Title: Classification of digital affine noncommutative geometries,
Abstract: It is known that connected translation invariant $n$-dimensional
noncommutative differentials $d x^i$ on the algebra $k[x^1,\cdots,x^n]$ of
polynomials in $n$-variables over a field $k$ are classified by commutative
algebras $V$ on the vector space spanned by the coordinates. This data also
applies to construct differentials on the Heisenberg algebra `spacetime' with
relations $[x^\mu,x^\nu]=\lambda\Theta^{\mu\nu}$ where $ \Theta$ is an
antisymmetric matrix as well as to Lie algebras with pre-Lie algebra
structures. We specialise the general theory to the field $k={\ \mathbb{F}}_2$
of two elements, in which case translation invariant metrics (i.e. with
constant coefficients) are equivalent to making $V$ a Frobenius algebras. We
classify all of these and their quantum Levi-Civita bimodule connections for
$n=2,3$, with partial results for $n=4$. For $n=2$ we find 3 inequivalent
differential structures admitting 1,2 and 3 invariant metrics respectively. For
$n=3$ we find 6 differential structures admitting $0,1,2,3,4,7$ invariant
metrics respectively. We give some examples for $n=4$ and general $n$.
Surprisingly, not all our geometries for $n\ge 2$ have zero quantum Riemann
curvature. Quantum gravity is normally seen as a weighted `sum' over all
possible metrics but our results are a step towards a deeper approach in which
we must also `sum' over differential structures. Over ${\mathbb{F}}_2$ we
construct some of our algebras and associated structures by digital gates,
opening up the possibility of `digital geometry'.