2 6d (2, 0) supergravity and reduction to 5d
In the section we review six-dimensional supergravity [1], and describe the reduction from
six dimensions to five. Specifically, the steps we take are the following.
• Describe the field content of off-shell (2, 0) supergravity.
• Take all fields to be independent of the fifth spatial dimension, and reduce all asso-
ciated representations from so(1, 5) to so(1, 4).
• Fix a gauge for the superconformal generators. Our choice is dictated by convenience
for five-dimensional calculations.
• Identify the five-dimensional supersymmetries as six-dimensional supersymmetries
combined with suitable local superconformal transformations to preserve our gauge
fixing conditions.
We adopt the convention that a six-dimensional index will be underlined to distinguish it
from its five-dimensional descendants. Further conventions for spinors etc. may be found
in the appendix.
Throughout our analysis we make use of the fact that our aim is to describe the cou-
pling of quantum field theories to supergravity backgrounds. Thus, in all supersymmetry
variations and constraint equations we drop all terms proportional to fermionic fields in
the supergravity multiplet.
2.1 (2, 0) supergravity
We begin with a brief discussion of the fields and structure of six-dimensional (2, 0) super-
gravity following [1]. The off-shell formulation that we utilize may be viewed as a gauge
theory for the 6d (2,0) superconformal group. The generators of this algebra and the
associated gauge fields are given in table 1.
Although the structure of the supergravity multiplet can be understood from the su-
perconformal group, an important feature is that not all of the fields appearing in table 1
are independent. There are constraints which relate ω
ab
µ
, f
a
µ
, and φ
αi
µ
to other fields in the
multiplet. We have need only of the following
ω
ab
µ
= 2e
ν[a
∂
[µ
e
b]
ν]
− e
ρ[a
e
b]σ
e
c
µ
∂
ρ
e
σc
+ 2e
[a
µ
b
b]
(2.1)
f
a
a
=
1
20
R,
where, in the above, R is the scalar curvature of the connection ω
ab
µ
. In the special case
where the dilation gauge field b
µ
vanishes, ω
ab
µ
is the spin connection of the six-dimensional
metric and R is the ordinary Ricci scalar curvature, however in a general background both
quantities receive corrections.
Another important feature of this supergravity is that off-shell closure of the super-
symmetry algebra can only be achieved provided that additional auxiliary fields are in-
cluded beyond the gauge fields indicated in table 1. The necessary fields are indicated in
table 2 below.
– 3 –
