"Random finite gauge matrix geometries: numerical and analytic aspects"
Spectral triples are noncommutative generalisations of Riemannian spin geometries, whose path-integral quantisation is of potential relevance in high energy physics (following results that started with Connes-Lott and evolved to the Standard Model of particles obtained by Chamseddine-Connes-Marcolli). We address a modest version of the quantisation problem by restricting ourselves to a certain class of finite-dimensional spectral triples known as matrix geometries; the study of the resulting path-integral is often referred to as 'random noncommutative geometry' or 'Dirac ensembles', and can be studied as a multi-matrix model. The aim of the present project is to study gauge theories on a matrix geometry background (gauge matrix geometries) and to build bridges between the numerical (our laboratory) and 'analytical' approaches. Concretely, we plan to define, analytically study and simulate localised states for gauge matrix geometries. We are also interested in knowing to which extent truncations of almost-commutative manifolds resemble gauge matrix geometries, and relate these to well-known results by M. Rieffel's "matrix algebra convergence to smooth spaces" paper series. (The ESI Junior Research Fellow will visit L Glaser.)
Duration of stay: March to June 2023