  Motivated by its potential use as a starting point for solving various
cosmological constant problems, we study F-theory compactified on the warped
product $\mathbb{R}_{\text{time}} \times S^3 \times Y_{8}$ where $Y_{8}$ is a
$Spin(7)$ manifold, and the $S^3$ factor is the target space of an $SU(2)$
Wess--Zumino--Witten (WZW) model at level $N$. Reduction to M-theory exploits
the abelian duality of this WZW model to an $S^3 / \mathbb{Z}_N$ orbifold. In
the large $N$ limit, the untwisted sector is captured by 11D supergravity. The
local dynamics of intersecting 7-branes in the $Spin(7)$ geometry is controlled
by a Donaldson--Witten twisted gauge theory coupled to defects. At late times,
the system is governed by a 1D quantum mechanics system with a ground state
annihilated by two real supercharges, which in four dimensions would appear as
"$\mathcal{N} = 1/2$ supersymmetry" on a curved background. This leads to a
cancellation of zero point energies in the 4D field theory but a split mass
spectrum for superpartners of order $\Delta m_\text{4D} \sim \sqrt{M_\text{IR}
M_\text{UV}}$ specified by the IR and UV cutoffs of the model. This is
suggestively close to the TeV scale in some scenarios. The classical 4D
geometry has an intrinsic instability which can produce either a collapsing or
expanding Universe, the latter providing a promising starting point for a
number of cosmological scenarios. The resulting 1D quantum mechanics in the
time direction also provides an appealing starting point for a more detailed
study of quantum cosmology.
