Differential Geometry Working Seminar

Spencer Whitehead, Department of Pure Mathematics, University of Waterloo

"Integrality theorems for symmetric instantons"

A symmetric instanton is a solution to the finite-energy anti-self-dual instanton equation on R^4 in which the connection commutes with some perscribed group of symmetries. This talk introduces the symmetric ADHM equations for structure group SU(N), and how index-theoretic methods can be used to derive integrality theorems for different symmetry groups.

In the case of N=2 and a discrete subgroup of SU(2), we derive the prime charge theorem, which imposes a canonical choice of representation for the equations at prime number charges.

In the case of N=2 for direct product subgroups of Spin(4), we use the index theorem to resolve a problem of Allen and Sutcliffe, showing that the minimal charge of a non-trivial instanton with the symmetries of the 600-cell is 119.

Time permitting, we will describe the notion of "quasi-irreducibility" for these objects, and its use in a path to a complete classification of symmetric instantons.

This seminar will be held jointly online and in person:

• Zoom link: https://uwaterloo.zoom.us/j/95873618652?pwd=OENSeFdERzUzV2NkM3hjQ0F1MzFYUT09
• Room: MC 5403