1. Double Trace Flows in dS/CFT
Diptarka Das
University of Kentucky
work with Sumit R. Das & Gautam Mandal
Great Lakes 2013
18th May 2013

2. Outline
Observation
The conformal group in d Euclidean dim is isomorphic to the Isometry
group in EAdSd+1 or dSd+1 .
Question
Do EAdS/CFT and dS/CFT (if it exists) capture the differences ? i.e.,
Given a gravity theory in EAdS or in dS will it predict different behaviors
of the dual Euclidean CFTs ? The objective is to make this question
more precise.
Content
1. Introduction
2. The dS/CFT dictionary
3. Holographic RG
4. Concluding remarks

3. Introduction
We focus on the Poincare patch of dSd+1 where, ds 2 =
Here, −∞ ≤ T ≤ 0, we will work in τ = −T .
L2
dS
T2
( dT 2 + dx 2 )

4. Introduction
There is a time-evolving wavefunction Ψ which is defined as,
Dφ(τ, x)exp (iS )
Ψ[φ0 (x), ] =
φ( ,x)=φ0 (x)
And it follows the real time evolution equation ( Schrodinger equation )
iGN
∂
˜
˜
ΨUV (φ, τ ) = H(τ )ΨUV (φ, τ )
∂τ
If there exists a dS/CFT correspondence then
< exp ρ
d d x (cut-off)−∆− φ0 (x)O+ (x) >st = Ψ[φ0 (x), ]
where,
ρ≡
Ld−1
dS
GN
∆± = d/2 ± ν, ν ≡
d 2 /4 − m2 L2
dS
We shall see a natural choice for the cut-off will be i .

5. Introduction
dS/CFT correspondence states
< exp ρ
d d x (i )−∆− φ0 (x)O+ (x) >st = Ψ[φ0 (x), ]
when we do the semi-classical calculation of the R.H.S we shall see why
the factors in the L.H.S are natural.
By Faulkner, Liu, Rangamani[arXiv:1010.4036 [hep-th]] and Heemskerk, Polchinski
parameters of Ψ are related to the couplings. In
particular the HJ limit of the Schrodinger equation gives the RG flow
equations.
[arXiv:1010.1264 [hep-th]]

6. Introduction
O2 deformation
We consider double trace deformation ( O2 ) of the CFT in d Euclidean
dimensions, ∆ is the dimension of O and 2ν = 2∆ − d
S → S0 +
f0
2
d d xO(x)2
We will consider relevant deformations, i.e, dim[O2 ] = 2∆− < d. Large
N arguments yield,
βλ = µ
dλ(µ)
= −2νλ + 2νbλ2
dµ
where,
f (µ) = λ(µ)µ2ν ,
O(k1 )O(k2 )
0
= G0 (k) = bk −2ν
Thus λ is dimensionless and b has the sign of the two point function. We
shall see how the same expression can be derived using holography.

7. The dS/CFT dictionary - Calculation of Ψ
We have
Ψ[φ0 (x), ] ∼ exp[iSon ]
The solution respecting the Bunch-Davies condition is
φ(τ, k) =
τ
d/2
(2)
Hν (kτ )
(2)
Hν (k )
iSon = −
i
2GN
At early times, k| |
φ0 (k)
(2)
[dk] Ld−1
dS
k Hν−1 (k )
∆−
+
φ0 (k)φ0 (−k)
d
(2)
( )
( )d Hν (k )
1 this becomes a bunch of harmonic oscillators

8. The dS/CFT dictionary
At late times k| |
iSon = −i
ρ2
2
[dk]
1
∆−
ρ2
φ (k)φ0 (−k)+
d 0
( )
2
[dk]φ0 (k)φ0 (−k)(i )−2∆− H(k)
where
H(k) = (i)d−1 −2ν
Γ(1 − ν)
Γ(1 + ν)
k
2
2ν
Now when we define,
exp ρ
d d x (i )−∆− φ0 (x)O+ (x)
st
= Ψ[φ0 (x), ]
Then Gst (k) = H(k). This agrees with the analytic continuation from
AdS. [Harlow, Stanford arXiv:1104.2621 [hep-th]]. And now there is only one
consistent alternative prescription which yields
O− O−
dS
alt
=−
1
= (i)d−1 O− O−
(2ν)2 Gst.
AdS
alt

9. The dS/CFT dictionary
The alternative quantization prescription ( factors in the Legendre
transform ) turns out to be
< exp
d d xJ(x)O− (x) >alt
=
×
Dφ0 (x) < exp ρ
exp ρ
dd x
J(x)
(i )−∆− φ0 (x)
2ν
From G0 (k) = bk −2ν we can now identify b as,
b=
d d x(i )−∆− φ0 (x)O+ (x) >st
22ν i 1−d Γ(1 + ν)
2ν Γ(1 − ν)

10. Holographic RG - Set-up
We follow the procedure developed in Heemskerk, Polchinski. We start
off by putting a floating cut-off at τ = . Then
Ψ[φ0 (x), ] =
˜
˜
˜
Dφ(x)ΨIR [φ, ]ΨUV [φ, φ0 ]
where
˜
˜
ΨIR [φ] = Ψ[φ(x), ]
˜
and from cut-off independence, ΨUV [φ, φ0 ] follows,
iGN
∂
˜
˜
ΨUV (φ, ) = H( )ΨUV (φ, )
∂
The idea is now to obtain an effective action of the dual theory at a finite
˜
cutoff by extending the dS/CFT relationship for ΨIR [φ, ],
e −Seff (
)
alt
=
˜
Dφ(x)
˜
DJ(x) ΨUV [φ, φ0 ]
exp −ρ
exp
dd x
J(x)
(i
2ν
˜
)−∆− φ(x)
d d xJ(x)O− (x)
alt

11. Holographic RG - Results
˜
We take a general quadratic form for ΨUV [φ, φ0 ] = exp[iK ], where
K=
1
GN
LdS
d
ddx −
1
˜
˜
g ( )φ2 + h( )φ + c( )
2LdS
The HJ equation gives,
−β(g ) = (i )
∂g
= g 2 + dg + m2 L2
dS
∂(i )
The zeroes of β(g ) are at g± = −∆± and alternative quantization means
we have to expand the coupling as
g = g− + δg
The beta function for δg is given by
β(δg ) = −(i )
∂δg
= −2ν(δg ) − (δg )2
∂(i )

12. Holographic RG - Results
The dS/CFT prescription gives us the relation,
f = −(i)d−1 (i )−2ν (2ν)2 g = −(i)d−1 (cut − off )2ν (2ν)2 g
Recall
f (µ) = λ(µ)µ2ν
Thus we can identify,
λ = −(i)d−1 (2ν)2 δg
For d = 3 we get,
β(λ) = −2νλ − λ2
This is the double trace coupling beta function for the Sp(n) theory in 3
dimensions. [LeClair, Neubert arXiv:0705.4657 [hep-th]]. Thus it agrees with the
proposed conjecture that the dual of dS4 is a SP(n) CFT3 . [Anninos,
Hartman, Strominger arXiv:1108.5735 [hep-th]]
When we use our result for b, the d-dimensional β function becomes
β(λ) = −2νλ + b
2−2ν Γ(1 − ν) 2
λ
2ν Γ(1 + ν)
Thus we recover the result which followed from the large N arguments in
the field theory side.

13. Results from AdS
From the AdS/CFT prescription, we have
bAdS =
22ν Γ(1 + ν)
2ν Γ(1 − ν)
The radial evolution equation satisfied by ΨUV ,AdS is
GN
∂
˜
˜
ΨUV ,AdS (φ, ) = −HAdS ( )ΨUV ,AdS (φ, )
∂( )
The form
ΨUV ,AdS = exp −
1
GN
LAdS
d
dd x
1
˜
˜
g ( )φ2 − h ( )φ − c ( )
2LAdS
leads to the flow equation
∂g
= −(g )2 + dg + m2 L2
AdS
∂
The flow equation for the departure from the fixed point δg is
β(δg ) = −
∂δg
= −2ν(δg ) + (δg )2
∂

14. Results from AdS
The relationship between the field theory dimensionless coupling and δg
is
λ = (2ν)2 δg = (2ν)22ν
Γ(1 + ν) 1
δg
Γ(1 − ν) bAdS
which leads once again to a beta function of the expected form.

15. Conclusions
dS/CFT allows us to calculate field theory β functions. For the double
trace case, the sign of the quadratic term in the β function equation
depends on the sign of the normalized two point function, i.e, b. Since
the treatment is quite general it will be interesting to see what happens
for multitrace deformations.
We saw from the direct on-shell action calculation that there is an overall
factor of (i)d−1 in front of the action. This is the same factor which
appears upon analytic continuation LAdS → iLdS from the AdS action.
This difference of phase has also been noticed in the collective action of
O(n) and Sp(n) which in d = 3 resulted in a overall minus sign.[ D.D, S.
R. Das, A. Jevicki, Q. Ye arXiv:1205.5776 [hep-th]]
The dS dual field theory cut-off was naturally identified with i .