Quantum Riemannian geometry of the discrete interval and q-deformation

We solve for quantum Riemannian geometries on the finite lattice interval $\bullet -\bullet -\cdots -\bullet$ with $n$ nodes (the Dynkin graph of type $A_n$) and find that they are necessarily $q$-deformed with $q=e^{\frac{\mathrm{ı}\pi }{n+1}}$. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel boundary effect' whereby, in order to admit a quantum-Levi Civita connection, the metric weight’ at any edge is forced to be greater pointing towards the bulk compared to towards the boundary, with ratio given by $(i+1{)}_q\mathrm{/}(i{)}_q$ at node $i$, where $(i{)}_q$ is a $q$-integer. The Christoffel symbols are also q-deformed. The limit $q\to 1$ likewise forces the quantum Riemannian geometry of the natural numbers $\mathbb{N}$ to have rational metric multiples $(i+1)\mathrm{/}i$ in the direction of increasing $i$. In both cases, there is a unique Ricci-scalar flat metric up to normalisation. Elements of quantum field theory and quantum gravity are exhibited for $n=3$ and for the continuum limit of the geometry of $\mathbb{N}$. The Laplacian for the scalar-flat metric becomes the Airy equation operator $\frac{1}{x}\frac{d^2}{d{x}^2}$ in so far as a limit exists. Scaling this metric by a conformal factor $e^{\psi (i)}$ gives a limiting Ricci scalar curvature proportional to $\frac{e^{-\psi }}{x}\frac{d^2\psi }{d{x}^2}$.