# Talk by Shahn Majid

On May 3rd, 2023, Shahn Majid (Queen Mary University of London, UK) gave a talk about **"Dirac operators built from quantum Riemannian geometries"** as part of the online workshop on **"Dirac equation between discrete and continuous: new trends and applications"** organized by Ginestra Bianconi (Queen Mary University of London) and Delio Mugnolo (FernUniversität in Hagen).

## Abstract

We outline the recent formalism of quantum Riemannian geometry (QRG), which works over any unital ‘coordinate algebra’ A and hence can interpolate between smooth functions on a manifold in the classical case, but equally the functions on the vertex set of a graph with noncommutative differential structure provided by the edges of the graph. We show how a

QRG together with a choice of spinor bundle with connection and a ‘Clifford action’ lead to a Dirac operator in the sense of a Connes spectral triple. We describe a natural choices of these for any QRG as an analogue of the classical Hodge-Dirac operator d + δ. The graph case recovers a previously proposed graph Dirac operator but the construction applies much

more widely, e.g. to A the algebra of 2 × 2 complex matrices.