\hat {\alpha }}\neq 0}
.
In one version of SUGRA (but certainly not
the only one), we have the following constraints
upon the torsion tensor:
T
α
_
^
β
_
^
γ
_
^
=
0
{\displaystyle T_{{\hat {\underline {\alpha
}}}{\hat {\underline {\beta }}}}^{\hat {\underline
{\gamma }}}=0}
T
α
^
β
^
μ
^
=
0
{\displaystyle T_{{\hat {\alpha }}{\hat {\beta
}}}^{\hat {\mu }}=0}
T
α
˙
^
β
˙
^
μ
^
=
0
{\displaystyle T_{{\hat {\dot {\alpha }}}{\hat
{\dot {\beta }}}}^{\hat {\mu }}=0}
T
α
^
β
˙
^
μ
^
=
2
i
σ
α
^
β
˙
^
μ
^
{\displaystyle T_{{\hat {\alpha }}{\hat {\dot
{\beta }}}}^{\hat {\mu }}=2i\sigma _{{\hat
{\alpha }}{\hat {\dot {\beta }}}}^{\hat {\mu
}}}
T
μ
^
α
_
^
ν
^
=
0
{\displaystyle T_{{\hat {\mu }}{\hat {\underline
{\alpha }}}}^{\hat {\nu }}=0}
T
μ
^
ν
^
ρ
^
=
0
{\displaystyle T_{{\hat {\mu }}{\hat {\nu
}}}^{\hat {\rho }}=0}
Here,
α
_
{\displaystyle {\underline {\alpha }}}
is a shorthand notation to mean the index
runs over either the left or right Weyl spinors.
The superdeterminant of the supervierbein,
|
e
|
{\displaystyle \left|e\right|}
, gives us the volume factor for M. Equivalently,
we have the volume 4|4-superform
e
μ
^
=
0
∧
⋯
∧
e
μ
^
=
3
∧
e
α
^
=
1
∧
e
α
^
=
2
∧
e
α
˙
^
=
1
∧
e
α
˙
^
=
2
{\displaystyle e^{{\hat {\mu }}=0}\wedge \cdots
\wedge e^{{\hat {\mu }}=3}\wedge e^{{\hat
{\alpha }}=1}\wedge e^{{\hat {\alpha }}=2}\wedge
e^{{\hat {\dot {\alpha }}}=1}\wedge e^{{\hat
{\dot {\alpha }}}=2}}
.
If we complexify the superdiffeomorphisms,
there is a gauge where
E
α
˙
^
μ
=
0
{\displaystyle E_{\hat {\dot {\alpha }}}^{\mu
}=0}
,
E
α
˙
^
β
=
0
{\displaystyle E_{\hat {\dot {\alpha }}}^{\beta
}=0}
and
E
α
˙
^
β
˙
=
δ
α
˙
β
˙
{\displaystyle E_{\hat {\dot {\alpha }}}^{\dot
{\beta }}=\delta _{\dot {\alpha }}^{\dot {\beta
}}}
. 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,
(
D
¯
2
−
8
R
)
f
{\displaystyle \left({\bar {D}}^{2}-8R\right)f}
is always a chiral superfield.
The action for a SUGRA theory with chiral
superfields X, is given by
S
=
∫
d
4
x
d
2
Θ
2
E
[
3
8
(
D
¯
2
−
8
R
)
e
−
K
(
X
¯
,
X
)
/
3
+
W
(
X
)
]
+
c
.
c
.
{\displaystyle S=\int d^{4}xd^{2}\Theta 2{\mathcal
{E}}\left[{\frac {3}{8}}\left({\bar {D}}^{2}-8R\right)e^{-K({\bar
{X}},X)/3}+W(X)\right]+c.c.}
where K is the Kähler potential and W is
the superpotential, and
E
{\displaystyle {\mathcal {E}}}
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,
while a constant shift to the superpotential
changes the effective cosmological constant.
As the effective Planck constant now depends
upon the value of the chiral superfield X,
we need to rescale the supervierbeins (a field
redefinition) to get a constant Planck constant.
This is called the Einstein frame.
== N = 8 supergravity in 4 dimensions ==
N=8 Supergravity is the most symmetric quantum
field theory which involves gravity and a
finite number of fields. It can be found from
a dimensional reduction of 11D supergravity
by making the size of 7 of the dimensions
go to zero. It has 8 supersymmetries which
is the most any gravitational theory can have
since there are 8 half-steps between spin
2 and spin -2. (A graviton has the highest
spin in this theory which is a spin 2 particle).
More supersymmetries would mean the particles
would have superpartners with spins higher
than 2. The only theories with spins higher
than 2 which are consistent involve an infinite
number of particles (such as String Theory
and Higher-Spin Theories). Stephen Hawking
in his A Brief History of Time speculated
that this theory could be the Theory of Everything.
However, in later years this was abandoned
in favour of String Theory. There has been
renewed interest in the 21st century with
the possibility that this theory may be finite.
== Higher-dimensional SUGRA ==
Higher-dimensional SUGRA is the higher-dimensional,
supersymmetric generalization of general relativity.
Supergravity can be formulated in any number
of dimensions up to eleven. Higher-dimensional
SUGRA focuses upon supergravity in greater
than four dimensions.
The number of supercharges in a spinor depends
on the dimension and the signature of spacetime.
The supercharges occur in spinors. Thus the
limit on the number of supercharges cannot
be satisfied in a spacetime of arbitrary dimension.
Some theoretical examples in which this is
satisfied are:
12-dimensional two-time theory
11-dimensional maximal SUGRA
10-dimensional SUGRA theories
Type IIA SUGRA: N = (1, 1)
IIA SUGRA from 11d SUGRA
Type IIB SUGRA: N = (2, 0)
Type I gauged SUGRA: N = (1, 0)
9d SUGRA theories
Maximal 9d SUGRA from 10d
T-duality
N = 1 Gauged SUGRAThe supergravity theories
that have attracted the most interest contain
no spins higher than two. This means, in particular,
that they do not contain any fields that transform
as symmetric tensors of rank higher than two
under Lorentz transformations. The consistency
of interacting higher spin field theories
is, however, presently a field of very active
interest.
== See also ==
== Notes ==
== References ==
=== Historical ===
Nath, P.; Arnowitt, R. (1975). "Generalized
Super-Gauge Symmetry as a New Framework for
Unified Gauge Theories". Physics Letters B.
56 (2): 177. Bibcode:1975PhLB...56..177N.
doi:10.1016/0370-2693(75)90297-x.
Freedman, D.Z.; van Nieuwenhuizen, P.; Ferrara,
S. (1976). "Progress Toward A Theory Of Supergravity".
Physical Review. D13 (12): 3214–3218. Bibcode:1976PhRvD..13.3214F.
doi:10.1103/physrevd.13.3214.
E. Cremmer, B. Julia and J. Scherk, "Supergravity
theory in eleven dimensions", Physics Letters
B76 (1978), pp. 409–412. scanned version
Freund, P.; Rubin, M. (1980). "Dynamics of
dimensional reduction". Physics Letters. B97
(2): 233–235. Bibcode:1980PhLB...97..233F.
doi:10.1016/0370-2693(80)90590-0.
Ali H. Chamseddine, R. Arnowitt, Pran Nath,
"Locally Supersymmetric Grand Unification",
" Phys. Rev.Lett.49:970,1982"
Green, Michael B.; Schwarz, John H. (1984).
"Anomaly Cancellation in Supersymmetric D=10
Gauge Theory and Superstring Theory". Physics
Letters B. 149: 117–122. Bibcode:1984PhLB..149..117G.
doi:10.1016/0370-2693(84)91565-x.
=== General ===
Bernard de Wit (2002), Supergravity
Nath, Pran, Supersymmetry, Supergravity and
Unification, Cambridge University Press, Cambridge,
(2016), ISBN 0-521-19702-3.
A Supersymmetry Primer (1998); updated in
2006.
Drees, Manuel, Godbole, Rohini, and Roy, Probir,
Theory & Phenomenology of Sparticles, World
Scientific, Singapore (2005), ISBN 9-810-23739-1.
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).
Wess, Julius; Bagger, Jonathan (1992). Supersymmetry
and Supergravity. Princeton University Press.
p. 260. ISBN 0-691-02530-4.
