nLab gravitino

Idea

In quantum field theory the term gravitino refers to the superpartner of the graviton, a Rarita-Schwinger field of spin $3/2$ that appears in supergravity.

In supergravity a field history is a connection on super spacetime locally given by a super Lie algebra-valued differential form

$$(E, \Omega, \Psi) \,\colon\, T X \longrightarrow \mathfrak{iso}\big(\mathbb{R}^{1,10\vert \mathbf{32}}\big)$$

on spacetime with values in the super Poincaré Lie algebra. Its components $\Psi$ in the spin representation $\mathbf{32} \subset \mathfrak{iso}\big(\mathbb{R}^{1,10\vert \mathbf{32}}\big)$ is the gravitino field.

The name derives from the fact that the other two components are identified in gravity with the graviton field.

Examples

Gravitino in 11d Supergravity

The Rarita-Scwinger-like equation of motion for the gravitino in D=11 N=1 supergravity is (on any chart)

(1)$$\Gamma^{a \, b_1 b_2} \, \rho_{b_1 b_2} \;=\; 0$$

(due to Cremmer, Julia & Scherk 1978, p. 411, cf. Castellani, D’Auria & Fré 1991, §III.8, p. 910),

where

• $\rho_{b_1 b_2}$ are the bosonic frame field components of the gravitino field strength:

  $$\mathrm{d}\, \Psi - \!\tfrac{1}{4} \omega^{a b} \Gamma_{a b} \psi \;=\; \rho_{b_1 b_2} E^{b_1} E^{b_2} + (\cdots) \Psi E \,,$$

  So for each value of the indices $b_i \in \{0, 1, \cdots, 10\}$ this is a smooth function from the chart to the real vector space underlying the irreducible real representation $\mathbf{32}$ of $Pin^+(1,10)$,

• $\Gamma^{a_1 \cdots a_p} \,\coloneqq\, \tfrac{1}{p!} \underset{\sigma \in Sym(p)}{\sum} sgn(\sigma) \Gamma^{a_{\sigma(1)}} \cdots \Gamma^{a_{\sigma(p)}}$ is the skew-symmetrized product of $p$ Clifford algebra basis elements in the irreducible real representation $\mathbf{32}$ of $Pin^+(1,10)$,

  here acting pointwise on the component spinors of $\rho$,

• the Einstein summation convention implies summation over repeated indices.

Proposition

(implications of 11d gravitino equation)

We have the following implications of the gravitino equation $\Gamma^{a b_1 b_2} \rho_{b_1 b_2} \;=\; 0$ (1) in D=11 supergravity:

(2)$$\Gamma^{b_1 b_2} \, \rho_{b_1 b_2} \;=\; 0$$

(3)$$\Gamma^{b_1} \, \rho_{b_1 b_2} \;=\; 0$$

(4)$$\Gamma^{a a'} \, \rho_{a' b} \;=\; - \rho^a{}_b$$

(5)$$\Gamma_{\!c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \;=\; - 2\rho_{c_1 c_2} \,.$$

Proof

Equation (2) follows immediately by Clifford contraction:

$$\Gamma_{\!a} \Gamma^{a b_1 b_2} \,\rho_{b_1 b_2} \;=\; 9 \, \Gamma^{b_1 b_2} \,\rho_{b_1 b_2}$$

Equation (3) follows by the contraction

$$\begin{array}{l} \Gamma_{\!c a} \Gamma^{a b_1 b_2} \, \rho_{b_1 b_2} \;=\; 18 \, \Gamma^{b} \rho_{ c b } \;+\; 8 \, \Gamma^{c b_1 b_2} \rho_{b_1 b_2} \end{array}$$

and using that the second summand vanishes by assumption (1).

For equation (4) we compute as follows:

$$\begin{array}{l} \Gamma^{a a'}\rho_{a' b} \\ \;=\; \tfrac{1}{2} \big( \Gamma^a \Gamma^{a'} \rho_{a' b} - \Gamma^{a'} \Gamma^a \rho_{a' b} \big) \\ \;=\; - \tfrac{1}{2} \Gamma^{a'} \Gamma^a \rho_{a' b} \\ \;=\; \tfrac{1}{2} \Gamma^a \Gamma^{a'} \rho_{a' b} - \eta^{a a'} \rho_{a' b} \\ \;=\; -\rho^a{}_b \,, \end{array}$$

where in the second and fourth step we used (3).

For (5) we consider this contraction:

$$\begin{array}{l} \Gamma_{c_1 c_2 a} \Gamma^{a b_2 b_3} \rho_{b_2 b_3} \\ \;=\; 16 \Gamma_{c_1}{}^{b} \rho_{c_2 b} - 16 \Gamma_{c_2}{}^{b} \rho_{c_1 b} - 18 \rho_{c_1 c_2} + 7 \Gamma_{c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \\ \;=\; 16 \rho_{c_1 c_2} - 16 \rho_{c_2 c_1} - 18 \rho_{c_1 c_2} + 7 \Gamma_{c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \\ \;=\; 14 \rho_{c_1 c_2} + 7 \Gamma_{c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \,, \end{array}$$

where in the second step we used (4).

Related concepts

• gravitino condensation

References

General

See also

• Wikipedia, Gravitino

Classification of long-range forces

Classification of possible long-range forces, hence of scattering processes of massless fields, by classification of suitably factorizing and decaying Poincaré-invariant S-matrices depending on particle spin, leading to uniqueness statements about Maxwell/photon-, Yang-Mills/gluon-, gravity/graviton- and supergravity/gravitino-interactions:

• Steven Weinberg, Feynman Rules for Any Spin. 2. Massless Particles, Phys. Rev. 134 (1964) B882 (doi:10.1103/PhysRev.134.B882)

• Steven Weinberg, Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass, Phys. Rev. 135 (1964) B1049 (doi:10.1103/PhysRev.135.B1049)

• Steven Weinberg, Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 (doi:10.1103/PhysRev.138.B988)

• Paolo Benincasa, Freddy Cachazo, Consistency Conditions on the S-Matrix of Massless Particles (arXiv:0705.4305)

• David A. McGady, Laurentiu Rodina, Higher-spin massless S-matrices in four-dimensions, Phys. Rev. D 90, 084048 (2014) (arXiv:1311.2938, doi:10.1103/PhysRevD.90.084048)

Review:

• Claus Kiefer, section 2.1.3 of: Quantum Gravity, Oxford University Press 2007 (doi:10.1093/acprof:oso/9780199585205.001.0001, cds:1509512)

• Daniel Baumann, What long-range forces are allowed?, 2019 (pdf)

As a dark matter candidate

Discussion of the gravitino as a dark matter candidate:

• John Ellis, Keith Olive, Supersymmetric Dark Matter Candidates (arXiv:1001.3651)

A proposal for super-heavy gravitinos as dark matter, by embedding D=4 N=8 supergravity into E10-U-duality-invariant M-theory:

• Krzysztof A. Meissner, Hermann Nicolai: Standard Model Fermions and Infinite-Dimensional R-Symmetries, Phys. Rev. Lett. 121 091601 (2018) [arXiv:1804.09606, doi:10.1103/PhysRevLett.121.091601]

• Krzysztof A. Meissner, Hermann Nicolai, Planck Mass Charged Gravitino Dark Matter, Phys. Rev. D 100, 035001 (2019) (arXiv:1809.01441)

following the proposal towards the end of

• Murray Gell-Mann, introductory talk at Shelter Island II, 1983 (pdf)

  in: Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

Further discussion:

• Krzysztof A. Meissner, Hermann Nicolai, Supermassive gravitinos and giant primordial black holes (arXiv:2007.11889)

• Krzysztof A. Meissner, Hermann Nicolai, Evidence for a stable supermassive gravitino with charge $2/3$? [arXiv:2303.09131]

• Adrianna Kruk, Michał Lesiuk, Krzysztof A. Meissner, Hermann Nicolai: Signatures of supermassive charged gravitinos in liquid scintillator detectors [arXiv:2407.04883]