Quantum Riemannian geometry of the discrete interval and q-deformation

Abstract

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 $q=e^{\imath\pi\over n+1}$. 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 $(i+1)_q/(i)_q$ 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 $\Bbb N$ to have rational metric multiples $(i+1)/i$ 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 $\Bbb N$. The Laplacian for the scalar-flat metric becomes the Airy equation operator ${1\over x}{d^2\over d x^2}$ 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 ${e^{-\psi}\over x}{d^2 \psi\over d x^2}$.