JHEP07(2017)086
Plugging φ into the action (2.1), the action just becomes a boundary term [20],
I = −(∆
0
− d/2)π
−d/2
Γ(∆
0
)
Γ(∆
0
− d/2)
Z
d
d
x d
d
x
0
φ
0
(x)φ
0
(x
0
)
|x − x
0
|
2∆
0
. (2.4)
An equivalent viewpoint, which will be more useful for us, is to think of the system as
consisting of the free scalar φ(z, x) for z ≥ 0, along with another field φ
0
(x) that lives on
the boundary and is linearly coupled to φ(z = 0, x). (So φ
0
(x) can be thought of a source
for φ). The action of the full system
I = I
free
+ I
source
(2.5)
where,
I
source
= α
d+1−∆
0
Z
d
d+1
x
√
g δ(z − ) φ(z, x)φ
0
(x) , (2.6)
where α is a constant, and we have regulated the boundary by moving it inward to z = .
For the purposes of this action, the allowed solutions are those that decay as z
∆
for small z.
The reason this is the same as the previous action (2.4) is the following: if we just think of
the interior of AdS, and place sources somewhere, then the field is given by the convolution
of the sources with the bulk two-point function G
bulk
(z, x |z
0
, x
0
) (which is constructed out
of the normalizable modes). In the limit that z
0
→ 0,
G
bulk
(z, x |z
0
, x
0
) →
1
2∆
0
− d
z
0∆
0
K
∆
0
(z, x |x
0
) , as z
0
→ 0 , (2.7)
so (2.5) implies (2.3), and in particular that α = 2∆
0
− d. The action (2.4) is the kind of
bilocal action we are seeking; it is the action for a generalized free field φ
0
of dimension
∆ = d −∆
0
. With standard quantization, as we have discussed, one can achieve ∆
0
≥ d/2,
where the lower bound is set by the BF bound on the mass. With alternate quantization [21]
this can be extended to ∆
0
≥ d/2 − 1.
2.1 A tower of auxiliary fields
For generalizing (2.5) it is useful to rewrite it as an inherently d dimensional theory. We
decompose the field in terms of its radial eigenfunctions, which are Bessel functions,
φ(z, t, x) =
Z
∞
0
dλ z
d/2
J
ν
(λz)
√
λ ϕ
λ
(t, x) , (2.8)
where ∆
0
= d/2 + ν, with ν =
p
m
2
+ d
2
/4. Inserting this expansion of φ(z, t, x) into the
action (2.5) and restoring the AdS scale L, we get,
I =
Z
d
d
x
Z
∞
0
dλ
1
2
(∂ϕ
λ
)
2
+ λ
2
L
−2
ϕ
2
λ
+ L
−ν−1
α λ
ν+
1
2
ϕ
λ
φ
0
, (2.9)
where we have defined α = 2
1−ν
/Γ(ν). So we have a tower of fields ϕ
λ
, of mass λ
2
L
−2
, all
of which are linearly coupled to φ
0
. Let us integrate out ϕ
λ
. This gives an effective action,
I
eff
=
1
2
Z
d
d
xd
d
x
0
φ
0
(x)K(x, x
0
) φ
0
(x
0
) , (2.10)
– 4 –
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