Classification of digital affine noncommutative geometries

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'.