superfields, superspace and supergravity

0.1 Superspace, superfields, supergravity and Lie superalgebras.

In general, a superfield–or quantized gravity field- has a highly reducible representation of the supersymmetry algebra, and the problem of specifying a supergravity theory can be defined as a search for those representationsPlanetmathPlanetmath that allow the construction of consistentPlanetmathPlanetmath local actions, perhaps considered as either quantum groupPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, or quantum groupoidPlanetmathPlanetmathPlanetmath, actions. Extending quantum symmetries to include quantized gravity fields–specified as ‘superfields’– is called supersymmetry in current theories of Quantum Gravity. Graded ‘Lie’ algebrasMathworldPlanetmathPlanetmathPlanetmathPlanetmath (or Lie superalgebrasPlanetmathPlanetmath) represent the quantum operator supersymmetries by defining these simultaneously for both fermion (spin 1/2) and boson (integer or 0 spin particles).

The quantized physical space with supersymmetric properties is then called a ‘superspace’, (another name for ‘quantized space with supersymmetry’) in Quantum Gravity. The following subsection defines these physical concepts in precise mathematical terms.

0.1.1 Mathematical definitions and propagation equations for superfields in superspace: Graded Lie algebras

Supergravity, in essence, is an extended supersymmetric theory of both matter and gravitation (viz. Weinberg, 1995 [1]). A first approach to supersymmetry relied on a curved ‘superspace’ (Wess and Bagger,1983 [3]) and is analogous to supersymmetric gauge theories (see, for example, SectionsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 27.1 to 27.3 of Weinberg, 1995). Unfortunately, a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath non–linear supergravity theory might be forbiddingly complicated and furthermore, the constraints that need be made on the graviton superfield appear somewhat subjective, (according to Weinberg, 1995). In a different approach to supergravity, one considers the physical componentsMathworldPlanetmathPlanetmath of the gravitational superfield which can be then identified based on ‘flat-space’ superfield methods (Chs. 26 and 27 of Weinberg, 1995). By implementing the gravitational weak-field approximation one obtains several of the most important consequences of supergravity theory, including masses for the hypothetical ‘gravitino’ and ‘gaugino particles’ whose existence might be expected from supergravity theories. Furthermore, by adding on the higher order terms in the gravitational constant to the supersymmetric transformationMathworldPlanetmathPlanetmath, the general coordinate transformations form a closed algebra and the Lagrangian that describes the interactions of the physical fields is then invariantMathworldPlanetmath under such transformations.The first quantization of such a flat-space superfield would obviously involve its ‘deformationMathworldPlanetmath’, and as a result its corresponding supersymmetry algebra becomes non–commutativePlanetmathPlanetmathPlanetmath.

0.1.2 Metric superfield

Because in supergravity both spinor and tensor fields are being considered, the gravitational fields are represented in terms of tetrads, eμa(x), rather than in terms of Einstein’s general relativistic metric gμν(x). The connectionsMathworldPlanetmathPlanetmath between these two distinct representations are as follows:

gμν(x)=ηabeμa(x)eγb(x), (0.1)

with the general coordinates being indexed by μ,ν, etc., whereas local coordinates that are being defined in a locally inertial coordinate systemMathworldPlanetmath are labeled with superscripts a, b, etc.; ηab is the diagonal matrixMathworldPlanetmath with elements +1, +1, +1 and -1. The tetrads are invariant to two distinct types of symmetryMathworldPlanetmathPlanetmath transformations–the local Lorentz transformations:

eμa(x)Λba(x)eμb(x), (0.2)

(where Λba is an arbitrary real matrix), and the general coordinate transformations:

xμ(x)μ(x). (0.3)

In a weak gravitational field the tetrad may be represented as:

eμa(x)=δμa(x)+2κΦμa(x), (0.4)

where Φμa(x) is small compared with δμa(x) for all x values, and κ=8πG, where G is Newton’s gravitational constant. As it will be discussed next, the supersymmetry algebra (SA) implies that the graviton has a fermionic superpartner, the hypothetical ‘gravitino’, with helicities ± 3/2. Such a self-charge-conjugate massless particle as the ‘gravitiono’ with helicities ± 3/2 can only have low-energy interactions if it is represented by a Majorana field ψμ(x) which is invariant under the gauge transformations:

ψμ(x)ψμ(x)+δμψ(x), (0.5)

with ψ(x) being an arbitrary Majorana field as defined by Grisaru and Pendleton (1977). The tetrad field Φμν(x) and the graviton field ψμ(x) are then incorporated into a term Hμ(x,θ) defined as the metric superfield. The relationships between Φμν(x) and ψμ(x), on the one hand, and the components of the metric superfield Hμ(x,θ), on the other hand, can be derived from the transformations of the whole metric superfield:

Hμ(x,θ)Hμ(x,θ)+Δμ(x,θ), (0.6)

by making the simplifying– and physically realistic– assumptionPlanetmathPlanetmath of a weak gravitational field (further details can be found, for example, in Ch.31 of vol.3. of Weinberg, 1995). The interactions of the entire superfield Hμ(x) with matter would be then described by considering how a weak gravitational field, hμν interacts with an energy-momentum tensor Tμν represented as a linear combinationMathworldPlanetmath of components of a real vector superfield Θμ. Such interaction terms would, therefore, have the form:

I=2κ𝑑x4[HμΘμ]D, (0.7)

( denotes ‘matter’) integrated over a four-dimensional (Minkowski) spacetime with the metric defined by the superfield Hμ(x,θ). The term Θμ, as defined above, is physically a supercurrent and satisfies the conservation conditions:

γμ𝐃Θμ=𝐃, (0.8)

where 𝐃 is the four-component super-derivative and X denotes a real chiral scalar superfield. This leads immediately to the calculation of the interactions of matter with a weak gravitational field as:

I=κd4xTμν(x)hμν(x), (0.9)

It is interesting to note that the gravitational actions for the superfield that are invariant under the generalized gauge transformations HμHμ+Δμ lead to solutions of the Einstein field equations for a homogeneousPlanetmathPlanetmathPlanetmath, non-zero vacuum energy density ρV that correspond to either a de Sitter space for ρV>0, or an anti-de Sitter space for ρV<0. Such spaces can be represented in terms of the hypersurface equation

x52±ημ,νxμxν=R2, (0.10)

in a quasi-Euclidean five-dimensional space with the metric specified as:

ds2=ημ,νxμxν±dx52, (0.11)

with ’+’ for de Sitter space and ’-’ for anti-de Sitter space, respectively.

The spacetime symmetry groups, or extended symmetry groupoidsPlanetmathPlanetmathPlanetmathPlanetmath, as the case may be– are different from the ‘classical’ Poincaré symmetry group of translationsMathworldPlanetmathPlanetmath and Lorentz transformations. Such spacetime symmetry groups, in the simplest case, are therefore the O(4,1) group for the de Sitter space and the O(3,2) group for the anti–de Sitter space. A detailed calculation indicates that the transition from ordinary flat space to a bubble of anti-de Sitter space is not favored energetically and, therefore, the ordinary (de Sitter) flat space is stable (viz. Coleman and De Luccia, 1980), even though quantum fluctuations might occur to an anti–de Sitter bubble within the limits permitted by the Heisenberg uncertainty principle.

0.2 Supersymmetry algebras and Lie (graded) superalgebras.

It is well known that continuous symmetry transformations can be represented in terms of a Lie algebraMathworldPlanetmath of linearly independentMathworldPlanetmath symmetry generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath tj that satisfy the commutation relationsMathworldPlanetmathPlanetmath:

[tj,tk]=ιΣlCjktl, (0.12)

Supersymmetry is similarly expressed in terms of the symmetry generators tj of a graded (‘Lie’) algebra which is in fact defined as a superalgebra) by satisfying relations of the general form:

tjtk-(-1)ηjηktktj=ιΣlCjkltl. (0.13)

The generators for which ηj=1 are fermionic whereas those for which ηj=0 are bosonic. The coefficientsMathworldPlanetmath Cjkl are structure constants satisfying the following conditions:

Cjkl=-(-1)ηjηkCjkl. (0.14)

If the generators j are quantum Hermitian operators, then the structure constants satisfy the reality conditions Cjk*=-Cjk . Clearly, such a graded algebraic structurePlanetmathPlanetmath is a superalgebra and not a proper Lie algebra; thus graded Lie algebras are often called ‘Lie superalgebras’.

The standard computational approach in QM utilizes the S-matrix approach, and therefore, one needs to consider the general, graded ‘Lie algebra’ of supersymmetry generators that commute with the S-matrix. If one denotes the fermionic generators by Q, then U-1(Λ)QU(Λ) will also be of the same type when U(Λ) is the quantum operator corresponding to arbitrary, homogeneous Lorentz transformations Λμν . Such a group of generators provide therefore a representation of the homogeneous Lorentz group of transformations 𝕃 . The irreducible representation of the homogeneous Lorentz group of transformations provides therefore a classification of such individual generators.

0.2.1 Graded ‘Lie Algebras’/Superalgebras.

A set of quantum operators QjkAB form an 𝐀,𝐁 representation of the group 𝐋 defined above which satisfy the commutation relations:

[𝐀,QjkAB]=-[ΣjJjjA,QjkAB], (0.15)


[𝐁,QjkAB]=-[ΣjJkkA,QjkAB], (0.16)

with the generators 𝐀 and 𝐁 defined by 𝐀(1/2)(𝐉±i𝐊) and 𝐁(1/2)(𝐉-i𝐊), with 𝐉 and 𝐊 being the Hermitian generators of rotationsMathworldPlanetmath and ‘boosts’, respectively.

In the case of the two-component Weyl-spinors Qjr the Haag–Lopuszanski–Sohnius (HLS) theoremMathworldPlanetmath applies, and thus the fermions form a supersymmetry algebra defined by the anti-commutation relations:

[Qjr,Qks*] =2δrsσjkμPμ, (0.17)
[Qjr,Qks] =ejkZrs,

where Pμ is the 4–momentum operator, Zrs=-Zsr are the bosonic symmetry generators, and σμ and 𝐞 are the usual 2×2 Pauli matricesMathworldPlanetmath. Furthermore, the fermionic generators commute with both energy and momentum operators:

[Pμ,Qjr]=[Pμ,Qjr*]=0. (0.18)

The bosonic symmetry generators Zks and Zks* represent the set of central charges of the supersymmetric algebra:

[Zrs,Ztn*]=[Zrs*,Qjt]=[Zrs*,Qjt*]=[Zrs*,Ztn*]=0. (0.19)

From another direction, the Poincaré symmetry mechanism of special relativity can be extended to new algebraic systems (Tanasă, 2006). In Moultaka et al. (2005) in view of such extensionsPlanetmathPlanetmathPlanetmathPlanetmath, consider invariant-free Lagrangians and bosonic multiplets constituting a symmetry that interplays with (Abelian) U(1)–gauge symmetry that may possibly be described in categoricalPlanetmathPlanetmath terms, in particular, within the notion of a cubical site (Grandis and Mauri, 2003).

We shall proceed to introduce in the next section generalizationsPlanetmathPlanetmath of the concepts of Lie algebras and graded Lie algebras to the corresponding Lie algebroids that may also be regarded as C*–convolution representations of quantum gravity groupoids and superfield (or supergravity) supersymmetries. This is therefore a novel approach to the proper representation of the non-commutative geometryPlanetmathPlanetmathPlanetmath of quantum spacetimes–that are curved (or ‘deformed’) by the presence of intense gravitational fields–in the framework of non-AbelianMathworldPlanetmathPlanetmath, graded Lie algebroids. Their correspondingly deformed quantum gravity groupoids (QGG) should, therefore, adequately represent supersymmetries modified by the presence of such intense gravitational fields on the Planck scale. Quantum fluctuations that give rise to quantum ‘foams’ at the Planck scale may be then represented by quantum homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of such QGGs. If the corresponding graded Lie algebroids are also integrable, then one can reasonably expect to recover in the limit of 0 the Riemannian geometry of General Relativity and the globally hyperbolic spacetime of Einstein’s classical gravitation theory (GR), as a result of such an integration to the quantum gravity fundamental groupoidMathworldPlanetmathPlanetmathPlanetmath (QGFG). The following subsection will define the precise mathematical concepts underlying our novel quantum supergravity and extended supersymmetry notions.


  • 1 S. Weinberg.: The Quantum TheoryPlanetmathPlanetmath of Fields. Cambridge, New York and Madrid: Cambridge University Press, Vols. 1 to 3, (1995–2000).
  • 2 A. Weinstein : Groupoids: unifying internal and external symmetry, Notices of the Amer. Math. Soc. 43 (7): 744-752 (1996).
  • 3 J. Wess and J. Bagger: Supersymmetry and Supergravity, Princeton University Press, (1983).
Title superfields, superspace and supergravity
Canonical name SuperfieldsSuperspaceAndSupergravity
Date of creation 2013-03-22 18:17:03
Last modified on 2013-03-22 18:17:03
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 23
Author bci1 (20947)
Entry type Feature
Classification msc 81R60
Classification msc 81R50
Classification msc 83C47
Classification msc 83C75
Classification msc 83C45
Classification msc 81P05
Synonym quantum gravity
Synonym quantum space-times
Related topic SupersymmetryOrSupersymmetries
Related topic NormedAlgebra
Related topic SupercategoriesPlanetmathPlanetmath
Related topic QuantumGravityTheories
Related topic SuperalgebroidsAndHigherDimensionalAlgebroids
Related topic AxiomaticTheoryOfSupercategories
Related topic LieSuperalgebra3
Related topic MetricSuperfields
Related topic SuperalgebroidsAndHigherDimensionalAlgebroids
Defines superspace
Defines superfields
Defines supergravity
Defines supersymmetry and L-superalgebras