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dS/CFT correspondence in two and three dimensions
Even though the AdS/CFT correspondence is by now well-understood, a similar correspondence for a de Sitter space has been quite a puzzle to establish. Such a development is of considerable interest in view of recent astronomical data suggesting that we live in a Universe of positive cosmological constant [37]. A concrete proposal for a dS/CFT correspondence was recently put forth by Strominger [38] and attracted much attention. According to this proposal, all observables in the bulk de Sitter space are generated by data specified on its asymptotic boundary which can be selected to be the Euclidean hypersurface in the infinite past. The isometries of the de Sitter space are mapped onto generators of the conformal group of the theory defined on the boundary . This CFT is hard to construct in general, but various features, such as conformal weights and masses, are known. In three-dimensional de Sitter space, the conformal group on the boundary is infinite and the central charge is known [39]. The CFT is a Liouville theory [40]. Dr. Siopsis and Scott Ness investigated the case of two-dimensional de Sitter space [41]. The asymptotic boundary is a circle, which upon a Wick rotation turns into time. The theory on the boundary is a conformal quantum mechanical model. They explicitly constructed this model for the case of a scalar particle, obtained the generators of the conformal group, calculated the eigenvalues of the Hamiltonian and the Green functions. Their method of solution was similar to the one discussed by de Alfaro, Fubini and Furlan (DFF) [7], even though the two Hamiltonians differed. It should be noted that the two-dimensional case is special in many respects and it is not clear how it can be generalized to de Sitter spaces of higher dimensionality. The conformal theory living on the boundary of de Sitter space is a quantum mechanical model. Thus, no regularization is needed to define it. In particular, the conformal anomaly is absent and the central charge is zero. This can be seen explicitly if one follows a procedure as in [38]. Moreover, the entire conformal algebra on the boundary is , again implying zero central charge. The thermodynamics one extracts from this model is also simple. For example, the entropy is easily seen to be zero if one uses the standard formula S = A/(4G), since the ``horizon'' is zero-dimensional (point). On the CFT side, note that there is only one scale, the temperature , unlike in higher dimensions, where the spatial volume provides one more scale and a dimensionless parameter may be defined (e.g., RT where R is the radius of space, if it is a sphere) and so thermodynamic quantities can be non-trivial functions of this parameter. Thus, in two dimensions, the entropy one calculates from CFT is a constant which must be zero by the third law of thermodynamics (which definitely applies to simple quantum mechanical systems). Nevertheless, the form of the generators of the conformal algebra and the construction of the Hilbert space is non-trivial as was demonstrated in [41]. Dr. Siopsis and Scott Ness are exploring the possibility of extending the discussion to de Sitter spaces of dimension higher than two and shed some light on the behavior of the conformal field theories on the boundary. In particular, they are considering a massive scalar field coupled to gravity in a three-dimensional Kerr-de Sitter background [42]. Performing a weak-field expansion, one obtains a Virasoro algebra in the northern and southern diamonds in static coordinates [39]. This algebra extends the algebra of isometries. The central charge is in agreement with the expected value . By considering the partition function of the system, one arrives at an expression for the entropy which should be in agreement with Cardy's formula for the density of states of a two-dimensional conformal field theory [43,39]. It would be interesting to also extend this discussion to higher dimensions. |
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