A framework for 3D Yang-Mills
Construction of Yang-Mills measures is considered to be an important step in the construction of quantum Yang-Mills theories. Yang-Mills measures have been constructed in 2D, but the problem remains open in dimensions three and higher. In this talk, I will present two kinds of abstract constructions of 3D Yang-Mills measures. One is easier, but with less desirable properties than the other. The easier construction is possible for essentially any 3D Yang-Mills theory. The more difficult construction is on a better state space with a richer class of measurable functions. We have been able to complete the second construction for U(1) theory, and we believe that it should also work for non-Abelian theories (this is work in progress). Both constructions build on a program of using the Yang-Mills heat flow to construct state spaces for Yang-Mills measures, initiated by Charalambous and Gross about ten years ago. All of this is based on ongoing joint work with Sky Cao. A preprint will be available soon.