JHEP01(2019)196
The Casimir equation for the reduced correlator H also takes separated form, and
its solution is similar to the naive OPE (2.10) but with the dimension shifted by 4 in
accordance to the z¯z denominator in the prefactor multiplying H:
H
{p
i
}
(z, ¯z, α, ¯α) =
X
`,∆,m,n
a
{p
i
}
(`, ∆, m, n)G
p
21
,p
34
`,∆+4
(z, ¯z)Z
p
21
,p
34
m,n
(α, ¯α). (2.19)
The representations appearing in H can be worked out from the α, ¯α dependence of the
coefficient of H in eq. (2.14), and are only those which appear in the tensor products of
[0, p
i
− 2, 0]; this gives the range
0 ≤ m ≤ min{p
i
−2}, max(|p
12
|, |p
34
|)+m ≤ n ≤ min(p
1
+p
2
, p
3
+p
4
)−m−4. (2.20)
While formally similar to the bosonic expansion enjoyed by G in eq. (2.10), we see that the
OPE for the susy-decomposed correlators f and H contains much fewer terms. For example,
for the simplest case of the 2222 correlator, eq. (2.10) contains six distinct R-symmetry
representations whereas there is a single one in eq. (2.19), involving only Z
(0,0)
0,0
(α, ¯α) = 1.
The full α dependence is then entirely produced by the chiral correlator f(z, α) and the
various factors in (2.14).
2.2 Lorentzian inversion formula
To determine the OPE data b
{p
i
}
j,m
and a
{p
i
}
`,∆,m,n
entering eqs. (2.17) and eqs. (2.19), our main
tool will be a recently derived inversion integral which plucks out the OPE coefficients
from the correlator. The key point is that this formula does not require to know the full
correlator, but just its “double discontinuity” which is much easier to compute.
Let us first deal with the global symmetry. Using that the Z
m,n
(α, ¯α) represent S
5
spherical harmonics, which are mutually orthogonal, it is straightforward to write down an
integral which projects the reduced correlator onto any desired representation. As noted,
the Z
m,n
’s are secretly Jacobi polynomials, with argument cos θ = 1 −
2
α
, which suggests
that the natural integration range is cos θ ∈ [−1, 1] or α, ¯α ∈ [1, ∞). One can then show
that the following orthogonality relation holds:
H
{p
i
}
(z, ¯z, m, n) =
8π
4
˜κ
r,s
2+
m+n
2
˜κ
r,s
1+
n−m
2
−1
×
Z
∞
1
dα
α
2
d¯α
¯α
2
α − ¯α
α¯α
2
Z
−r,−s
m,n
(α, ¯α)H
{p
i
}
(z, ¯z, α, ¯α) (2.21)
where r = p
21
, s = p
34
, and
˜κ
r,s
h
=
Γ(h +
1
2
r)Γ
h −
1
2
r
Γ
h +
1
2
s
Γ
h −
1
2
s
2π
2
Γ(2h − 1)Γ(2h)
=
r
r,s
h
r
−r,−s
h
2π
2
(2h − 1)
, (2.22)
r
r,s
h
=
Γ
h +
1
2
r
Γ
h +
1
2
s
Γ(2h − 1)
. (2.23)
This diagonalizes the S
5
part of the OPE (2.19), leaving only sums over dimension and spin:
H
{p
i
}
(z, ¯z, m, n) =
X
`,∆
a
{p
i
}
(`, ∆, m, n)G
(p
21
,p
34
)
`,∆+4
(z, ¯z). (2.24)
– 7 –
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