254. Continuing dS/CFT – the correspondence. Part 1

Posted by Dmitry

February 12, 2009

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When I’ve discussed dS/CFT correspondence last time, I listed several criticisms of it, but probably had to explain in the first place what is the essence of dS/CFT

According to Bousso, Maloney and Strominger, the correspondence works as follows.

de Sitter in global coordinates

First of all, de Sitter space does not have a nice spatial infinity like AdS space (where dual field theory degrees of freedom might live) – but it has I_- and I_+ infinities, in the global coordinates

$ds^2=-d\tau^2+{\rm cosh}^2\taud\Omega_{d-1}^2$ (1)

corresponding to $\tau=-\infty$ and $\tau=+\infty$ (see the Figure above). As follows from the form of the metric (1), topology of space at $I_\pm$ is the one of a sphere $S_{d-1}$ .

We naturally would like to map physics in the bulk to the physics localized on infinities $I_\pm$ . It is actually hard to do that, since for an observer in the bulk, say, located near the throat of de Sitter space $\tau=0$ both infinities at $I_\pm$ are behind horizon.

It is especially clear to see in the static coordinate system

$ds^2=-(1-r^2)dt^2+\frac{dr^2}{1-r^2}+r^2d\Omega_{d-2}^2$ . (2)

Penrose diagram of de Sitter space

On the Penrose diagram above our bulk observer is located at r=0 . Infinities $I_\pm$ in turn correspond to $r=\infty$ and, as we see, they are separated from the bulk observer by event horizon at r=1 . There is no way physics at I $_\pm$ can influence physics at r=0 . From my opinion, this is the heart of the problem, the reason why all attempts to construct dS/CFT correspondence failed so far.

However, the relation between physics at $I_\pm$ and in the bulk is not that meaningless. Let us consider a free massive scalar field in de Sitter space and calculate its two point Green function. It should be de Sitter invariant and can therefore only depend on the invariant interval separating two points in de Sitter space. We can show that there are generally two linearly independent contributions into this Green function: one has UV singularity usual for all field theories, even in Minkowski space, another – has singularity when one takes antipodal points in de Sitter. However, for any observer, his antipodal point in de Sitter space is behind horizon!

How can it be so? The resolution of the puzzle comes from introducing time dependence into the problem. If we want to somehow prepare de Sitter space (say, smoothly evolve it from Minkowski by increasing the expectation value of the inflaton field ), there is no way the second linearly independent term with antipodal singularity might appear (it contradicts causality). Therefore, we simply have to put the coefficient in front of this term to zero by hands.

On the other hand, suppose that de Sitter always existed Then, antipodal singularity was always there, it always influenced the physics inside horizon. We cannot remove the term with antipodal singularity by hands and, as a result, acquire the whole family of Allen-Mottola de Sitter invariant vacua. The same logic is applied to relation between physics in the bulk and at $I_\pm$ .

To be continued.

AdS/CFT, Cosmology, Journal club, Lectures on de Sitter spacetime, Master Postdoc, Quantum field theory, String theory

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