Joint Analysis / Geometric Analysis Seminar

Bruce DriverUniversity of California at San Diego
On $YM_2$ and the Makeenko-Migdal equations

Friday, September 8, 2017 - 2:30pm
Malott 224

For a connection form $A$ on $\mathbb{R}_2$ with values in the Lie algebra of a compact group $K$; a character of $K $applied to the holonomy around a closed loop $\sigma$ in the plane, gives a function $W_\sigma(A)$ of the connection form. Yang-Mills quantum field theory places a measure on the set of such connection forms. Ken Wilson has shown that much of the physical content of the theory is encoded in the behavior of the expectation of products of two (or more) such functions $W_{\sigma_1}(A)\cdot W_{\sigma_2}(A)$.

There is a large literature aimed at gaining insight into the behavior of this expectation as a function of the two curves. In this talk I will explain what happens to this function when the two curves arise from splitting a single curve at their intersection point. With the guidance of the heuristic machinery invented by physicists Makeenko and Migdal we will see that a particular kind of differential calculus in infinite dimensional curve space can be used to make rigorous some of their ideas and surprising identities.