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

 $g_{\mu\nu}(x)=\eta_{ab}~{}e^{a}_{\mu}(x)e^{b}_{\gamma}(x)~{},$ (1.1)
 $e^{a}_{\mu}(x)\longmapsto\Lambda^{a}_{b}(x)e^{b}_{\mu}(x)~{},$ (1.2)

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

 $x^{\mu}\longmapsto(x^{\prime})^{\mu}(x)~{}.$ (1.3)

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

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

where $\Phi^{a}_{\mu}(x)$ is small compared with $\delta^{a}_{\mu}(x)$ 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:

 $\psi_{\mu}(x)\longmapsto\psi_{\mu}(x)+\delta_{\mu}\psi(x)~{},$ (1.5)

with $\psi(x)$ being an arbitrary Majorana field as defined by Grisaru and Pendleton (1977). The tetrad field $\Phi_{\mu\nu}(x)$ 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 $\Phi_{\mu_{\nu}}(x)$ 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:

 $H_{\mu}(x,\theta)\longmapsto H_{\mu}(x,\theta)+\Delta_{\mu}(x,\theta)~{},$ (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 $\Theta^{\mu}$. Such interaction terms would, therefore, have the form:

 $I_{\mathcal{M}}=2\kappa\int dx^{4}[H_{\mu}\Theta^{\mu}]_{D}~{},$ (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 $\Theta^{\mu}$, as defined above, is physically a supercurrent and satisfies the conservation conditions:

 $\gamma^{\mu}\mathbf{D}\Theta_{\mu}=\mathbf{D}~{},$ (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:

 $I_{\mathcal{M}}=\kappa\int d^{4}xT^{\mu\nu}(x)h_{\mu\nu}(x)~{},$ (1.9)

It is interesting to note that the gravitational actions for the superfield that are invariant under the generalized gauge transformations $H_{\mu}\longmapsto H_{\mu}+\Delta_{\mu}$ 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

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

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

 $ds^{2}=\eta_{\mu,\nu}x^{\mu}x^{\nu}\pm dx^{2}_{5}~{},$ (1.11)

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

Title metric superfields MetricSuperfields 2013-03-22 18:19:08 2013-03-22 18:19:08 bci1 (20947) bci1 (20947) 9 bci1 (20947) Topic msc 83E50 msc 83C45 supergravity fields SuperfieldsSuperspace SpinNetworksAndSpinFoams supergravity field