Non-linear Laplace equation, de Sitter vacua and information geometry

Three exact solutions say φ0 of massless scalar theories on Euclidean space, i.e. D = 6 φ, D = 4 φ and D = 3 φ models are obtained which share similar properties. The information geometry of their moduli spaces coincide with the Euclidean AdS7, AdS5 and AdS4 respectively on which φ0 can be described as a stable tachyon. In D = 4 we recognize that the SU(2) instanton density is proportional to φ40. The original action S[φ] written in terms of new scalars φ̃ = φ− φ0 is shown to be equivalent to an interacting scalar theory on D-dimensional de Sitter background. AdS/CFT correspondence [1], as a bulk/boundary correspondence, is a quantitative realization of the holographic principle. In [2] Witten showed that the metric on the boundary of the AdS space is well-defined only up to a conformal transformation and the correlation functions of the CFT on the boundary are given by the dependence of the supergravity action on the asymptotic behavior at infinity, see also [3]. Using the metric ds2 = 1 x0 (dx0 2 +dx1 2 + · · ·+dxd) for the Euclidean AdSd+1 Witten showed that the generating function for CFT correlators, I[φ] = ln 〈exp ∫ φO〉 is I[φ] = ∫ dydz φ0(~y)φ0(~z) |~y − ~z|+ , (1) where φ0 here, is some scalar field on the boundary, determined by the asymptotic behavior of scalar fields Φ in the bulk: Φ ∼ x0φ0 as x0 → 0. Here, λ+ is the larger root of the equation λ(λ+ d) = m2. These results led us to a classical interpretation for EAdS/CFT correspondence as the relation between the solutions of the Klein-Gordon equation φ(x0, ~x) (bulk fields) and the Cauchy data φ(0, ~x) (boundary fields) [5]. In fact under the conformal transformation δμν → gμν = x0δμν that gives the EAdSd+1 metric mentioned above in terms of δμν , the metric of the D-dimensional flat Euclidean space Rd+1, massless fields φ on Rd+1 transform to massive scalars Φ = x0 +φ with mass −d2−1 4 . From the classical equation of motion δS[φ] = 0 one can determine φ(x0, ~x) in terms of the Cauchy data φ0(~x) = φ(0, ~x). Inserting the solution in S[φ] one obtains I[φ] given in Eq.(1). By the same method, though only for scalars with ∗e-mail: [email protected] specific mass m2 = d 2 −1 4 , the correlation functions of the boundary operators in dSd+1/CFTd correspondence that Strominger [4] explicitly calculated for d = 2 and proposed for general d can be obtained [5]. By generalizing the method to free spinors the boundary term to be added to the bulk Dirac action necessary for AdS/CFT and dS/CFT correspondence [6] are obtained for general free massive spinors in (A)dS space [7]. What can one learn about AdS/CFT correspondence if one uses this method for interacting scalar theories instead of free scalars? The scalar field theories that can be considered are massless D = 6 φ3, D = 4 φ4 and D = 3 φ6 models [7, 8] given by the action, S[φ] = ∫