metric superfields



This is a topic entry on metric superfields in quantum supergravity and the mathematical cncepts related to spinor and tensor fields.

1 Metric superfields: spinor and tensor fields

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 the general relativistic metric gμν(x). The connectionsMathworldPlanetmath between these two distinct representations are as follows:

gμν(x)=ηabeμa(x)eγb(x), (1.1)

with the general coordinatesMathworldPlanetmathPlanetmath 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 invariantMathworldPlanetmath to two distinct types of symmetryMathworldPlanetmathPlanetmathPlanetmath transformationsMathworldPlanetmath–the local Lorentz transformations:

eμa(x)Λba(x)eμb(x), (1.2)

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

xμ(x)μ(x). (1.3)

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

eμa(x)=δμa(x)+2κΦμa(x), (1.4)

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), (1.5)

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), (1.6)

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, (1.7)

( 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, (1.8)

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), (1.9)

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, (1.10)

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

ds2=ημ,νxμxν±dx52, (1.11)

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

Note The presentationMathworldPlanetmathPlanetmath above follows the exposition by S. Weinberg in his book on “Quantum Field Theory” (2000), vol. 3, Cambridge University Press (UK), in terms of both concepts and mathematical notations.

Title metric superfields
Canonical name MetricSuperfields
Date of creation 2013-03-22 18:19:08
Last modified on 2013-03-22 18:19:08
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 9
Author bci1 (20947)
Entry type Topic
Classification msc 83E50
Classification msc 83C45
Synonym supergravity fields
Related topic SuperfieldsSuperspace
Related topic SpinNetworksAndSpinFoams
Defines supergravity field