Quantum Riemannian Geometry of the A_n graph

Abstract: We solve for quantum Riemannian geometries on the finite lattice interval • − • − · · · − • with $n$ nodes (the Dynkin graph of type $A_n$) and find that they are necessarily $q$-deformed with $q$ a root of unity. 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 $q \to 1$ is the quantum Riemannian geometry of the natural numbers $N$ with rational metric multiples $(i + 1)/i$ in the direction of increasing $i$. In both cases there is a unique metric up to normalisation with zero Ricci scalar curvature. Elements of QFT and quantum gravity are exhibited for $n = 3$ and for the continuum limit of the geometry of $N$. The Laplacian for the scaler-flat metric becomes the Airy equation operator $(1/ x) d^2/ dx^2$ in so far as a limit exists. The talk is based on joint work with J. Argota-Quiroz available on arXiv: 2204.12212 (math.QA).

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home