Differential Geometry Working Seminar

Spencer Whitehead, Department of Pure Mathematics, University of Waterloo

"Symmetric Instantons"

Ansatze to the anti-self-dual instanton equations allow one to easily generate approximately 5/8ths of the solutions at any given charge, but to find the remaining instantons is a more difficult problem. An approach developed by Sutcliffe at the beginning of the 21st century is to impose symmetry conditions on the gauge potential of an instanton. In the case of a SU(2) instanton, we will see how these symmetry conditions can be used with the ADHM construction to turn the problem into one of the representation theory of the symmetry group, and how one can use this theory to construct instantons having the symmetry of the dodecahedron and rhombitruncated icosidodecahedron of lower charges than the corresponding JNR instanton. We will then see how the theory is affected when the structure group is changed from SU(2) to SU(N), and the symmetry group is changed from a discrete subgroup of SO(3) to a discrete subgroup of SO(4). Finally, we will look at how in this setting, the equivariant index theorem can be used to derive a class of 'integrality theorems' restricting the instanton solutions.

Zoom meeting:

• Meeting ID: 958 7361 8652
• Passcode: 577854