# PlanetPhysics/Superfields2

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### Metric superfields

In general, superfields are physically understood as quantized gravity fields that admit a highly reducible representation of a supersymmetry algebra. The problem of specifying a supergravity theory can be then defined as a search for those representations that allow the construction of consistent local actions, perhaps considered as either quantum group, or quantum groupoid, actions. Extending quantum symmetries to include quantized gravity fields--specified as `superfields'-- is called supersymmetry in quantum gravity theories. A first approach to supersymmetry relied on a curved `superspace' (Wess and Bagger,1983) and is analogous to supersymmetric gauge theories (see, for example, sections 27.1 to 27.3 of Weinberg, 1995).

#### Metric superfield

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 Einstein's general relativistic metric $g_{\mu \nu }(x)$. The connections between these two distinct representations are as follows:
   $g_{\mu \nu }(x)=\eta _{ab}~e_{\mu }^{a}(x)e_{\gamma }^{b}(x)~,$
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:
   $e_{\mu }^{a}(x)\longmapsto \Lambda _{b}^{a}(x)e_{\mu }^{b}(x)~,$
(where $\Lambda _{b}^{a}$ is an arbitrary real matrix), and the general coordinate transformations:
   $x^{\mu }\longmapsto (x')^{\mu }(x)~.$
In a weak gravitational field the tetrad may be represented as:
   $e_{\mu }^{a}(x)=\delta _{\mu }^{a}(x)+2\kappa \Phi _{\mu }^{a}(x)~,$
where $\Phi _{\mu }^{a}(x)$ is small compared with $\delta _{\mu }^{a}(x)$ for all $x$ values, and $\kappa =\surd 8\pi G$, 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:
   $\psi _{\mu }(x)\longmapsto \psi _{\mu }(x)+\delta _{\mu }\psi (x)~,$
with $\psi (x)$ being an arbitrary Majorana field as defined by Grisaru and Pendleton (1977). The tetrad field </math>\Phi _{\mu \nu}(x)$andthegravitonfield$\psi _\mu(x)$arethenincorporatedintoaterm<math>H_{\mu }(x,\theta )$ defined as the metric superfield . The relationships between </math>\Phi _{\mu _ \nu}(x)$and$\psi _\mu(x)$,ontheonehand,andthecomponentsofthemetricsuperfield<math>H_{\mu }(x,\theta )$, on the other hand, can be derived from the transformations of the whole metric superfield:
   $H_{\mu }(x,\theta )\longmapsto H_{\mu }(x,\theta )+\Delta _{\mu }(x,\theta )~,$
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 </math>T^{\mu \nu}$representedasalinearcombinationofcomponentsofareal[[../Vectors/|vector]]superfield<math>\Theta ^{\mu }$. Such interaction terms would, therefore, have the form:
   $I_{\mathcal {M}}=2\kappa \int dx^{4}[H_{\mu }\Theta ^{\mu }]_{D}~,$
(${\mathcal {M}}$ denotes `matter') integrated over a four-dimensional (Minkowski) spacetime with the metric defined by the superfield $H_{\mu }(x,\theta )$. The term $\Theta ^{\mu }$, as defined above, is physically a supercurrent and satisfies the conservation conditions:
   $\gamma ^{\mu }\mathbf {D} \Theta _{\mu }=\mathbf {D} ~,$
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:
   $I_{\mathcal {M}}=\kappa \int d^{4}xT^{\mu \nu }(x)h_{\mu \nu }(x)~,$
It is interesting to note that the gravitational actions for the superfield that are invariant under the generalized gauge transformations $H_{\mu }\longmapsto H_{\mu }+\Delta _{\mu }$ 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 $\rho _{V}<0$. Such spaces can be represented in terms of the hypersurface equation
   $x_{5}^{2}\pm \eta _{\mu ,\nu }x^{\mu }x^{\nu }=R^{2}~,$
in a quasi-Euclidean five-dimensional space with the metric specified as:
   $ds^{2}=\eta _{\mu ,\nu }x^{\mu }x^{\nu }\pm dx_{5}^{2}~,$
with '$+$' for de Sitter space and '$-$' for anti-de Sitter space, respectively.