An M-theory dS maximum from Casimir energies on Riemann-flat manifolds

We initiate the study of flux compactifications on non-supersymmetric Riemann-flat manifolds (RFM's) with Casimir energy. While curvature and other corrections are suppressed in RFM's, the inclusion of Casimir energies allows one to evade standard dS no-go theorems, and the absence of orientifolds or other singular sources means that the construction is completely captured by ten or eleven-dimensional supergravity. We obtain a fully explicit formula for the Casimir stress-energy in a general RFM, including its ten or eleven-dimensional profile. The Casimir energy localizes in particular loci of the RFM, which we call ``Casimir branes''. The tension of Casimir branes sometimes cancels exactly, due to a spacetime analog of worldsheet Atkin-Lehner symmetry. We use Casimir energies to construct an explicit $dS_5$ maximum solution of a flux compactification of M-theory on a specific 6-dimensional RFM. The resulting solution is scale-separated, has a vacuum energy of $10^{-8}$ in five-dimensional Planck units, the Hubble radius is $10^4$ Planck lengths, and the light fields have masses of order $H$. This is a fully explicit, top-down de Sitter maximum in M-theory, with precisely computable vacuum energy. While the solution is not parametric, it is under very good control: higher derivative and loop corrections to the vacuum energy are suppressed in powers of a small parameter $\delta V/V\sim 10^{-5}$, and M2 and M5-brane instantons are negligible. In short, the solution survives all known corrections. Nevertheless, it might be sensitive to more exotic ones, such as e.g. loops of 11d Planckian virtual black holes if there were a large enough number of them. We also extend the Ewald numerical method for lattice sums to arbitrary dimensions and develop an efficient numerical implementation.}