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).
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.