JHEP07(2018)030
Type BPS ∆ Spin so(8)
(A, 0) (long) 0 ≥ ∆
0
+ j + 1 j [a
1
a
2
a
3
a
4
]
(A, 1) 1/16 ∆
0
+ j + 1 j [a
1
a
2
a
3
a
4
]
(A, 2) 1/8 ∆
0
+ j + 1 j [0a
2
a
3
a
4
]
(A, 3) 3/16 ∆
0
+ j + 1 j [00a
3
a
4
]
(A, +) 1/4 ∆
0
+ j + 1 j [00a
3
0]
(A, −) 1/4 ∆
0
+ j + 1 j [000a
4
]
(B, 1) 1/8 ∆
0
0 [a
1
a
2
a
3
a
4
]
(B, 2) 1/4 ∆
0
0 [0a
2
a
3
a
4
]
(B, 3) 3/8 ∆
0
0 [00a
3
a
4
]
(B, +) 1/2 ∆
0
0 [00a
3
0]
(B, −) 1/2 ∆
0
0 [000a
4
]
conserved 5/16 j + 1 j [0000]
Table 1. Multiplets of osp(8|4) and the quantum numbers of their corresponding superconformal
primary operator. The conformal dimension ∆ is written in terms of ∆
0
≡ a
1
+a
2
+(a
3
+a
4
)/2. The
Lorentz spin can take the values j = 0, 1/2, 1, 3/2, . . .. Representations of the so(8) R-symmetry
are given in terms of the four so(8) Dynkin labels, which are non-negative integers.
In the rest of this paper we will only consider the four-point function of O
Stress
(~x, Y ).
Superconformal invariance implies that it takes the form
hO
Stress
(~x
1
, Y
1
)O
Stress
(~x
2
, Y
2
)O
Stress
(~x
3
, Y
3
)O
Stress
(~x
4
, Y
4
)i
=
(Y
1
· Y
2
)
2
(Y
3
· Y
4
)
2
|~x
12
|
2
|~x
34
|
2
A(U, V ; σ, τ) ,
A(U, V ; σ, τ) =
X
M ∈ osp(8|4)
λ
2
M
G
M
(U, V ; σ, τ) ,
(2.2)
where
U ≡
~x
2
12
~x
2
34
~x
2
13
~x
2
24
, V ≡
~x
2
14
~x
2
23
~x
2
13
~x
2
24
, σ ≡
(Y
1
· Y
3
)(Y
2
· Y
4
)
(Y
1
· Y
2
)(Y
3
· Y
4
)
, τ ≡
(Y
1
· Y
4
)(Y
2
· Y
3
)
(Y
1
· Y
2
)(Y
3
· Y
4
)
,
(2.3)
G
M
are superconformal blocks, and λ
2
M
are the OPE coefficients squared for each supermul-
tiplet M. In table 2 we list the M that may appear in this four-point function, following
the constraints discussed in [52]. Since these are the only multiplets we will consider in this
paper, we denote the short multiplets other than the stress-tensor as (B, +) and (B, 2),
the semi-short multiplets as (A, 2)
j
and (A, +)
j
where j is the spin, and the long multiplet
as (A, 0)
j,n,q
, where n = 0, 1, . . . denotes the leading order twist 2n + 2 and q = 0 , . . . n
denotes the distinct operators with the same leading order quantum numbers.
Of particular importance will be the OPE coefficient for the stress-tensor multiplet. In
the conventions of [11], if we normalize O
Stress
such that the OPE coefficient of the identity
operator is λ
Id
= 1, then
λ
2
Stress
=
256
c
T
,
(2.4)
– 4 –
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