Introduction to supergravity 2015: Exercise 1.
Institute for Theoretical Physics, Masaryk University,
611 37 Brno, Czech Republic
Here we treat the supergravity multiplet as a theory invariant under the global supersymmetry,
therefore the supersymmetric parameter is global
ξα = fermionic global parameter (no space-time dependence). (1)
This is possible in the linearized level. The supersymmetry transformations of the linearized free
supergravity multiplet are
δξψmα =
1
2
∂nhkm σk
α ˙ρ¯σn ˙ρρ
− σn
α ˙ρ¯σk ˙ρρ
ξρ, (2)
δξhmk = −
i
2
¯ψm ˙α¯σ ˙αα
k ξα −
i
2
¯ψk ˙α¯σ ˙αα
m ξα
+
i
2
¯ξ˙α¯σ ˙αα
k ψmα +
i
2
¯ξ˙α¯σ ˙αα
m ψkα. (3)
The conventions can be found in the book of Wess and Bagger [1].
1. Find δξ
¯ψm ˙α. This can be found by the definition δξ
¯ψm ˙α = (δξψmα)∗
.
2. Calculate [δξ, δη] hmn. This will have the form
[δξ, δη] hmn = Br
∂rhmn + ∂mΓn + ∂nΓm. (4)
Find Br and Γn and explain what is their physical meaning.
3. Calculate [δξ, δη] ψmα. This will have the form
[δξ, δη] ψmα = Cr
∂rψmα + ∂mGα + ˜rmα, (5)
where ˜rmα will vanish on-shell. Find Cr, Gα and ˜rmα and explain what is their physical
meaning.
4. Off-shell the graviton (hmn) has 6 degrees of freedom while the gravitino (ψmα) has 12 offshell
degreed of freedom. How does this relate to the fact that we need to use the equations
of motions to close the supersymmetry algebra?
References
[1] J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton, USA: Univ. Pr. (1992)
259 p
1