metric superfields

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This is a topic entry on metric superfields in quantum supergravityand 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 beingconsidered, the gravitational fields are represented in terms oftetrads, $e_{\mu }^a(x),$ rather than in terms of the generalrelativistic metric $g_{\mu \nu }(x)$. The connections betweenthese two distinct representations are as follows:

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

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:

$$e_{\mu }^a(x)⟼{\mathrm{\Lambda }}_b^a(x)e_{\mu }^b(x),$$(1.2)

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

$$x^{\mu }⟼{(x^{\prime })}^{\mu }(x).$$(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 ${\delta }_{\mu }^a(x)$ 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:

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

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:

$$H_{\mu }(x,\theta )⟼H_{\mu }(x,\theta )+{\mathrm{\Delta }}_{\mu }(x,\theta ),$$(1.6)

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:

$$I_{ℳ}=2\kappa \int 𝑑x^4{[H_{\mu }{\mathrm{\Theta }}^{\mu }]}_D,$$(1.7)

($ℳ$ 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:

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

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:

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

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

$$x_5^2\pm {\eta }_{\mu ,\nu }{x}^{\mu }{x}^{\nu }=R^2,$$(1.10)

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

$$d{s}^2={\eta }_{\mu ,\nu }{x}^{\mu }{x}^{\nu }\pm d{x}_5^2,$$(1.11)

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

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