on spacetime with values in the super Poincaré Lie algebra. Its components Ψ\Psi in the spin representation 32⊂𝔦𝔰𝔬(ℝ 1,10|32)\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)Γ ab 1b 2ρ b 1b 2=0 \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

ρ b 1b 2\rho_{b_1 b_2} are the bosonic frame field components of the gravitino field strength:

dΨ−14ω abΓ abψ=ρ b 1b 2E b 1E b 2+(⋯)ΨE, \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∈{0,1,⋯,10}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 32\mathbf{32} of Pin + ( 1 , 10 ) Pin^+(1,10) ,
Γ a 1⋯a p≔1p!∑σ∈Sym(p)sgn(σ)Γ a σ(1)⋯Γ a σ(p)\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 pp Clifford algebra basis elements in the irreducible real representation 32\mathbf{32} of Pin + ( 1 , 10 ) Pin^+(1,10) ,

here acting pointwise on the component spinors of ρ\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 Γ ab 1b 2ρ b 1b 2=0\Gamma^{a b_1 b_2} \rho_{b_1 b_2} \;=\; 0 (1) in D=11 supergravity:

(2)Γ b 1b 2ρ b 1b 2=0 \Gamma^{b_1 b_2} \, \rho_{b_1 b_2} \;=\; 0

(3)Γ b 1ρ b 1b 2=0 \Gamma^{b_1} \, \rho_{b_1 b_2} \;=\; 0

(4)Γ aa′ρ a′b=−ρ a b \Gamma^{a a'} \, \rho_{a' b} \;=\; - \rho^a{}_b

(5)Γ c 1c 2 b 1b 2ρ b 1b 2=−2ρ c 1c 2. \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:

Γ aΓ ab 1b 2ρ b 1b 2=9Γ b 1b 2ρ b 1b 2 \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

Γ caΓ ab 1b 2ρ b 1b 2=18Γ bρ cb+8Γ cb 1b 2ρ b 1b 2 \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:

Γ aa′ρ a′b =12(Γ aΓ a′ρ a′b−Γ a′Γ aρ a′b) =−12Γ a′Γ aρ a′b =12Γ aΓ a′ρ a′b−η aa′ρ a′b =−ρ a b, \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:

Γ c 1c 2aΓ ab 2b 3ρ b 2b 3 =16Γ c 1 bρ c 2b−16Γ c 2 bρ c 1b−18ρ c 1c 2+7Γ c 1c 2 b 1b 2ρ b 1b 2 =16ρ c 1c 2−16ρ c 2c 1−18ρ c 1c 2+7Γ c 1c 2 b 1b 2ρ b 1b 2 =14ρ c 1c 2+7Γ c 1c 2 b 1b 2ρ b 1b 2, \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).

gravitino condensation

References

General

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 SS-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 2/32/3? [arXiv:2303.09131]
Adrianna Kruk, Michał Lesiuk, Krzysztof A. Meissner, Hermann Nicolai: Signatures of supermassive charged gravitinos in liquid scintillator detectors [arXiv:2407.04883]

Last revised on October 27, 2024 at 10:30:35. See the history of this page for a list of all contributions to it.