# metric superfields

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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*, ${e}_{\mu}^{a}(x),$ rather than in terms of the general
relativistic metric ${g}_{\mu \nu}(x)$. The connections^{} between
these two distinct representations are as follows:

$${g}_{\mu \nu}(x)={\eta}_{ab}{e}_{\mu}^{a}(x){e}_{\gamma}^{b}(x),$$ | (1.1) |

with the general coordinates^{} being indexed by $\mu ,\nu ,$ etc.,
whereas local coordinates that are being defined in a locally
inertial 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 of
symmetry^{} transformations^{}–the local Lorentz transformations:

$${e}_{\mu}^{a}(x)\u27fc{\mathrm{\Lambda}}_{b}^{a}(x){e}_{\mu}^{b}(x),$$ | (1.2) |

(where ${\mathrm{\Lambda}}_{b}^{a}$ is an arbitrary real matrix), and the general coordinate transformations:

$${x}^{\mu}\u27fc{({x}^{\prime})}^{\mu}(x).$$ | (1.3) |

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

$${e}_{\mu}^{a}(x)={\delta}_{\mu}^{a}(x)+2\kappa {\mathrm{\Phi}}_{\mu}^{a}(x),$$ | (1.4) |

where ${\mathrm{\Phi}}_{\mu}^{a}(x)$ is small compared with ${\delta}_{\mu}^{a}(x)$ for
all $x$ values, and $\kappa =\sqrt{8\pi G}$, 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 $\pm $ 3/2. Such a self-charge-conjugate massless
particle as the gravitiono with helicities $\pm $ 3/2 can only have
*low-energy* interactions if it is represented by a Majorana
field ${\psi}_{\mu}(x)$ which is invariant under the gauge
transformations:

$${\psi}_{\mu}(x)\u27fc{\psi}_{\mu}(x)+{\delta}_{\mu}\psi (x),$$ | (1.5) |

with $\psi (x)$ being an arbitrary Majorana field as defined by
Grisaru and Pendleton (1977). The tetrad field ${\mathrm{\Phi}}_{\mu \nu}(x)$ and the graviton field ${\psi}_{\mu}(x)$ are then
incorporated into a term ${H}_{\mu}(x,\theta )$ defined as the
*metric superfield*. The relationships between ${\mathrm{\Phi}}_{{\mu}_{\nu}}(x)$ and ${\psi}_{\mu}(x)$, on the one hand, and the components
of the metric superfield ${H}_{\mu}(x,\theta )$, on the other hand,
can be derived from the transformations of the whole metric
superfield:

$${H}_{\mu}(x,\theta )\u27fc{H}_{\mu}(x,\theta )+{\mathrm{\Delta}}_{\mu}(x,\theta ),$$ | (1.6) |

by making the simplifying– and physically realistic– assumption^{}
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 ${H}_{\mu}(x)$ with matter would be then
described 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 real
vector superfield ${\mathrm{\Theta}}^{\mu}$. Such interaction terms would,
therefore, have the form:

$${I}_{\mathcal{M}}=2\kappa \int \mathit{d}{x}^{4}{[{H}_{\mu}{\mathrm{\Theta}}^{\mu}]}_{D},$$ | (1.7) |

($\mathcal{M}$ 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, is
physically a *supercurrent* and satisfies the conservation
conditions:

$${\gamma}^{\mu}\mathbf{D}{\mathrm{\Theta}}_{\mu}=\mathbf{D},$$ | (1.8) |

where $\mathbf{D}$ 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:

$${I}_{\mathcal{M}}=\kappa \int {d}^{4}x{T}^{\mu \nu}(x){h}_{\mu \nu}(x),$$ | (1.9) |

It is interesting to note that the gravitational actions for the
superfield that are invariant under the generalized gauge
transformations ${H}_{\mu}\u27fc{H}_{\mu}+{\mathrm{\Delta}}_{\mu}$ lead to
solutions of the Einstein field equations for a homogeneous^{},
non-zero vacuum energy density ${\rho}_{V}$ that correspond to either
a de Sitter space for ${\rho}_{V}>0$, or an anti-de Sitter space for
$$. Such spaces can be represented in terms of the
hypersurface equation

$${x}_{5}^{2}\pm {\eta}_{\mu ,\nu}{x}^{\mu}{x}^{\nu}={R}^{2},$$ | (1.10) |

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

$$d{s}^{2}={\eta}_{\mu ,\nu}{x}^{\mu}{x}^{\nu}\pm d{x}_{5}^{2},$$ | (1.11) |

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

Note
The presentation^{} 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 |