In supersymmetry, pure 4D N = 1 {\displaystyle {\mathcal {N}}=1} supergravity describes the simplest four-dimensional supergravity, with a single supercharge and a supermultiplet containing a graviton and gravitino. The action consists of the Einstein–Hilbert action and the Rarita–Schwinger action. The theory was first formulated by Daniel Z. Freedman, Peter van Nieuwenhuizen, and Sergio Ferrara, and independently by Stanley Deser and Bruno Zumino in 1976. The only consistent extension to spacetimes with a cosmological constant is to anti-de Sitter space, first formulated by Paul Townsend in 1977. When additional matter supermultiplets are included in this theory, the result is known as matter-coupled 4D N = 1 {\displaystyle {\mathcal {N}}=1} supergravity.
Flat spacetime To describe the coupling between gravity and particles of arbitrary spin, it is useful to use the vielbein formalism of general relativity. This replaces the metric by a set of vector fields e a = e a μ ∂ μ {\displaystyle e_{a}=e_{a}^{\mu }\partial _{\mu }} indexed by flat indices a {\displaystyle a} such that
g μ ν = e μ a e ν b η a b . {\displaystyle g_{\mu \nu }=e_{\mu }^{a}e_{\nu }^{b}\eta _{ab}.}
In a sense the vielbeins are the square root of the metric. This introduces a new local Lorentz symmetry on the vielbeins e μ a → e μ b Λ a
b ( x ) {\displaystyle e_{\mu }^{a}\rightarrow e_{\mu }^{b}\Lambda ^{a}{}_{b}(x)} , together with the usual diffeomorphism invariance associated with the spacetime indices μ {\displaystyle \mu } . This has an associated connection known as the spin connection ω μ a b {\displaystyle \omega _{\mu }^{ab}} defined through ∇ μ e a = ω μ
b
a e b {\displaystyle \nabla _{\mu }e_{a}=\omega _{\mu }{}^{b}{}_{a}e_{b}} , it being a generalization of the Christoffel connection to arbitrary spin fields. For example, for spinors the covariant derivative is given by
D μ = ∂ μ + 1 4 ω μ a b γ a b , {\displaystyle D_{\mu }=\partial _{\mu }+{\frac {1}{4}}\omega _{\mu }^{ab}\gamma _{ab},}
where γ a {\displaystyle \gamma _{a}} are gamma matrices satisfing the Dirac algebra, with γ a b = γ [ a γ b ] {\displaystyle \gamma _{ab}=\gamma _{[a}\gamma _{b]}} . These are often contracted with vielbeins to construct γ μ = e μ a γ a {\displaystyle \gamma _{\mu }=e_{\mu }^{a}\gamma _{a}} which are in general position-dependent fields rather than constants. The spin connection has an explicit expression in terms of the vielbein and an additional torsion tensor which can arise when there is matter present in the theory. A vanishing torsion is equivalent to the Levi-Civita connection. The pure N = 1 {\displaystyle {\mathcal {N}}=1} supergravity action in four dimensions is the combination of the Einstein–Hilbert action and the Rarita–Schwinger action
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