Gauge theory studies principal bundle connections on a principal Lie group bundle. These connections correspond to gauge fields in physics, such as an electromagnetic field, and the Lie group of the principal bundle corresponds to the symmetries of the physical system. The space of connections satisfying certian non-linear partial differential equations is useful in low-dimensional topology. In fact, in Donaldson theory, the collection of Yang-Mills connections gives topological/differential invariants of 4-dimensional manifolds. In 1994, a new set of partial differential equations is introduced by Seiberg and Witten in their study of string theory, which led to a revolution in the field of low dimensional topology.
This project is to explore various new topological invariants using topological field theory and dualities arising from gauge theory and string theory