# Pure 4D N = 1 supergravity
**Pure 4D N = 1 supergravity** is the locally supersymmetric extension of general relativity in four spacetime dimensions that contains, besides the gravitational field, a single spin-3/2 gravitino and no matter multiplets. It was the first supergravity theory constructed, in the spring of 1976, by [Daniel Z. Freedman](https://www.edgepedia.org/daniel-z-freedman), [Peter van Nieuwenhuizen](https://www.edgepedia.org/peter-van-nieuwenhuizen), and [Sergio Ferrara](https://www.edgepedia.org/sergio-ferrara), and the result was recovered shortly afterwards by Stanley Deser and Bruno Zumino in a simpler form based on the first-order (Palatini) formalism of general relativity[1](https://cern-courier.web.cern.ch/a/the-many-lives-of-supergravity/). The number N counts the independent local supersymmetries; N = 1 means one gravitino, the gauge field of local supersymmetry, just as the photon is the gauge field of an internal circle rotation[1](https://cern-courier.web.cern.ch/a/the-many-lives-of-supergravity/).
| Key fact | Detail |
|---|---|
| Field content | Vierbein \( e_\mu^{\,a} \) and one gravitino \( \psi_\mu \); on-shell the multiplet carries 2 + 2 degrees of freedom, a massless spin-2 graviton and its spin-3/2 partner[2](https://arxiv.org/abs/hep-th/0204035) |
| Discovery | Built in spring 1976 by Freedman, van Nieuwenhuizen, and Ferrara; recovered by Deser and Zumino at CERN the same year[1](https://cern-courier.web.cern.ch/a/the-many-lives-of-supergravity/) • [3](https://cds.cern.ch/record/438874/files/197605095.pdf) |
| Off-shell closure | Auxiliary fields are needed because the gauge algebra does not close off-shell with only the vierbein and gravitino; the minimal sets carry six auxiliary bosonic degrees of freedom, matching the gravitino's 12 off-shell against the vielbein's 6 plus 6 auxiliary[2](https://arxiv.org/abs/hep-th/0204035) • [4](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036) |
| Minimal formulations | Old minimal (Stelle–West, Ferrara–van Nieuwenhuizen, 1978): complex scalar plus non-gauge vector; new minimal (1981): gauge vector plus antisymmetric tensor; non-minimal set of 20 + 20 fields due to Breitenlohner (1977)[4](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036) • [5](https://www.sciencedirect.com/science/article/abs/pii/0370269381907784) • [6](https://cds.cern.ch/record/135456/files/198202139.pdf) |
| Ultraviolet behavior | The four-particle S-matrix is one-loop finite; pure supergravity is renormalizable at one and two loops but fails at three; the one- and two-loop finiteness statement applies to pure supergravity; supergravity coupled to matter can have one-loop divergences[7](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.37.1662) • [8](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.38.527) • [9](https://link.springer.com/article/10.12942/lrr-2002-5) • [16](https://authors.library.caltech.edu/records/rtett-rwj78) |
| Cosmological constant | With only graviton and gravitino the theory is anti-de Sitter (negative cosmological constant); a de Sitter version with positive constant was derived in 2015 using a nilpotent goldstino multiplet[10](https://ds-supergravity.org/index.html) • [11](https://arxiv.org/abs/1507.08264) |
## The on-shell theory
The theory raises supersymmetry to a local gauge symmetry and combines it with general coordinate transformations, so it necessarily extends general relativity[2](https://arxiv.org/abs/hep-th/0204035). Because spinor fields are present, it is formulated in the vielbein (Cartan) form of general relativity rather than with a metric, and it is invariant under local Lorentz transformations[2](https://arxiv.org/abs/hep-th/0204035). The two fields are the vierbein \( e_\mu^{\,a} \), which carries \( D(D-3)/2 = 2 \) on-shell degrees of freedom in four dimensions, and the gravitino \( \psi_\mu \), a Rarita–Schwinger spinor-vector field with 2 on-shell degrees of freedom[2](https://arxiv.org/abs/hep-th/0204035).
Deser and Zumino showed that the simplest candidate action, the sum of the Einstein action and the action for a massless, minimally coupled Rarita–Schwinger Majorana field, is consistent: it is free of the usual higher-spin inconsistencies and invariant under local supersymmetry transformations[3](https://cds.cern.ch/record/438874/files/197605095.pdf). Their combined Lagrangian is
\[ \mathcal{L} = -\tfrac{1}{2} e R - \tfrac{i}{2} \epsilon^{\lambda\mu\nu\rho} \bar{\psi}_\lambda \gamma_5 \gamma_\mu D_\nu \psi_\rho, \]
with \( e = \det e_\mu{}^a \), written in the first-order formalism in which vierbeins and connection coefficients are treated independently; minimal coupling in this sense implies torsion[3](https://cds.cern.ch/record/438874/files/197605095.pdf). The commutator of two local supersymmetry transformations contains a general coordinate transformation[3](https://cds.cern.ch/record/438874/files/197605095.pdf).
The gauge algebra of this basic formulation is open: the commutator of two supersymmetry transformations closes only on the equations of motion, and the composite parameters on the right-hand side are field dependent, unlike in [Yang–Mills theory](https://www.edgepedia.org/yang-mills-theory) or standard general relativity[2](https://arxiv.org/abs/hep-th/0204035).
## Off-shell formulations and auxiliary fields
**Auxiliary fields** restore off-shell closure. They carry no physical degrees of freedom, are eliminated algebraically by their equations of motion, and their elimination reproduces the on-shell formulation[2](https://arxiv.org/abs/hep-th/0204035). Counting fixes how many are needed: the vielbein has 6 off-shell degrees of freedom and the gravitino 12, so six extra bosonic degrees of freedom are required for the off-shell matching \( n_B = n_F \)[4](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036) • [12](https://cds.cern.ch/record/2049333/files/arXiv%3A1509.01500.pdf).
Three main formulations exist. The old minimal set, due to Stelle and West and to Ferrara and van Nieuwenhuizen in 1978, uses a complex scalar (2 degrees of freedom) and a real vector field not associated with a gauge symmetry (4 degrees of freedom)[4](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036) • [6](https://cds.cern.ch/record/135456/files/198202139.pdf). The new minimal formulation, presented in 1981, uses an axial vector \( A_\mu \) and an antisymmetric tensor \( a_{\mu\nu} \) as auxiliary fields, again providing six off-shell degrees of freedom and thus a minimal set[5](https://www.sciencedirect.com/science/article/abs/pii/0370269381907784); in modern terms these are a gauge 2-form carrying 3 degrees of freedom and a gauge vector carrying 3[4](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036). The non-minimal set of Breitenlohner (1977) contains 20 + 20 fields[6](https://cds.cern.ch/record/135456/files/198202139.pdf). An intermediate 16 + 16 set was also studied by de Wit and van Nieuwenhuizen[6](https://cds.cern.ch/record/135456/files/198202139.pdf).
The two minimal formulations are inequivalent as off-shell theories, and non-minimal off-shell formulations exist as well[13](https://www.ilorentz.org/IL-publications/dissertations/sources/Bergshoeff_1983.pdf). Off-shell formulations matter in practice because they make the supersymmetry transformation laws model-independent and closed without using the equations of motion, which facilitates quantization and is crucial for the spontaneous breaking of supersymmetry[4](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036). They also underpin the coupling to chiral and gauge matter multiplets[2](https://arxiv.org/abs/hep-th/0204035).
## Quantum behavior
**Ultraviolet divergences** are softened but not eliminated by supersymmetry. An explicit calculation showed that the four-particle S-matrix of pure supergravity is one-loop finite[7](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.37.1662). A counterterm analysis to all loop orders found that pure supergravity is renormalizable at one and two loops but fails at three loops, while supergravity coupled to matter is one-loop nonrenormalizable[8](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.38.527). Supersymmetry [Ward identities](https://www.edgepedia.org/ward-identities) underpin these restrictions on possible counterterms[9](https://link.springer.com/article/10.12942/lrr-2002-5).
The gravity coupling \( k = \sqrt{32\pi G_N} \) carries dimensions of length, so divergences proportional to terms with extra derivatives cannot be absorbed into the original Lagrangian; this is why supergravity, like general relativity, is treated as an effective field theory below the Planck scale of \( 10^{19} \) GeV[9](https://link.springer.com/article/10.12942/lrr-2002-5). In 2013 a four-loop four-point calculation identified the first known ultraviolet divergence in a pure ungauged supergravity theory in four dimensions[14](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.231302).
## Cosmological constant and de Sitter supergravity
With no fields beyond the graviton and gravitino, D = 4, N = 1 supergravity is known to be anti-de Sitter, with a negative cosmological constant[10](https://ds-supergravity.org/index.html). A positive cosmological constant resisted a straightforward supersymmetric treatment for decades. In 2015, Eric Bergshoeff, Daniel Z. Freedman, Renata Kallosh, and Antoine Van Proeyen derived an explicit de Sitter supergravity action using superconformal methods: the supergravity multiplet couples to a nilpotent [goldstino](https://www.edgepedia.org/goldstino) multiplet, and in the unitary gauge where the goldstino vanishes one recovers pure supergravity with a positive cosmological constant[11](https://arxiv.org/abs/1507.08264). The classical equations of motion, with all fermions vanishing, then have de Sitter space as a maximally symmetric solution[11](https://arxiv.org/abs/1507.08264).
## Place among supergravity theories
Pure N = 1 supergravity is the smallest member of a family bounded by representation theory. Standard supergravity theories carry at most 32 real supercharges, because dimensional reduction to four dimensions must not produce fields of spin 5/2 or higher, and this limits the spacetime dimension to \( D \leq 11 \)[2](https://arxiv.org/abs/hep-th/0204035). Four-dimensional theories admit at most eight gravitinos, and Gell-Mann noted that the maximal pure 4D supergravity, with every force tied to the ordinary graviton, would contain this number of gravitinos[1](https://cern-courier.web.cern.ch/a/the-many-lives-of-supergravity/). Pure N = 1 supergravity also arises as the low-energy limit of string compactifications, and its coupling to matter multiplets is the framework of supergravity unification models[2](https://arxiv.org/abs/hep-th/0204035) • [15](https://cds.cern.ch/record/2750005).
## References
1. [The many lives of supergravity, CERN Courier (2016)](https://cern-courier.web.cern.ch/a/the-many-lives-of-supergravity/)
2. [Friedemann Brandt, Lectures on supergravity, Fortschr. Phys. 50 (2002) 1126](https://arxiv.org/abs/hep-th/0204035)
3. [S. Deser and B. Zumino, Consistent supergravity, Phys. Lett. B 62 (1976) 335](https://cds.cern.ch/record/438874/files/197605095.pdf)
4. [Lucrezia Ravera, On the Role of Torsion and Higher Forms in Off-Shell Supergravity, Fortschr. Phys. (2023)](https://onlinelibrary.wiley.com/doi/full/10.1002/prop.202300036)
5. [An alternative minimal off-shell version of N = 1 supergravity, Phys. Lett. B (1981)](https://www.sciencedirect.com/science/article/abs/pii/0370269381907784)
6. [M. F. Sohnius, The new minimal formulation of N = 1 supergravity and its tensor calculus, CERN preprint TH.3205 (1982)](https://cds.cern.ch/record/135456/files/198202139.pdf)
7. [M. T. Grisaru, P. van Nieuwenhuizen, and J. A. M. Vermaseren, One-Loop Renormalizability of Pure Supergravity, Phys. Rev. Lett. 37 (1976) 1662](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.37.1662)
8. [S. Deser, J. H. Kay, and K. S. Stelle, Renormalizability Properties of Supergravity, Phys. Rev. Lett. 38 (1977) 527](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.38.527)
9. [John F. Donoghue, Perturbative Quantum Gravity and its Relation to Gauge Theory, Living Rev. Relativity 5 (2002)](https://link.springer.com/article/10.12942/lrr-2002-5)
10. [de Sitter supergravity project page](https://ds-supergravity.org/index.html)
11. [E. A. Bergshoeff, D. Z. Freedman, R. Kallosh, and A. Van Proeyen, Pure de Sitter Supergravity, Phys. Rev. D 92 (2015) 085040](https://arxiv.org/abs/1507.08264)
12. [D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara-related off-shell degree-of-freedom counting, CERN-PH-TH/2015-214](https://cds.cern.ch/record/2049333/files/arXiv%3A1509.01500.pdf)
13. [Eric Bergshoeff, doctoral dissertation (1983)](https://www.ilorentz.org/IL-publications/dissertations/sources/Bergshoeff_1983.pdf)
14. [Z. Bern et al., Ultraviolet Properties of Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.231302)
15. [Supersymmetry unification, naturalness, and discovery prospects at HL-LHC and HE-LHC, CERN Document Server](https://cds.cern.ch/record/2750005)
16. [Renormalizability Properties of Supergravity](https://authors.library.caltech.edu/records/rtett-rwj78)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Particles and nuclei › Particle physics › Beyond-Standard-Model particle hypotheses › Heavy and weak-scale BSM particles › Gravitino*
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