Facilitator: Pierre Martinetti

Noncommutative geometry provides a geometrical interpretation of the Lagrangian of the standard model coupled to gravity. If this interpretation actually says something about physics, then it is quite natural to wonder what “picture” of space (-time) emerges from the spectral triple of the standard model, and if there are some phenomenological consequences. Here is a list of topics and questions that could serve as a starting point for discussions on that matter.

**Metric interpretation of the Higgs field**: an originality of the spectral triple construction is to be (as far as I know) the only approach to noncommutative spaces in which a notion of distance is available. The latter is a generalization to the noncommutative setting of the Monge-Kantorovich distance in the theory of optimal transport (and gives back the usual Riemannian distance when applied to the canonical spectral triple of a Riemannian manifold). Applied to the spectral triple of the standard model, this distance provides a metric interpretation of the Higgs field, as the component of the metric in a discrete internal dimension (i.e. between two copies of spacetime). Is there any phenomenological interpretation, or is this just a mathematical curiosity ?

**Particle physics on noncommutative space**: could the noncommutative structure of spacetime interfer with particle physics, namely does it make sense to imagine particles propagating in a two-sheet space, and if yes what are the possible phenomenological consequences ? More specifically, is there a way such that the geometry of the standard model induces a modification of the dispersion relation (e.g. similar to the ones intensively studied by people focusing on the phenomenological aspect of quantum gravity) ?

**Quantized spacetime**: many approaches to quantum gravity deal in some way or another with the notion of quantized space. Most often the latter is intended as a space in which geometrical quantities such as volume, area and length are described by operators whose spectrum is discrete (e.g. in loop quantum gravity or in the Doplicher, Fredenhagen, Roberts model). The Planck length then emerges as the minimum measurable physical length. Can one infer a notion of minimal length from the spectral triple of the standard model ? Is there a definition of area and volume in noncommutative geometry ?

**Emergent geometry**: a quite fashionable notion in the literature is that of “emergent geometry”. It is expected that in some high energy regime the notion of geometry somehow disappears, and that space-time as we now it should be thought as a collective (statistical) effect of more fundamental (possibly discrete) degrees of freedom. What are the possibilities offered by spectral triple to describe such “pre-geometric” phase of the universe ?