Strings and Quantum Gravity

THEORETICAL HIGH ENERGY PHYSICS
Strings and Quantum Gravity

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RESEARCH > (A)dS/CFT Correspondence and Cosmology > dS/AdS duality
The dS/CFT correspondence bears a striking formal resemblance to its AdS counterpart [1] suggesting that the former be derived from the latter by some kind of analytic continuation [46]. As was pointed out in [39], one needs to exercise care in such extrapolations. If one analyzes the behavior of the respective Green functions carefully, one discovers that dS Green functions may not be obtained by a double analytic continuation of their AdS counterparts.
Dr. Siopsis put forth a different proposal of extrapolating from AdS to dS spaces [47]. He established a duality between the two spaces which interchanged the role of coordinates and momenta for a scalar field. He thus showed that a massive mode in dS space was dual to a tachyonic mode in AdS space. This was based on the following basic observation. A (D+1)-dimensional AdS space (AdS_D+1) is defined within a flat (D+2)-dimensional space as the hypersurface

$\begin{displaymath} X_0^2 - X_1^2 - \dots - X_D^2 + X_{D+1}^2 = \ell^2 \end{displaymath}$

where $\ell^2 > 0$ . A particle of mass m moving in this space has a trajectory on the mass-shell hypersurface in momentum space

$\begin{displaymath} P_0^2 - P_1^2 - \dots - P_D^2 + P_{D+1}^2 = m^2 \end{displaymath}$

After an analytic continuation, $X_{D+1}\to iX_{D+1}$ (and correspondingly, $P_{D+1}\to -iP_{D+1}$ ), one obtains the Euclidean AdS space (EAdS). The mass-shell condition then reads

$\begin{displaymath} -P_0^2 + P_1^2 + \dots + P_D^2 + P_{D+1}^2 = -m^2 \end{displaymath}$

The form of this constraint is identical to the constraint for massive modes (m² > 0). Thus, if we interchange the role of coordinates and momenta, we arrive at the same theory in (E)AdS space.
On the other hand, for tachyonic modes (m² < 0) [48], the above is the defining equation of the dS hyperboloid. Thus, by interchanging the roles of momenta and coordinates in the (D+2)-dimensional space, a duality is established between tachyonic modes in EAdS space and massive scalars in dS space [47]. This duality explains why in dS space one obtains Green functions that are similar to their EAdS counterparts but for tachyonic modes, even though the two inner products are different, due to the different roles of the timelike direction. It would be interesting to extend the results to other modes and include spin. This is currently under investigation.

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