JHEP07(2018)072
components T
±
ab
, and both are real and independent.
7
The other independent bosonic fields
are the dilatation gauge field b
µ
, the SO(1, 1)
R
gauge field A
µ
, and the SU(2)
R
gauge field
V
µ
i
j
. The fermionic fields are the gravitini ψ
µ
i
and an auxiliary fermion, the dilatino, χ
i
.
All fermions in the theory are symplectic Majorana spinors in Euclidean signature, and
can be further decomposed into their chiral left- and right-handed components using the
Weyl projectors (1 ±γ
5
)/2. Note that these chiral components are not related by complex
conjugation in Euclidean signature.
The various transformation rules, as well as the explicit expressions for the composite
fields and the weights of the various fields under dilatations and (chiral) SO(1, 1)
R
trans-
formations, can be found in [50]. For the purpose of our analysis, we will need the trans-
formations for the fermionic fields of the Weyl multiplet under Q- and S-supersymmetry,
δψ
µ
i
= 2D
µ
i
+
1
16
iT
ab
γ
ab
γ
µ
i
−iγ
µ
η
i
, (2.2)
δχ
i
=
1
24
iγ
ab
/DT
ab
i
+
1
6
R(V)
ab
i
j
γ
ab
j
−
1
3
R(A)
ab
γ
ab
γ
5
i
+D
i
+
1
24
T
ab
γ
ab
η
i
, (2.3)
where
i
and η
i
are the parameters of Q- and S-supersymmetry, respectively, and R(V)
ab
i
j
,
R(A)
ab
are the R-symmetry curvatures. The derivative D
µ
is covariant with respect to all
bosonic gauge transformations except special conformal boosts, while D
µ
is fully superco-
variant. Note that in a bosonic background these two derivatives coincide, except when
acting on b
µ
which is the only independent field that transforms under special conformal
boosts. For later reference, we note
D
µ
i
=
∂
µ
−
1
4
ω
µ
ab
γ
ab
+
1
2
b
µ
+
1
2
A
µ
γ
5
i
+
1
2
V
µ
i
j
j
, (2.4)
where ω
µ
ab
becomes the spin-connection in the gauge-fixed Poincar´e theory.
2.2 Matter multiplets
In addition to the Weyl multiplet, we also consider a number of matter multiplets. The Eu-
clidean vector multiplets involve two real scalar fields
8
X
+
and X
−
, a symplectic Majorana
spinor Ω
i
, the gauge field W
µ
and an auxiliary SU(2)
R
triplet Y
ij
,
V
I
= (X
I
+
, X
I
−
, W
I
µ
, Ω
i I
, Y
ij I
) , (2.5)
where the index I runs from 0 to n
V
(including the compensating multiplet needed to
gauge-fix the conformal theory to Poincar´e supergravity). The auxiliary field is subject to
the (pseudo-)reality condition Y
I
ij
:= (Y
ij I
)
∗
= ε
ik
ε
jl
Y
kl I
. The Euclidean transformation
rules under Q- and S-supersymmetry are
δX
I
±
= ±i ¯
i±
Ω
i I
±
,
δW
I
µ
= ¯
i+
γ
µ
Ω
i I
−
− 2i X
I
−
ψ
µ
i
+
− ¯
i−
γ
µ
Ω
i I
+
− 2i X
I
+
ψ
µ
i
−
,
δΩ
i I
±
= −2i /DX
I
±
i
∓
−
1
2
ˆ
F (W )
∓I
ab
−
1
4
X
I
∓
T
∓
ab
γ
ab
i
±
− ε
kj
Y
ik I
j
±
+ 2 X
I
±
η
i
±
,
δY
ij I
= 2 ε
k(i
¯
k+
/DΩ
j) I
−
− 2 ε
k(i
¯
k−
/DΩ
j) I
+
, (2.6)
7
We use the Euclidean conventions of [50], summarized in appendix A.1.
8
As opposed to a single complex scalar in Lorentzian signature.
– 6 –
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