CutiePieHole

1y

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2tU
′′∂z¯ +
1
2
U
′
∂t, (2.3)
where U = U(z) is holomorphic and the prime denotes differentiation. In order to obtain a
real vector field one must add the complex conjugate, however for notational simplicity we
suppress this addition in (2.3) and all subsequent formula. In general the metric transforms
under a diffeomorphism as the Lie derivative
δζ gmn = −Lζ gmn. (2.4)
For ζ parametrized by U as in (2.3), (2.4) becomes
δU gzz = −
ℓ
2
2
U
′′′
,
δU gzz¯ = δU gzt = δU gtt = 0. (2.5)
The change (2.5) in the metric satisfies (2.2) and so (2.3) generates an asymptotic symmetry
of de Sitter space on Iˆ−.
A special case of (2.3) is
U = α + βz + γz2
, (2.6)
5
where α, β, γ are complex constants. In this case U
′′′ vanishes, and the metric is therefore
invariant. These transformations generate the SL(2, C) global isometries of 2+1 de Sitter.
In conclusion the asymptotic symmetry group is the conformal group of the complex
plane, and the isometry group is SL(2, C) subgroup of the asymptotic symmetry group.
3. The Boundary Stress Tensor
Brown and York 25 have given a general prescription for defining a stress tensor
associated to a boundary of a spacetime. Our treatment parallels the discussion of AdS3
in 26. We will be interested in the boundary t → −∞ at Iˆ−. In the case at hand the
stress tensor is4
T
µν = −
4π
√γ
δS
δγµν
, (3.1)
where γ is the induced metric on the boundary. The action is
S =
1
16πG Z
M
d
3x
√
−g(R −
2
ℓ
2
) + 1
8πG Z
∂M
√
γK +
1
8πGℓ Z
∂M
√
γ + Smatter, (3.2)
K here is the trace of the extrinsic curvature defined by Kµν = −∇(µnν) = −
1
2Lnγµν with
n
µ
the outward pointing unit normal. The second integral in (3.2) is the usual gravitational
surface term. The third integral is a surface counterterm required for finiteness of T at
an asymptotic boundary, and uniquely fixed by locality and general covariance 26. The
matter action is assumed not to be relevant near the boundary and will henceforth be
suppressed.
The sign in the definition (3.1) leads to a positive mass5
for Schwarzschild-de Sitter
32. It would be interesting to see if (3.1) reproduces the canonical operator product
expansion for a 2D CFT stress tensor.
Using (3.2) to evaluate (3.1) we learn that, for solutions of the bulk equations of
motion,
T
µν =
1
4G

Kµν − (K +
1
ℓ
)γ
µν
. (3.3)
This vanishes for de Sitter space on Iˆ− in the coordinates (2.1). For more general asymptotically dS3 spacetimes (obeying (2.2)) (3.3) implies
Tzz =
1
4G

Kzz +
1
ℓ
γzz
. (3.4)
4 Our conventions in this section are those of 25,26 except for a factor of −2π in this equation,
in order to conform with the more standard conventions 31 for 2D CFT stress tensor.
5 More precisely, a positive value of the AD mass L0 + L¯0, as defined below in equation 7.4. 6
4. The Central Charge
A central charge can be associated to dS3 by a***yzing the behavior of the stress
tensor on I
−. We follow related discussions for AdS3 21,26 (which in turn followed the
earlier work 33). Under the conformal transformations (2.3), one finds
δU Tzz = −
ℓ
8G
U
′′′
. (4.1)
This transformation identifies the central charge as6
c =
3ℓ
2G
. (4.2)
It is presumably also possible to derive c by an alternate method 34,33,26 which
relates it to the trace anomaly by using spherical rather than planar spatial sections.
5. The Plane and Iˆ− Correlators
In the previous sections we have seen that the conformal group of euclidean R2 has
an action on Iˆ−. One therefore expects that appropriately rescaled gravity correlators
restricted to Iˆ− will be those of a euclidean 2D conformal field theory. In this section we
verify this expectation for the case of a massive scalar.
Consider a scalar field of mass m with wave equation
m2
ℓ
2φ = ℓ
2∇2φ = −∂
2
t φ + 2∂tφ + 4e
2t
∂z∂z¯φ. (5.1)
Near Iˆ− the last term in (5.1) is negligible and solutions behave as
φ ∼ e
h±t
, t → −∞, (5.2)
where
h± = 1 ±
p
1 − m2ℓ
2. (5.3)
We first consider the case 0 < m2
ℓ
2 < 1 so that h± are real and h− < 1 < h+. As a
boundary condition on Iˆ− we demand
lim
t→−∞
φ(z, z, t ¯ ) = e
h−tφ−(z, z¯), (5.4)
6 This equation was arrived at from a different perspective in 9, and a related equation
appears in 11.
7
with corrections suppressed by at least one power of e
2t
. The dS3 /CFT2 correspondence
proposes, in direct a***ogy with the AdS/CFT correspondence, that φ− is dual to an
operator Oφ of dimension h+ in the boundary CFT. The two point correlator of Oφ is (up
to normalization) the quadratic coefficient of φ− in the expression7
lim
t→−∞ Z
Iˆ−
d
2
zd2
v

e
−2(t+t
′
)φ(t, z, z¯)
↔∂ tG(t, z, z¯;t
′
, v, v¯)
↔∂ t
′φ(t
′
, v, v¯)

t=t
′
. (5.5)
G here is the de Sitter invariant Green function given in the appendix. Near Iˆ− it reduces
to
lim
t,t′→−∞
G(t, z, z¯;t
′
, v, v¯) = c+e
h+(t+t
′
)
|z − v|
2h+
+
c−e
h−(t+t
′
)
|z − v|
2h−
, (5.6)
where c± are constants. Inserting (5.4) and (5.6), (5.5) is proportional to8
Z
Iˆ−
d
2
zd2
vφ−(z, z¯)|z − v|
−2h+ φ−(v, v¯). (5.7)
We conclude that the dual operator Oφ obeys
hOφ(z, z¯)Oφ(v, v¯)i =
const.
|z − v|
2h+
, (5.8)
as is appropriate for an operator of dimension h+.
It should be noted that the boundary conditions (5.4) are not the most general. There
are also solutions with the subleading behavior φ+(z, z¯)e
h+t at Iˆ−. Including these would
lead to an additional term in (5.7) proportional to
Z
Iˆ−
d
2
zd2
vφ+(z, z¯)|z − v|
−2h− φ+(v, v¯), (5.9)
which might be associated with an operator of dimension h−. In the next section we will
find that this extra boundary condition can imposed on the second boundary at I
+, which
is not within the coordinate patch O− covered in the planar coordinates (2.1). Alternately
7 This is equivalent to the usual expression used in AdS/CFT (except of course with a different
boundary and G), as can be readily seen from the formula for the bulk Green function G in terms
of the bulk-to-boundary Green function. A clear discussion can be found in 35. We have avoided
use of the bulk-to-boundary Green function because in the de Sitter case we need to keep track
of both terms in (5.6).
8 Similar holographic expressions were derived for general coset spaces in 36, but the case of
de Sitter space was not explicitly considered.
8
this second set of independent fields might be eliminated by imposing a suitable boundary
condition at the future horizon t → ∞, which would lead to a different Green function.9
What happens if m2
ℓ
2 > 1? In that case the conformal weight h− is no longer real.
h± are complex conjugates with real part equal to unity. Of course the appearance of
imaginary conformal weights suggests that the dual CFT is not unitary. This might mean
that consistent theories of de Sitter quantum gravity have no stable scalars with m2
ℓ
2 > 1. On the other hand we know of no obvious reason that the dual CFT needs to be unitary.
The preceding discussion is reminiscent of Liouville theory and indeed suggests that
the boundary CFT has a Liouville-like form. In some discussions of Liouville theory
operators with complex dimensions are encountered. Further there are various kinds of
operators (called macroscopic and microscopic in 37) which may or may not be allowed
depending on the context.
6. The Sphere and I
± Correlators
An alternate form of the dS3 metric is
ds2
ℓ
2
= −dτ 2 + 4 cosh2
τ
dwdw¯
(1 + ww¯)
2
(6.1)
w = tan θ
2
e
iφ here is a complex coordinate on the round sphere. This metric describes dS3
as a contracting/expanding two-sphere. These coordinates cover the entire space which
has future and past S
2 boundaries I
±, as depicted in figure 2. In general we can consider
correlators with points on either or both of the boundaries. North Pole South Pole
I
I
Fig. 2: Lines of constant τ in global spherical coordinates are the spacelike
two-spheres indicated by dashed lines.
9
I thank J. Maldacena for this suggestion.
9
We begin with the two-point correlator with both points on I
−. We wish to compute
the spherical a***og of (5.5), which is
lim τ→−∞ Z
I−
d
2wd2
v
p
h(w)h(v)

e
−2(τ+τ
′
)φ(τ, w, w¯)
↔∂ τG(τ, w, w¯; τ
′
, v, v¯)
↔∂ τ
′φ(τ
′
, v, v¯)

τ=τ
′
,
(6.2)
where h(w) = 2(1 + ww¯)
−2
is the measure on the sphere. Near I
−, φ can be decomposed
as
lim τ→−∞
φ(τ, w, w¯) = φ
in

• (w, w¯)e
  h+τ + φ
  in
  −(w, w¯)e
  h−τ
  . (6.3)
  The superscripts ”in” (”out”) are used to denote quantities on I
  − (I
  +). As seen in the
  appendix, the propagator behaves as
  lim
  τ,τ′→−∞
  G(τ, w, w¯; τ
  ′
  , v, v¯) = c+e
  h+(τ+τ
  ′
  )
  (1 + ww¯)
  h+ (1 + vv¯)
  h+
  |w − v|
  2h+
• c−e
  h−(τ+τ
  ′
  )
  (1 + ww¯)
  h− (1 + vv¯)
  h−
  |w − v|
  2h−
  .
  (6.4)
  (6.2) is then proportional to
  Z
  I−
  d
  2
  vd2w
  p
  h(v)h(w)

c+φ
in
−(w, w¯)∆h+ (w, w¯; v, v¯)φ
in
−(v, v¯)

• c−φ
  in
• (w, w¯)∆h− (w, w¯; v, v¯)φ
  in
• (v, v¯)
  
  
  .
  (6.5)
  ∆h± here is the two point function for a conformal field of dimension h± on the sphere:
  ∆h± =
  (1 + ww¯)(1 + vv¯)
  |w − v|
  2
  h±
  , (6.6)
  including the normalization factor from the Weyl anomaly on a curved geometry. Hence
  we see that, as in the planar case, the two-point scalar correlators can be identified with
  correlators of conformal fields of dimension h±, except that now they are on the sphere
  rather than the plane. A similar expression holds for the boundary at I
  +.
  Life becomes more interesting when we put one point on I
  − and one on I
  +. Then we
  must compute
  lim τ→−∞ Z
  I−
  d
  2w
  Z
  I+
  d
  2
  v
  p
  h(w)h(v)

e
2(τ−τ
′
)φ(τ, w, w¯)
↔∂ τG(τ, w, w¯; τ
′
, v, v¯)
↔∂ τ
′φ(τ
′
, v, v¯)

τ=−τ
′
.
(6.7)
For this case, as shown in the appendix, the relevant limit of the Green function is
lim
τ→−∞,τ′→+∞
G(τ, w, w¯; τ
′
, v, v¯) = c+ cos(πh+)e
h+(τ−τ
′
)∆h+ (w, w¯; −
1
v¯
, −
1
v
) + (h+ ↔ h−).
(6.8)
1
It is convenient to define the inverted boundary field at I
+
φ˜out

• (v, v¯) = φ
  out
• (−
  1
  v¯
  , −
  1

JACKIE ARTIST

1y

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O O o a i e

1y

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JACKIE ARTIST

1y

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JACKIE ARTIST

1y

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Do you?

CutiePieHole

1y

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1y

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CutiePieHole

1y

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JACKIE ARTIST

1y

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And they cited that document.

JACKIE ARTIST

1y

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JACKIE ARTIST

1y

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Then stumbled upon some website that explains physics in a simple manner.

And they cited that document.

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1y

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1y

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1y

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