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 thoserepresentations that allow the construction of consistent local actions, perhaps consideredas either quantum group, or quantum groupoid, actions. Extending quantum symmetries to includequantized gravity fields–specified as ‘superfields’– is called supersymmetry in current theories of Quantum Gravity. Graded ‘Lie’ algebras (or Lie superalgebras) represent the quantum operator supersymmetriesby 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 definesthese 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 ofboth 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, forexample, Sections 27.1 to 27.3 of Weinberg, 1995). Unfortunately,a complete non–linear supergravity theory might be forbiddinglycomplicated and furthermore, the constraints that need be made onthe graviton superfield appear somewhat subjective,(according to Weinberg, 1995). In a different approach to supergravity,one considers the physical components of the gravitational superfieldwhich can be then identified based on ‘flat-space’ superfield methods(Chs. 26 and 27 of Weinberg, 1995). By implementing the gravitationalweak-field approximation one obtains several of the most importantconsequences of supergravity theory, including masses for thehypothetical ‘gravitino’ and ‘gaugino particles’ whose existence might beexpected from supergravity theories. Furthermore, by adding on thehigher order terms in the gravitational constant to thesupersymmetric transformation, the general coordinatetransformations form a closed algebra and the Lagrangian thatdescribes the interactions of the physical fields is then invariantunder such transformations.The first quantization of such a flat-spacesuperfield would obviously involve its ‘deformation’, and as a result its correspondingsupersymmetry algebra becomes non–commutative.

0.1.2 Metric superfield

Because in supergravity both spinor and tensor fields are beingconsidered, the gravitational fields are represented in terms oftetrads, $e_{\mu }^a(x),$ rather than in terms of Einstein’s generalrelativistic metric $g_{\mu \nu }(x)$. The connections betweenthese two distinct representations are as follows:

$$g_{\mu \nu }(x)={\eta }_{ab}{e}_{\mu }^a(x)e_{\gamma }^b(x),$$(0.1)

with the general coordinates being indexed by $\mu ,\nu ,$ etc.,whereas local coordinates that are being defined in a locallyinertial coordinate system are labeled with superscripts a, b,etc.; ${\eta }_{ab}$ is the diagonal matrix with elements +1, +1,+1 and -1. The tetrads are invariant to two distinct types ofsymmetry transformations–the local Lorentz transformations:

$$e_{\mu }^a(x)⟼{\mathrm{\Lambda }}_b^a(x)e_{\mu }^b(x),$$(0.2)

(where ${\mathrm{\Lambda }}_b^a$ is an arbitrary real matrix), and the generalcoordinate transformations:

$$x^{\mu }⟼{(x^{\prime })}^{\mu }(x).$$(0.3)

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

$$e_{\mu }^a(x)={\delta }_{\mu }^a(x)+2\kappa {\mathrm{\Phi }}_{\mu }^a(x),$$(0.4)

where ${\mathrm{\Phi }}_{\mu }^a(x)$ is small compared with ${\delta }_{\mu }^a(x)$ forall $x$ values, and $\kappa =\sqrt{8\pi G}$, where G is Newton’sgravitational constant. As it will be discussed next, thesupersymmetry algebra (SA) implies that the graviton has afermionic superpartner, the hypothetical ‘gravitino’, withhelicities $\pm$ 3/2. Such a self-charge-conjugate masslessparticle as the ‘gravitiono’ with helicities $\pm$ 3/2 can only havelow-energy interactions if it is represented by a Majoranafield ${\psi }_{\mu }(x)$ which is invariant under the gaugetransformations:

$${\psi }_{\mu }(x)⟼{\psi }_{\mu }(x)+{\delta }_{\mu }\psi (x),$$(0.5)

with $\psi (x)$ being an arbitrary Majorana field as defined byGrisaru and Pendleton (1977). The tetrad field ${\mathrm{\Phi }}_{\mu \nu }(x)$ and the graviton field ${\psi }_{\mu }(x)$ are thenincorporated into a term $H_{\mu }(x,\theta )$ defined as themetric superfield. The relationships between ${\mathrm{\Phi }}_{{\mu }_{\nu }}(x)$ and ${\psi }_{\mu }(x)$, on the one hand, and the componentsof the metric superfield $H_{\mu }(x,\theta )$, on the other hand,can be derived from the transformations of the whole metricsuperfield:

$$H_{\mu }(x,\theta )⟼H_{\mu }(x,\theta )+{\mathrm{\Delta }}_{\mu }(x,\theta ),$$(0.6)

by making the simplifying– and physically realistic– assumptionof a weak gravitational field (further details can be found, forexample, in Ch.31 of vol.3. of Weinberg, 1995). The interactionsof the entire superfield $H_{\mu }(x)$ with matter would be thendescribed by considering how a weak gravitational field,$h_{{\mu }_{\nu }}$ interacts with an energy-momentum tensor $T^{\mu \nu }$ represented as a linear combination of components of a realvector superfield ${\mathrm{\Theta }}^{\mu }$. Such interaction terms would,therefore, have the form:

$$I_{ℳ}=2\kappa \int 𝑑x^4{[H_{\mu }{\mathrm{\Theta }}^{\mu }]}_D,$$(0.7)

($ℳ$ denotes ‘matter’) integrated over a four-dimensional(Minkowski) spacetime with the metric defined by the superfield$H_{\mu }(x,\theta )$. The term ${\mathrm{\Theta }}^{\mu }$, as defined above, isphysically a supercurrent and satisfies the conservationconditions:

$${\gamma }^{\mu }𝐃{\mathrm{\Theta }}_{\mu }=𝐃,$$(0.8)

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

$$I_{ℳ}=\kappa \int d^{4}x{T}^{\mu \nu }(x)h_{\mu \nu }(x),$$(0.9)

It is interesting to note that the gravitational actions for thesuperfield that are invariant under the generalized gaugetransformations $H_{\mu }⟼H_{\mu }+{\mathrm{\Delta }}_{\mu }$ lead tosolutions of the Einstein field equations for a homogeneous,non-zero vacuum energy density ${\rho }_V$ that correspond to eithera de Sitter space for ${\rho }_V>0$, or an anti-de Sitter space for. Such spaces can be represented in terms of thehypersurface equation

$$x_5^2\pm {\eta }_{\mu ,\nu }{x}^{\mu }{x}^{\nu }=R^2,$$(0.10)

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

$$d{s}^2={\eta }_{\mu ,\nu }{x}^{\mu }{x}^{\nu }\pm d{x}_5^2,$$(0.11)

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

The spacetime symmetry groups, or extended symmetry groupoids, as the case maybe– are different from the ‘classical’ Poincaré symmetry groupof translations and Lorentz transformations. Such spacetimesymmetry groups, in the simplest case, are therefore the$\mathrm{O}(4,1)$ group for the de Sitter space and the $\mathrm{O}(3,2)$ groupfor the anti–de Sitter space. A detailed calculation indicatesthat the transition from ordinary flat space to a bubble ofanti-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 fluctuationsmight occur to an anti–de Sitter bubble within the limitspermitted by the Heisenberg uncertainty principle.

0.2 Supersymmetry algebras and Lie (graded) superalgebras.

It is well known that continuous symmetry transformationscan be represented in terms of a Lie algebra of linearlyindependent symmetry generators $t_j$ that satisfy thecommutation relations:

$$[t_j,t_k]=\iota {\mathrm{\Sigma }}_l{C}_{jk}{t}_l,$$(0.12)

Supersymmetry is similarly expressed in terms of the symmetrygenerators $t_j$ of a graded (‘Lie’) algebra which is infact defined as a superalgebra) by satisfying relations of thegeneral form:

$$t_j{t}_k-{(-1)}^{{\eta }_j{\eta }_k}{t}_k{t}_j=\iota {\mathrm{\Sigma }}_l{C}_{jk}^l{t}_l.$$(0.13)

The generators for which ${\eta }_j=1$ are fermionic whereas thosefor which ${\eta }_j=0$ are bosonic. The coefficients $C_{jk}^l$are structure constants satisfying the following conditions:

$$C_{jk}^l=-{(-1)}^{{\eta }_j{\eta }_k}{C}_{jk}^l.$$(0.14)

If the generators ${}_j$ are quantum Hermitian operators, then thestructure constants satisfy the reality conditions $C_{jk}^{*}=-C_{jk}$ . Clearly, such a graded algebraic structure is a superalgebraand not a proper Lie algebra; thus graded Lie algebras are often called‘Lie superalgebras’.

The standard computational approach in QM utilizes the S-matrixapproach, and therefore, one needs to consider the general,graded ‘Lie algebra’ of supersymmetry generators thatcommute with the S-matrix. If one denotes the fermionic generatorsby $Q$, then $U^{-1}(\mathrm{\Lambda })QU(\mathrm{\Lambda })$ will also be of thesame type when $U(\mathrm{\Lambda })$ is the quantum operator correspondingto arbitrary, homogeneous Lorentz transformations ${\mathrm{\Lambda }}^{{\mu }_{\nu }}$ . Such a group of generators provide therefore arepresentation of the homogeneous Lorentz group of transformations$𝕃$ . The irreducible representation of the homogeneousLorentz group of transformations provides therefore aclassification of such individual generators.

0.2.1 Graded ‘Lie Algebras’/Superalgebras.

A set of quantum operators $Q_{jk}^{AB}$ form an $𝐀,𝐁$ representation of the group $𝐋$ defined abovewhich satisfy the commutation relations:

$$[𝐀,Q_{jk}^{AB}]=-[{\mathrm{\Sigma }}_j^{\prime }{J}_{j{j}^{\prime }}^A,Q_{j^{\prime }k}^{AB}],$$(0.15)

and

$$[𝐁,Q_{jk}^{AB}]=-[{\mathrm{\Sigma }}_{j^{\prime }}{J}_{k{k}^{\prime }}^A,Q_{j{k}^{\prime }}^{AB}],$$(0.16)

with the generators $𝐀$ and $𝐁$ defined by$𝐀\equiv (1/2)(𝐉\pm i𝐊)$ and$𝐁\equiv (1/2)(𝐉-i𝐊)$, with$𝐉$ and $𝐊$ being the Hermitian generators ofrotations and ‘boosts’, respectively.

In the case of the two-component Weyl-spinors $Q_{jr}$ theHaag–Lopuszanski–Sohnius (HLS) theorem applies, and thus thefermions form a supersymmetry algebra defined by theanti-commutation relations:

$[Q_{jr},Q_{ks}^{*}]$$=2{\delta }_{rs}{\sigma }_{jk}^{\mu }{P}_{\mu },$(0.17)

$[Q_{jr},Q_{ks}]$$=e_{jk}{Z}_{rs},$

where $P_{\mu }$ is the 4–momentum operator, $Z_{rs}=-Z_{sr}$are the bosonic symmetry generators, and ${\sigma }_{\mu }$ and$𝐞$ are the usual $2\times 2$ Pauli matrices.Furthermore, the fermionic generators commute with both energy andmomentum operators:

$$[P_{\mu },Q_{jr}]=[P_{\mu },Q_{jr}^{*}]=0.$$(0.18)

The bosonic symmetry generators $Z_{ks}$ and $Z_{ks}^{*}$represent the set of central charges of the supersymmetricalgebra:

$$[Z_{rs},Z_{tn}^{*}]=[Z_{rs}^{*},Q_{jt}]=[Z_{rs}^{*},Q_{jt}^{*}]=[Z_{rs}^{*},Z_{tn}^{*}]=0.$$(0.19)

From another direction, the Poincaré symmetry mechanism ofspecial relativity can be extended to new algebraic systems(Tanasă, 2006). In Moultaka et al. (2005) in view of suchextensions, consider invariant-free Lagrangians and bosonicmultiplets constituting a symmetry that interplays with (Abelian)$\mathrm{U}(1)$–gauge symmetry that may possibly be described incategorical terms, in particular, within the notion of acubical site (Grandis and Mauri, 2003).

We shall proceed to introduce in the next section generalizationsof the concepts of Lie algebras and graded Lie algebras to thecorresponding Lie algebroids that may also be regarded asC*–convolution representations of quantum gravitygroupoids and superfield (or supergravity) supersymmetries. Thisis therefore a novel approach to the proper representation of thenon-commutative geometry of quantum spacetimes–that arecurved (or ‘deformed’) by the presence of intensegravitational fields–in the framework of non-Abelian,graded Lie algebroids. Their correspondingly deformedquantum gravity groupoids (QGG) should, therefore, adequatelyrepresent supersymmetries modified by the presence of such intensegravitational fields on the Planck scale. Quantum fluctuationsthat give rise to quantum ‘foams’ at the Planck scale may be thenrepresented by quantum homomorphisms of such QGGs. If thecorresponding graded Lie algebroids are also integrable,then one can reasonably expect to recover in the limit of $\mathrm{\hslash }\to 0$ the Riemannian geometry of General Relativity andthe globally hyperbolic spacetime of Einstein’s classicalgravitation theory (GR), as a result of such an integration to thequantum gravity fundamental groupoid (QGFG). The followingsubsection will define the precise mathematical conceptsunderlying our novel quantum supergravity and extendedsupersymmetry notions.