The standard model as gravity on a `non-associative' manifold
Mathematics
Final Report Abstract
Low energy physics is accurately described by a particular effective field theory - the so called standard model of particle physics coupled to Einstein gravity. But the effective field theory framework leaves certain basic questions unanswered, and gives us less guidance than we would like about what might come beyond the standard model. What for example determines the basic list of particles seen in experiment, and what sets the choice of fundamental forces? An intriguing possibility is that some of the patterns and features observed in the standard model might derive from simple geometric relations, and that certain open questions might be answered by looking more closely at the geometric structure that appears to underlie the standard model. The idea that a geometric structure might underlie the particle content observed in experiment, has been attractive ever since Einstein showed that gravity has a beautiful geometric description. The geometry used to describe gravity theories (Riemannian geometry), however, is not the appropriate setting for capturing all of the internal details of particle theories (although people have tried - Kaluza-Klein theory for example, and other higher dimensional theories with compactified dimensions). A more general notion of geometry is required, and this is the basic idea behind the ‘noncommutative’ geometry approach to the standard model. Just as the ‘external’ space-time of a gravity theory is coordinatized by an algebra of coordinates, if there is geometric structure underlying the standard model, then its ‘internal’ topology should be captured by an algebra of coordinates. In noncommutative geometry the assumption is that this coordinate algebra should be non-commutative, i.e. that the standard model should be thought of as pure gravity on a ‘noncommutative’ manifold. It is possible to find a noncommutative geometry that captures most of the details of the standard model. Unfortunately, however, this geometry fails to match all of the observed phenomenology, and leaves key theoretical questions unanswered. In this project we dropped the assumption that the coordinate algebra of the standard model should be noncommutative. Preliminary work appeared to indicate that the geometry underlying the physics of the standard model might instead be more accurately coordinatized by a certain commutative but nonassociative Jordan algebra, and that such a construction should neatly avoid some of the remaining conceptual pitfalls plaguing the associative NCG SM, as well as necessarily point towards new physics. The ultimate goal of the project was to build a concrete realization of the standard model of particle physics, formulated as a ‘Jordan’ nonassociative geometry, and to investigate its consequences, both with regards to the generalization from noncommutative geometry to nonassociative geometry, and with regards to exploring new physics.
