A 0-dimensional Field Theory of Noncommutative Geometry
In noncommutative geometry (NCG), spectral triples generalize manifolds. In physics, a complete formulation of NCG beyond the classical setting of Chamseddine-Connes, in which the use of spectral triples leads to the Standard Model, would shed light upon quantum gauge theories and quantum gravity. While an approach in terms of "path integrals over geometries" is considerably challenging, using Barrett's characterization of fuzzy geometries (in some sense, discrete noncommutative manifolds) leads to a well-defined path integral that can be computed as a matrix ensemble whose measure has the peculiarity of presenting products of traces of noncommutative polynomials. In this talk I address how to obtain gauge theories in this setting and address the functional renormalization equation for the multimatrix models that these matrix integrals motivate. Based on 1912.13288; 2007.10914; 2102.06999; 2105.01025 and 2111.02858.