JHEP11(2019)115
The free part of the Lagrangian L
free
reads:
L
free
=
1
2
h
µν
G
µν
−
1
2
¯
ψ
µα(n), I
R
µα(n), I
, (2.18)
where G
µν
and R
µα(n), I
are “the left hand sides” of the free EoMs:
G
µν
≡ h
µν
− 2∂
(µ
∂
ρ
h
ν)ρ
+ ∂
µ
∂
ν
h
′
− η
µν
h
′
− ∂
ρ
∂
σ
h
ρσ
,
R
µα(n), I
≡ γ
µνρ
∂
ν
ψ
ρ
α(n), I
.
(2.19)
The cubic Lagrangian L
cubic
consists of two different kind of vertices: hhh-type gravita-
tional self coupling and hψψ-type cross coupling:
L
cubic
= L
hhh
+ L
hψψ
. (2.20)
The graviton cubic self coupling is well known; it is given by (see, for example, [
14]):
L
hhh
= κ(h
µρ
h
νσ
∂
µ
∂
ν
h
ρσ
− h
µν
h
ρσ
∂
µ
∂
ν
h
ρσ
+ h
µν
∂
ρ
h
µσ
∂
ρ
h
σ
ν
−
1
2
h
µν
∂
µ
h
ρσ
∂
ν
h
ρσ
−h
µν
∂
ρ
h
µν
∂
ρ
h
′
−
1
2
h
µρ
h
ν
ρ
∂
µ
∂
ν
h
′
−
1
4
h
′
∂
µ
h
νρ
∂
µ
h
νρ
+ h
µν
∂
ρ
h
ρµ
∂
ν
h
′
−h
µν
∂
ρ
h
ρ
µ
∂
σ
h
σ
ν
+
1
2
h
µν
h
′
∂
µ
∂
ν
h
′
+
1
2
h
′
∂
µ
h
µρ
∂
ν
h
ν
ρ
+
1
4
h
′
∂
µ
h
′
∂
µ
h
′
). (2.21)
The cubic cross couplings, on the other hand, derive from the fermion-bilinear term in (
2.7).
As we see in appendix
A, the 1-curl of the fermion, ∂
[µ
ψ
ν]α(n), I
, itself is proportional to the
free EoMs. Then the couplings arising from the first term in the covariant 1-curl (
2.13) are
trivial in that they can be field redefined away. The nontrivial cubic couplings therefore
come only from the spin-connection terms in the covariant 1-curl (
2.13). They amount to:
L
hψψ
= −
1
4
κ
¯
ψ
µα(n), I
γ
µνρ
nη
σλ|αβ
+
1
4
γ
σλ
η
αβ
ψ
νβ
α(n−1), I
h
σλk ρ
, (2.22)
where h
σλk ρ
≡ ∂
σ
h
λρ
− ∂
λ
h
σρ
is the graviton 1-curl, and η
σλ|αβ
≡
1
2
η
σα
η
λβ
− η
σβ
η
λα
.
On account of the identity (
C.1) the tensor structure inside the parentheses in eq. (2.22)
can then be simplified, thanks to the γ-trace condition (
2.12). The final result is:
L
hψψ
= −
1
8
n +
1
2
κ
¯
ψ
µα(n), I
γ
µνρ
γ
σλ
ψ
ν
α(n), I
h
σλk ρ
. (2.23)
The higher-order couplings L
higher-order
in the action (
2.17) do not stop at any finite
order, but they are unique and can also be worked out in principle. This article, however,
does not require any explicit knowledge of these terms. It is important to note that given
the cubic couplings, there exists at least one fully nonlinear consistent theory, namely the
generalized hypergravity that we are considering in this section.
Finally, starting from eq. (
2.6) we would like to write down the generalized hypersym-
metry transformations of the metric-like fluctuations defined in eqs. (
2.9) and (2.11). It is
follows that the graviton transforms as:
δh
µν
=
1
2
n +
1
2
κ ¯ǫ
α(n), I
γ
(µ
ψ
ν)
α(n), I
+ O(κ
2
), (2.24)
– 6 –
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