Lecture 20
8.821 Holographic Duality
Fall 2014
Chapter 3: Duality Toolbox
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 20
3.1.6: EUCLIDEAN CORRELATION FUNCTIONS
Let us consider Euclidean space. The basic observables of a CFT is correlation functions of local operators. From
path integral formalism, we know a way to calculate them is by generating functional:
ZCF T [φ(x)] = he
´
dd xφ(x)O (x)
iE
(1)
where subscript E denotes the path integral defined in Euclidean space and φ(x) and O(x) should be considered as
the collection of all boundary operators and their sources.
The simplest case is φ(x) = 0, then ZCF T [0] reduces to the partition function of CFT, which, by AdS/CFT
correspondence, should be equal to the gravity dual partition function Zgravity . Since we have discussed the
correspondence:
O(x) ⇐⇒ Φn (x, z)
φ(x) ⇐⇒ Φnn (x, z)|∂AdS
(2)
(3)
where n is short for normalizable and nn is for non-normalizable, we should expect
ZCF T [φ(x)] = Zbulk [Φ|∂AdS = φ(x)]
(4)
where Φ|∂AdS = φ(x) really means Φ(z, x) → φ(x)z d−∆ when z → 0. Generally we do not know how to define
Zbulk [Φ|∂AdS ], but one can define it in semi-classical limit: gs → 0 and α0 → 0. The path integral formalism gives
ˆ
Zbulk [Φ|∂AdS ] =
DΦeSE [Φ]
(5)
Φ|∂AdS =φ(x)
ˆ
1
√
dd+1 x g(R + Lmatter )
(6)
2
2κ
In the semi-classical limit, the leading order partition function is given by its stationary point, namely the classical
solution:
Zbulk = exp(SE [Φc ])
(7)
where the action is
SE [Φ] = −
where Φc is the classical solution to equation of motion that satisfies the boundary conditions. As we know before,
this limit corresponds to N → ∞ and λ → ∞ limit in the boundary theory. In particular, SE [Φc ] ∝ 2κ1 2 ∝ N 2 is
consistent with the leading behavior of a large N gauge theory.
Thus in N → ∞, λ → ∞ limit, we have the following crucial formula
log ZCF T [φ] = SE [Φc ]Φc → z a φ(x)
(z → 0)
(8)
where a depends on dimension and spin of an operator, e.g. a = d − ∆ for scalar operator of dimension ∆; a = 0
for conserved current; a = 2 for stress tensor.
Remarks
1.
2.
´
In the generating functional, φ in dd xφ(x)O(x) should be considered as infinitesimal correction, i.e.
log ZCF T [φ] should be considered as a power series of φ(x). Similar for the right hand side of (8), where
the non-normalizable mode of Φ should be infinitesimal and created perturbatively.
Both sides of (8) are in fact divergent. The left hand side has usual UV divergences of a QFT, which
by IR/UV connection, corresponds to the volume divergences of AdS on the right hand side. Thus in
order to obtain finite answer, we need to renormalize them. The approach is to set a cutoff of z = when z → 0 and add a local functional contact term. To be specific, we have
(R)
SE [Φc ] = SE [Φc ]|z= + Sct [Φc ()]
1
(9)
Lecture 20
3.
8.821 Holographic Duality
Fall 2014
With renormalized action, general connected n-point function is given by functional derivative with
respect to φ:
(R)
(R)
δ n SE [Φc ]
δ n log ZCF T
=
(10)
hO1 (x1 ) · · · On (xn )ic =
δφ1 (x1 ) · · · δφn (xn ) δφ1 (x1 ) · · · δφn (xn ) φ=0
4.
φ=0
Especially 1-point function is nonzero in the presence of source φ.
(R)
hO(x)iφ =
δS
(R)
(R)
δSE [Φc ]
δS [Φc ]
= lim z d−∆ E
z→0
δΦc (z, x)
δφ(x)
(11)
[Φ ]
Note δΦEc (z,x)c ∼ Πc (z, x) is the canonical momentum conjugate to Φ treating z as “time” (later we will
show this for scalar field). Now we get
hO(x)iφ ∼ lim z d−∆ Π(R)
c (z, x)
(12)
z→0
One can show that for Φ(z, x) → z d−∆ A(x) + z ∆ B(x) when z → 0,
hO(x)iφ = 2νB(x)
d
(ν = ∆ − )
2
(13)
namely the 1-point function corresponds to the normalizable mode function. One can check that hO(x)iφ
and B(x) do have the same transformation property under scaling: hO(λx)iφ = λ−∆ hO(x)iφ and
z ∼0
Φ(λz, λx) = Φ(z, x) =⇒ λ∆ z ∆ B(λx) = z ∆ B(x) =⇒ B(λx) = λ−∆ B(x).
Let us calculate a massive scalar field as an example to show the above equality is valid. The Euclidean action is
given by
ˆ
1
√
SE = −
dd xdz g(g M N ∂M Φ∂N Φ + m2 Φ2 )
(14)
2
and metric is
ds2 =
R2
(dz 2 + dx2 )
z2
dx2 ≡ ηµν dxµ dxν
(15)
(R)
Our goal is to find SE [Φc ] for limz→0 z ∆−d Φc (x) = φ(x). Take the functional derivative with respect to ∂z Φ to
√
Lagrangian, we get the canonical momentum as Π = − gg zz ∂z Φ. The equation of motion is
√
−∂M ( gg M N ∂N Φ) + m2 Φ = 0
(16)
We write Φ = Φ(k, z)eikx and plug into the equation of motion to get
z d+1 ∂z (z 1−d ∂z Φ) − k 2 z 2 Φ − m2 R2 Φ = 0
(17)
This equation can be exactly solved, but we will not need it until the last step. Since we are calculating the effective
action with classical solution Φ, we get
ˆ
ˆ
1
1 ∞
√
√
SE [Φc ] = −
dd+1 xΦc (−∂M ( gg M N ∂N Φ) + m2 Φ) −
dzdd x∂M ( gg M N ∂N Φc Φc )
2
2 0
∞
ˆ
1
=
dd xΠc (z, x)Φc (z, x)
2
0
∞
ˆ
dd k
1
Φ
(k,
z)Π
(
=
−k,
z)
(18)
c
c
2
(2π)d
0
δS
(R)
[Φ ]
where in the second line we used the equation of motion and it shows that δΦEc (z,x)c ∼ Πc (z, x) in the limit z → 0.
First look at its behavior as z → ∞. Since we should require the regularity of Φ in the bulk, Φ must be finite
√
when z → ∞, so is ∂z Φ. However, we know Π = − gg zz ∂z Φ ∼ z 1−d ∂z Φ → 0 when z → ∞. Then the term in
infinity vanishes. For z → 0, the behavior is
Φc → A(x)z d−∆ + B(x)z ∆
2
(19)
Lecture 20
and
8.821 Holographic Duality
Fall 2014
√
Π = − gg zz ∂z Φ → −(d − ∆)A(x)z −∆ − ∆B(x)z ∆−d
d
2,
d
2
(20)
d
2
the we see Φc Πc is divergent because of z d−2∆ term (for ∆ = requires
If we restrict ourselves with ∆ = + ν ≥
special treatment), so is SE [Φc ]. Then we need to introduce a contact term to cancel it. As we discussed before,
the renormalized action is
ˆ
1
dd xΠc Φc + Sct [Φc (, x)]
(21)
SE [Φc ] = −
2
where the contact term can be chosen as
Sct [Φc (, x)] =
1
2
ˆ
z=
dd k
f (k 2 )Φc (k, z)Φc (−k, z)
(2π)d
(22)
where f (k 2 ) is some analytic function in k 2 such that to cancel the divergence and then taking the limit → 0. For
this purpose, we will need to consider subleading terms in (19). Consider the basis of solutions Φ1 and Φ2 defined
by
Φ1 → z d−∆
Φ2 → z ∆
(z → 0)
(23)
For small z,
Φ1 = z d−∆ (1 + a1 k 2 z 2 + · · · )
∆
2 2
Φ2 = z (1 + b1 k z + · · · )
(24)
(25)
where each expansion is analytic in k 2 because the equation of motion depends only on k 2 . The canonical momenta
basis are given by
Π1 → −(d − ∆)z −∆
Π2 → −∆z
∆−d
(26)
(27)
Our general solution should be expanded upon those basis as
Φc = A(k)Φ1 + B(k)Φ2
Πc = A(k)Π1 + B(k)Π2
(28)
(29)
Substitute this to the action, we get
SE [Φc ]|z= = −
1
2
ˆ
dd k
A2 Π1 Φ1 + B 2 Π2 Φ2 + AB(Π1 Φ2 + Φ1 Π2 )
d
(2π)
(30)
where the notation is A2 ≡ A(k)A(−k) as so on. Note in above formula, the first term is divergent, the second is
vanishing and the last one is finite. This implies that we can choose our contact term as
ˆ
1
dd k Π1 2
Sct =
Φ
2 z= (2π)d Φ1
ˆ
1
dd k Π1
=
(AΦ1 + BΦ2 )2
2 z= (2π)d Φ1
ˆ
1
dd k
2
2 Π1 2
=
A
Φ
(31)
Φ
Π
+
2ABΠ
Φ
+
B
1 1
1 2
2 z= (2π)d
Φ1 2
where Π1 /Φ1 |z= is an analytic function in k 2 . We see the first term is the divergent to cancel the one in SE , the
second term is finite and the last term is vanishing. Sum them together and we get the renormalized action:
ˆ
ˆ
1
dd k
1
dd k
(R)
SE [Φc ] =
A(−k)B(k)(Π
Φ
−
Φ
Π
)
=
2νA(−k)B(k)
(32)
1
2
1
2
2 z= (2π)d
2
(2π)d
Imposing the boundary condition A(k) = φ(k) and the regularity of Φ at z → ∞ will fix the ration χ(k) = B(k)/A(k)
(for this part we have to solve the equation of motion explicitly), we see
ˆ
dd k
1
(R)
2νχ(k)φ(k)φ(−k)
(33)
SE [Φc ] =
2
(2π)d
3
Lecture 20
8.821 Holographic Duality
Fall 2014
where χ(k) is independent on φ(k) and which implies
hO(k)iφ = 2νχ(k)φ(k) = 2νB(k)
(34)
hO(x)iφ = 2νB(x)
(35)
or in coordinate space:
From this formula, we can also calculate 2-point function
(R)
δ 2 SE
= 2νχ(k) = hO(x)iφ /φ(k)
GE (k) =
δφ(k)δφ(−k) (36)
φ=0
In other words, this is a linear response of 1-point function to the source φ(k),
hO(x)iφ = GE (k)φ(k)
(37)
If we solve out the equation of motion explicitly, we can find χ such that we can calculate the two point function
from bulk point of view. The solution is Bessel function if we require regularity at z → ∞,
Φc ∝ z d/2 Kν (kz)
(38)
as z → 0,
Kν (kz) =
Γ(ν)
2
kz
2
−ν
(1 + · · · ) +
Γ(−ν)
2
kz
2
ν
(1 + · · · )
(39)
2ν
2ν
k
Γ(−ν) k
=⇒ GE (k) = 2ν
2
Γ(ν)
2
(40)
which implies
B
Γ(−ν)
χ(k) =
=
A
Γ(ν)
We can do the Fourier transformation back to coordinate space and find
GE (x) ∝
1
|x|2∆
which is consistent with the 2-point function of CFT of a scalar with scaling dimension ∆.
4
(41)
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Fall 2014
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