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Supergravity

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Modern theory of gravitation that combines supersymmetry and general relativity
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Intheoretical physics,supergravity (supergravity theory;SUGRA for short) is a modernfield theory that combines the principles ofsupersymmetry andgeneral relativity; this is in contrast to non-gravitational supersymmetric theories such as theMinimal Supersymmetric Standard Model. Supergravity is thegauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with thePoincaré algebra andsuperalgebra, called thesuper-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way.[1]

Gravitons

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Like all covariant approaches to quantum gravity,[2] supergravity contains a spin-2 field whose quantum is thegraviton. Supersymmetry requires the graviton field to have asuperpartner. This field hasspin 3/2 and its quantum is thegravitino. The number of gravitino fields is equal to the number of supersymmetries.

History

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Gauge supersymmetry

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The first theory of local supersymmetry was proposed byDick Arnowitt andPran Nath in 1975[3] and was calledgauge supersymmetry.

Supergravity

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The first model of 4-dimensional supergravity (without this denotation) was formulated by Dmitri Vasilievich Volkov and Vyacheslav A. Soroka in 1973,[4] emphasizing the importance of spontaneous supersymmetry breaking for the possibility of a realistic model. Theminimal version of 4-dimensional supergravity (with unbroken local supersymmetry) was constructed in detail in 1976 byDan Freedman,Sergio Ferrara andPeter van Nieuwenhuizen.[5] In 2019 the three were awarded a specialBreakthrough Prize in Fundamental Physics for the discovery.[6] The key issue of whether or not the spin 3/2 field is consistently coupled was resolved in the nearly simultaneous paper, byDeser andZumino,[7] which independently proposed the minimal 4-dimensional model. It was quickly generalized to many different theories in various numbers ofdimensions 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 certainhigher-dimensional supergravity theories viadimensional reduction (e.g. N=1, 11-dimensional supergravity is dimensionally reduced on T7 to 4-dimensional, ungauged,N = 8 supergravity). The resulting theories were sometimes referred to asKaluza–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 describeelectromagnetism coupled togravity.

mSUGRA

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mSUGRA means minimal SUper GRAvity. The construction of a realistic model of particle interactions within theN = 1 supergravity framework where supersymmetry (SUSY) breaks by a superHiggs mechanism carried out byAli Chamseddine,Richard Arnowitt andPran Nath in 1982. Collectively now known as minimal supergravity Grand Unification Theories (mSUGRA GUT), gravity mediates the breaking of SUSY through the existence of ahidden sector. mSUGRA naturally generates the Soft SUSY breaking terms which are a consequence of the Super Higgs effect. Radiative breaking of electroweak symmetry throughRenormalization 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, it is widely investigated inparticle physics.

See also:Gravity-Mediated Supersymmetry Breaking in the MSSM

11D: the maximal SUGRA

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Main article:Eleven-dimensional supergravity

One of these supergravities, the 11-dimensional theory, generated considerable excitement as the first potential candidate for thetheory of everything. This excitement was built on four pillars, two of which have now been largely discredited:

  • Werner Nahm showed[8] 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[citation needed] gives this emphasis.
  • In 1981Ed Witten showed[9] 11 as the smallest number of dimensions big enough to contain thegauge groups of theStandard Model, namelySU(3) for thestrong interactions andSU(2) timesU(1) for theelectroweak interactions.[citation needed] Many techniques exist to embed the standard model gauge group in supergravity in any number of dimensions like the obligatory gauge symmetry intype I andheterotic string theories, and obtained intype II string theory bycompactification on certainCalabi–Yau manifolds. TheD-branes engineer gauge symmetries too.
  • In 1978Eugène Cremmer,Bernard Julia andJoël Scherk (CJS) found[10] 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 known and quantum-mechanically inequivalent reduce to the CJS theory when one imposes the classical equations of motion. However, in the mid-1980sBernard de Wit andHermann 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 1980Peter Freund andM. 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.[11] The noncompact dimensions have to form ananti-de Sitter space. There are many possible compactifications, but theFreund-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 byPeter van Nieuwenhuizen,Sergio Ferrara andDaniel Z. Freedman.

The end of the SUGRA era

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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:[citation needed]

  • The compact manifolds which were known at the time and which contained the standard model were not compatible with supersymmetry, and could not holdquarks orleptons. One suggestion was to replace the compact dimensions with the 7-sphere, with the symmetry groupSO(8), or the squashed 7-sphere, with symmetry groupSO(5) timesSU(2).
  • Until recently, the physicalneutrinos seen in experiments were believed to be massless, and appeared to be left-handed, a phenomenon referred to as thechirality 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 oforbifoldconformal field theories in the late 1980s.
  • Supergravity models generically result in an unrealistically largecosmological constant in four dimensions, and that constant is difficult to remove, and so requirefine-tuning. This is still a problem today.
  • Quantization of the theory led to quantum field theorygauge 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 involvingsuperstrings. However, by moving to 10 dimensions one loses the sense of uniqueness of the 11-dimensional theory.[12]

The core breakthrough for the 10-dimensional theory, known as thefirst superstring revolution, was a demonstration byMichael B. Green,John H. Schwarz andDavid Gross that there are only three supergravity models in 10 dimensions which have gauge symmetries and in which all of the gauge andgravitational anomalies cancel. These were theories built on the groupsSO(32) andE8×E8{\displaystyle E_{8}\times E_{8}}, thedirect product of two copies ofE8. Today we know that, usingD-branes for example, gauge symmetries can be introduced in other 10-dimensional theories as well.[13]

The second superstring revolution

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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 manyCalabi–Yaus to compactify on, many more thanYau had estimated, as he admitted in December 2005 at the23rd 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 stringperturbation 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 thesecond superstring revolution occurred.Joseph Polchinski realized that obscure string theory objects, calledD-branes, which he discovered six years earlier, equate to stringy versions of thep-branes known in supergravity theories. String theory perturbation didn't restrict thesep-branes. Thanks to supersymmetry, p-branes in supergravity gained understanding well beyond the limits of string theory.

Armed with this newnonperturbative 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 namedM-theory. Furthermore, he argued that M-theory'slong wavelength limit, i.e. when the quantum wavelength associated to objects in the theory appear much larger than the size of the 11th dimension, needs 11-dimensional supergravity descriptors that fell out of favor with thefirst 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

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The term "low energy limits" labels some 10-dimensional supergravity theories. These arise as the massless,tree-level approximation of string theories. Trueeffective 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 possibleUV completion of supergravity;[citation needed] supergravity research is useful independent of those relations.

4DN = 1 SUGRA

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Main article:4D N = 1 supergravity

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[which?] to T. This map is thevierbein. The local Lorentz symmetry has agauge connection associated with it, thespin connection.

The following discussion will be in superspace notation, as opposed to the component notation, which isn't manifestly covariant under SUSY. There are actuallymany 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 anR4|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 aMajorana spinor. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handedWeyl 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 byα˙{\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 }}}. Seevan der Waerden notation for more details.M=(μ,α,α˙){\displaystyle M=(\mu ,\alpha ,{\dot {\alpha }})}. The supervierbein is denoted byeNM^{\displaystyle e_{N}^{\hat {M}}}, and the spin connection byωM^N^P{\displaystyle \omega _{{\hat {M}}{\hat {N}}P}}. Theinverse supervierbein is denoted byEM^N{\displaystyle E_{\hat {M}}^{N}}.

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

eNM^(x,θ¯,θ)∗=eN∗M^∗(x,θ,θ¯){\displaystyle e_{N}^{\hat {M}}(x,{\overline {\theta }},\theta )^{*}=e_{N^{*}}^{{\hat {M}}^{*}}(x,\theta ,{\overline {\theta }})} whereμ∗=μ{\displaystyle \mu ^{*}=\mu },α∗=α˙{\displaystyle \alpha ^{*}={\dot {\alpha }}}, andα˙∗=α{\displaystyle {\dot {\alpha }}^{*}=\alpha } andω(x,θ¯,θ)∗=ω(x,θ,θ¯){\displaystyle \omega (x,{\overline {\theta }},\theta )^{*}=\omega (x,\theta ,{\overline {\theta }})}.

Thecovariant derivative is defined as

DM^f=EM^N(∂Nf+ωN[f]){\displaystyle D_{\hat {M}}f=E_{\hat {M}}^{N}\left(\partial _{N}f+\omega _{N}[f]\right)}.

Thecovariant 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 ofR 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 superfieldX is a superfield which satisfiesD¯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˙^}=cα˙^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 coefficientsc.

Unlike nonSUSY GR, thetorsion has to be nonzero, at least with respect to the fermionic directions. Already, even in flat superspace,Dα^eα˙^+D¯α˙^eα^≠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:

Tα_^β_^γ_^=0{\displaystyle T_{{\hat {\underline {\alpha }}}{\hat {\underline {\beta }}}}^{\hat {\underline {\gamma }}}=0}
Tα^β^μ^=0{\displaystyle T_{{\hat {\alpha }}{\hat {\beta }}}^{\hat {\mu }}=0}
Tα˙^β˙^μ^=0{\displaystyle T_{{\hat {\dot {\alpha }}}{\hat {\dot {\beta }}}}^{\hat {\mu }}=0}
Tα^β˙^μ^=2iσα^β˙^μ^{\displaystyle T_{{\hat {\alpha }}{\hat {\dot {\beta }}}}^{\hat {\mu }}=2i\sigma _{{\hat {\alpha }}{\hat {\dot {\beta }}}}^{\hat {\mu }}}
Tμ^α_^ν^=0{\displaystyle T_{{\hat {\mu }}{\hat {\underline {\alpha }}}}^{\hat {\nu }}=0}
Tμ^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.

Thesuperdeterminant of the supervierbein,|e|{\displaystyle \left|e\right|}, gives us the volume factor for M. Equivalently, we have the volume 4|4-superformeμ^=0∧⋯∧eμ^=3∧eα^=1∧eα^=2∧eα˙^=1∧eα˙^=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 whereEα˙^μ=0{\displaystyle E_{\hat {\dot {\alpha }}}^{\mu }=0},Eα˙^β=0{\displaystyle E_{\hat {\dot {\alpha }}}^{\beta }=0} andEα˙^β˙=δα˙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. Iff is any superfield,(D¯2−8R)f{\displaystyle \left({\bar {D}}^{2}-8R\right)f} is always a chiral superfield.

The action for a SUGRA theory with chiral superfieldsX, is given by

S=∫d4xd2Θ2E[38(D¯2−8R)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.}

whereK is theKähler potential andW is thesuperpotential, andE{\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 effectivePlanck constant, while a constant shift to the superpotential changes the effectivecosmological constant. As the effective Planck constant now depends upon the value of the chiral superfieldX, we need to rescale the supervierbeins (a field redefinition) to get a constant Planck constant. This is called theEinstein frame.

N = 8 supergravity in 4 dimensions

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N = 8 supergravity is the mostsymmetric 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 hisA Brief History of Time speculated that this theory could be theTheory 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

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Main article:Higher-dimensional supergravity

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 aspinor 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:

The supergravity theories that have attracted the most interest contain no spins higher than two. This means, in particular, that they do not contain any fields that transform as symmetric tensors of rank higher than two under Lorentz transformations. The consistency of interacting higher spin field theories is, however, presently a field of very active interest.

See also

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References

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  1. ^Van Nieuwenhuizen, P. (1981). "Supergravity".Physics Reports.68 (4):189–398.Bibcode:1981PhR....68..189V.doi:10.1016/0370-1573(81)90157-5.
  2. ^Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity".arXiv:gr-qc/0006061.
  3. ^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.
  4. ^Volkov, D.V.; Soroka, V.A. (1973). "Higgs effect for Goldstone particles with spin 1/2".JETP Letters.16 (11):438–440.Bibcode:1973JETPL..18..312V.doi:10.1007/BFb0105271.
  5. ^Freedman, D.Z.; van Nieuwenhuizen, P.; Ferrara, S. (1976). "Progress Toward A Theory Of Supergravity".Physical Review D.13 (12):3214–3218.Bibcode:1976PhRvD..13.3214F.doi:10.1103/physrevd.13.3214.
  6. ^"Supergravity scientists share $3M US Breakthrough Prize".CBC News.
  7. ^Deser, S.; Zumino, B. (1976)."Consistent Supergravity".Physics Letters B.62 (3):335–337.Bibcode:1976PhLB...62..335D.doi:10.1016/0370-2693(76)90089-7.
  8. ^Nahm, Werner (1978)."Supersymmetries and their representations".Nuclear Physics B.135 (1):149–166.Bibcode:1978NuPhB.135..149N.doi:10.1016/0550-3213(78)90218-3.
  9. ^Witten, Ed (1981). "Search for a realistic Kaluza-Klein theory".Nuclear Physics B.186 (3):412–428.Bibcode:1981NuPhB.186..412W.doi:10.1016/0550-3213(81)90021-3.
  10. ^E. Cremmer, B. Julia and J. Scherk, "Supergravity theory in eleven dimensions",Physics LettersB76 (1978)pp 409-412,
  11. ^Peter G.O. Freund; Mark A. Rubin (1980). "Dynamics of dimensional reduction".Physics Letters B.97 (2):233–235.Bibcode:1980PhLB...97..233F.doi:10.1016/0370-2693(80)90590-0.
  12. ^Duff, M. J. (1998). "A Layman's Guide to M-theory".arXiv:hep-th/9805177.
  13. ^Blumenhagen, R.;Cvetic, M.; Langacker, P.; Shiu, G. (2005)."Toward Realistic Intersecting D-Brane Models".Annual Review of Nuclear and Particle Science.55 (1):71–139.arXiv:hep-th/0502005.Bibcode:2005ARNPS..55...71B.doi:10.1146/annurev.nucl.55.090704.151541.S2CID 15148429.

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