As far as I understand, string phenomenology is usually concerned with compactifications of string theory, M-theory or F-theory in which the uncompactified dimensions form a 4-dimensional Minkowski spacetime. However, we know our actual universe has a positive cosmological constant hence its asymptotics are that of a De Sitter spacetime. On the intuitive level it makes sense to me, since the microscopic physics should have little to do with spacetime asymptotics. However, from another point of view I see a problem.
It seems to me that a cosmological constant in the effective 4-dimensional field theory requires a non-vanishing Ricci tensor in the compactified dimensions. For example, the classical case study for anti-De Sitter string theory is AdS_4 x S_6. The compactified dimensions form the sphere, a manifold with positive curvature, compensating the negative curvature of AdS.
This non-vanishing Ricci tensor seems to require different topology from a vanishing Ricci tensor. Hence all standard compactifications like Calabi-Yau manifolds, G2 manifolds etc. don't seem to be compatible with a non-vanishing cosmological constant.
What am I missing here?
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