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OVV in higher dimensions?

Brett McInnes proposes a generalization of the Hartle-Hawking approach to the vacuum selection problem pioneered by Ooguri, Vafa, and Verlinde (OVV) and described by this blog article to higher dimensions. McInnes identifies the existence of two possible Lorentz geometries associated with one Euclidean geometry as the key idea of the OVV paradigm. He argues that the higher-dimensional geometries must have flat compact sections which is certainly a non-trivial and possibly incorrect statement:

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