| Name: | Peter Kilway |
|---|---|
| Home Institution: | University of Notre Dame |
| Email: | pkilway[at]nd.edu |
| Project: | Vortices in Geometry and Physics |
| Project Mentor: | Paul Feehan |
Project Description
Let M be a Riemannian manifold and E a vector bundle over M. The Yang-Mills Equations are a collection of PDEs for a connection A on the vector bundle E which are important in modern particle physics. The Yang-Mills Equations are: $$d_A * F_A = 0$$ Here, \(F_A\) is the associated curvature 2-form of the connection A, * is the Hodge star operator on 2-forms, and \(d_A\) is the differential associated to A. The goal of this project is to find more solutions to special cases of the Yang-Mills equations, starting with the Anti-Self-Dual Equations for a connection on a vector bundle E over a simply-connected orientable Riemannian 4-manifold M, which are given by: $$*F_A = -F_A$$ And then the Vortex Equations for a connection on a Hermitian vector bundle E over a complex Kahler manifold (generality may vary): [define way later]
One method of finding solutions is via a method of "gluing" which is outlined below: [To be elaborated on]Weekly Research Log
Week 1: 5/27-5/30
Before and during the week, I read sections 2.1.1, 2.1.2, and 2.1.3 of Donaldson and Kronheimer's book on 4-manifolds in order to aquire important information about Connections on Vector Bundles, the Curvature Form of a Connection, and Anti-Self-Dual Connections on simply-connected orientable Riemannian 4-Manifolds. I met with Paul Feehan on Thursday. Over the course of this meeting, we discussed the basics of mathematical gauge theory, reviewing, refining, and elaborating on the material I had read beforehand. We also defined the energy functional on connections over a vector bundle, and stated the Yang-Mills Equations for a connection on a vector bundle. At the end of the meeting, we briefly discussed the practice of handling PDEs on manifolds and discussed the various equations that I will (potentially) attempt to solve, in certain specific cases, over the course of the REU. These equations are: The Anti-Self-Dual Equations for a connection on a vector bundle over a simply-connected orientable Riemannian 4-manifold, the Seiberg-Witten Monopole Equations, and (special cases of) the Vortex Equations for a Connection on a Hermitian vector bundle over a Kahler manifold. At the present moment I only know the Anti-Self-Dual Equations and will be focusing on studying them for the immediate future. Next week I hope to parse through section 4.2 of Donaldson and Kronheimer's book. I hope to establish a routine involving cooking for myself, going on walks around campus, and exercising regularly over the course of week 2, and my non-academic priorities all involve this goal. My roommates and I have begun hosting board game nights.
Week 2
Week 3
Project References
- Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Clarendon Press.