This week’s whiteboard is a joint effort between José Figueroa-O’Farrill and PhD student Paul Reynolds, who study objects in mathematical physics called instantons. These are special solutions to the Yang-Mills field equations which play important roles both in Physics and Mathematics. For example, in quantum physics instantons describe the process of tunnelling, where particles can get through barriers that classically they would be unable to surmount. In mathematics, they were instrumental in Donaldson’s work on 4-dimensional manifolds, for which he was awarded the Fields Medal.
Instantons appear in all sorts of dimensions, and José and Paul are specifically interested in 8 dimensions. Four-dimensional instantons are intimately related to the quaternions, whereas the 8-dimensional instantons are intimately related to the octonions.
The first bit of the blackboard (to the left) explains that the 8-dimensional instanton equations (which consist of 7 independent conditions) can be viewed as the components of a momentum mapping. This generalises a similar fact about 4-dimensional instantons. In the 4-dimensional case the geometry is that of a hyperkähler momentum mapping, but the geometry of the 8-dimensional equations remains unknown.
Hyperkähler manifolds can be constructed by taking the quaternions and quotienting out by certain group actions. Paul is trying to do something similar with the octonians: trying to find the right quotients that will give interesting manifolds which fit the geometry of the 8-dimensional instantons. The part of the board to the right describes his attempts at finding the geometry of a quotient in a special case.