The standard model as gravity on a `non-associative' manifold

• Applicant: Dr. Shane Farnsworth
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields ; Mathematics
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 392887939

At low energies, the laws of physics appear to be accurately described by a particulareffective field theory (EFT), the so-called standard model of particle physics (SM) coupled to Einstein gravity. But the EFT framework leaves certain basic questions about the SM unanswered, and gives us less guidance than we would like about whatmight come beyond the SM. What for example determines the basic list of fermion fields, and why do they transform in the particular representations (and with the particular charges) that they do? Similarly, what determines the basic list of scalar fields, their representations and charges? An intriguing framework from which to address such questions is that of non-commutative geometry (NCG).Connes NCG is a generalization of Riemannian geometry which provides an elegant framework for describing Yang-Mills theories, coupled to Einstein-Hilbert gravity. The basic idea behind NCG is to shift focus away from topological spaces and manifolds, to instead focus on the algebra of functions defined over them (the algebra of coordinates). This simple idea allows one to explore geometries where one has only the algebra and there is no classical notion of the underlying space whatsoever. In particular, and rather remarkably, the NCG framework yields a striking reinterpretation of the SM as gravity on a "non-commutative" manifold. In this way it provides new geometric meaning for many features of the SM, and gives a tighter and more elegant setting for formulating the SM particle content and Lagrangian.In collaboration with Latham Boyle of the Perimeter Institute for Theoretical Physics in Canada, I initiated a generalization of the NCG formalism to also include those geometries which are "non-associative". Remarkably, this non-associative generalization of the NCG formalism has been of most use in the associative setting, where it has elucidated the role of certain NCG axioms, and led to a more apt description of the symmetries of a NCG. In this way it has provided solutions to a number of outstanding conceptual issues suffered by the associative NCG SM. Looking beyond the associative setting however, our preliminary work indicates that there exists a complete reformulation of the associative NCG SM, constructed instead as a commutative non-associative geometry. Our current investigations show that such a construction will neatly avoid some of the remaining conceptual pitfalls that plague the associative NCG SM, and will necessarily point to new physics. I am requesting DFG funding in order to build a concrete realization of the standard model of particle physics, formulated as a non-associative geometry (NAG), and to investigate its consequences.

• DFG Programme: Research Grants
• International Connection: Canada
• Cooperation Partner: Professor Dr. Latham Boyle